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HCF or Highest Common Factor refers to a set of two or more different numbers that can be divided by exactly or a common number. HCF is also called as GCD/GCM - Greatest Common Divisor or Greatest Common Measure.

LCM or Least Common Multiple refers to the smallest quantity of number that is divisible by two or more given quantities of number without a remainder: 12is the least common multiple of 2, 3, 4, and 6. LCM is also called as Lowest Common Multiple. LCM is the smallest number that is multiple of two or more numbers.

HCF and LCM are calculated by either factorization method or division method.

If number 'a' divided another number 'b' exactly, then we say that 'a' is a factor of 'b'.
In this case, 'b' is called a multiple of 'a'.

H.C.F can be calculated by any of the following methods,

Factorization Method:
Express each one of the numbers as product of prime factors.
The product of least powers of common prime factors gives HCF.

Division Method:
Two find the HCF of two given numbers, divide the largest by the small number, then divide the dividend by the remainder. Repeat this until remainder is 0. The last dividend is the HCF of the two numbers.
Least Commmon Factor(LCF) can be calculated by the following method,

Factorization Method:
Express each one of the numbers as product of prime factors.
The product of highest powers of all prime factors gives LCF.

Example:
Calculate the HCF and LCM for the given set of numbers.
Greatest Common Measure of 18,36,72

Solution:
Step1: Express the numbers as product of prime factors.
18 = 2× 3 ²
36 = 2 ² × 3 ²
72 = 2 ³ × 3 ²

Step2: Take the prime numbers with least power and is present in all sets.
2 and 3 are the prime number common to all given numbers.
The least power of 2 in the set is - 2
The least power of 3 in the set is - 3 ²

Step3: Product of the numbers taken.
3 ² × 2 = 9 × 2 = 18

So 18 is the GCD (Greatest Common Divisor) of the numbers.
Least Common Multiple or Factor of 36, 90, 72.

Step1: Express the numbers as product of prime factors.
36 = 2 ² × 3 ²
90 = 2 × 5 × 3 ²
72 = 2 ³ × 3 ²

Step2: Take the prime numbers with highest power for all prime numbers.
2, 3,5 are the prime number identified.
The highest power of 2 in the set is - 2 ³
The highest power of 3 in the set is - 3 ²
The highest power of 5 in the set is - 5

Step3: Product of the numbers taken.
3 ² × 5 × 2 ³ = 9 × 5 × 8 = 360

So 360 is the LCM (Least Common Multiple) of the numbers.

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HCF or Highest Common Factor refers to a set of two or more different numbers that can be divided by exactly or a common number. HCF is also called as GCD/GCM - Greatest Common Divisor or Greatest Common Measure.

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Welcome to the our Reverse Percentages Calculator.

Here you will our reverse percentage calculator which will help you to find the original number after a percentage increase of decrease.

Reverse Percentages Calculator 1

Reverse Percentage Calculator 1 - How It Works

This calculator will help you to find the original number, when you have been given the final number and the percentage of the original number that made it.

E.g. if you know that 70% of a number is 210 then it will find the original number. (Answer: 300)

Simply enter the final number in the first box and the percentage in the 2nd box.

Click on the 'Find Original Number' box to work out what the original number was!

Answers are rounded to 2dp.

Reverse Percentages Calculator 2

Reverse Percentage Calculator 2 - How It Works

This calculator will help you to find the original number, when you have been given the final number and the percentage increase or decrease.

Example: if a toy in a sale marked 20% off costs $210, what is the original price. (Answer: $262.50)

Simply enter the final number in the first box and the percentage in the 2nd box and whether it was increased or decreased.

Click on the 'Find Original Number' box to work out what the original number was!

Answers are rounded to 2dp.

How to find Reverse Percentages

Read on below if you want to find out how to find out reverse percentages work.

Reverse percentages are used when the percentage and the final number is given, and the original number needs to be found.

Step 1) Get the percentage of the original number.

If the percentage is an increase then add it to 100, if it is a decrease then subtract it from 100.

Step 2) Multiply the final number by 100.

Step 3) Divide this number by the percentage.

Examples of Reverse Percentages

Example 1) 35% of a number is 320. What is the number?

Step 1) The percentage of the original number is 35%

Step 2) 320 x 100 = 32,000

Steo 3) 32,000 ÷ 35 = 914.2857...

Answer: the original number was 914.29 to 2dp.

You can use calculator 1 to solve this problem.

Example 2) A car is reduced by 20% in price to $40,000. What was the original price?

Step 1) The percentage of the original number is 100 - 20 = 80%

Step 2) 40,000 x 100 = 4,000,000

Step 3) 4,000,000 ÷ 80 = 50,000

Answer: the original car cost $50,000

You can use calculator 2 to solve this problem.

Example 3) Sally invests money in some shares. Five years later, she sells them for $7200 at a profit of 25% of their original value. How much did she spend on the shares?

Step 1) The percentage of the original number is 100 + 25 = 125%

Step 2) 7200 x 100 = 720,000

Step 3) 720,000 ÷ 125 = 5760

Answer: the amount she spent on shares was $5760.

You can use calculator 2 to solve this problem.

More Recommended Math Resources

Take a look at some more of our resources similar to these.

Online Percentage Practice Zone

Our online percentage practice zone gives you a chance to practice finding percentages of a range of numbers.

You can choose your level of difficulty and test yourself with immediate feedback!

Percentage Of Calculator

This is the calculator to use if you want to find a percentage of a number.

Simple choose your number and the percentage and the calculator will do the rest.

Percentage Increase Calculator

If you need to find the percentage increase/decrease between 2 numbers, then this calculator is the one you need.

Simple input the original number, then the new number and the calculator will tell you the percentage increase or decrease.

Converting Percentage Calculator

If you need to convert a percentage to a fraction - either a decimal fraction or a fraction in its simplest form, then use this calculator.

You can also use the fraction to percentage calculator for converting any fraction into a percentage.

This calculator takes a percentage and converts it at the click of a button.

Looking for more math calculators?

We have a wide range of free math calculators to help you.

Most of our calculators show you their working out so that you can see exactly what they have done to get the answer.

Our calculator hub page contains links to all of our calculators!

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The Math Salamanders hope you enjoy using these free printable Math worksheets and all our other Math games and resources.

We welcome any comments about our site or worksheets on the Facebook comments box at the bottom of every page.

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Welcome to the our Reverse Percentages Calculator. Here you will our reverse percentage calculator which will help you to find the original number after a percentage increase of decrease. This calculator will help you to find the original number, when you have been given the final number and the percentage of the original number that made it.

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Compound Interest is calculated on the initial payment and also on the interest of previous periods.

Example: Suppose you give \$100 to a bank which pays you 10% compound interest at the end of every year. After one year you will have \$100 + 10% = \$110, and after two years you will have \$110 + 10% = \$121.

Click here to view some problem which can be solved by using this calculator.

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Compound Interest Calculator with step by step explanations

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Compound Interest is calculated on the initial payment and also on the interest of previous periods. Example: Suppose you give \$100 to a bank which pays you 10% compound interest at the end of every year. After one year you will have \$100 + 10% = \$110, and after two years you will have \$110 + 10% = \$121.

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Calculator Use

Use this prime numbers calculator to find all prime factors of a given integer number up to 10 trillion. This calculator presents:

1. Prime factors of a number

2. Prime decomposition in exponential form

3. CSV (comma separated values) list of prime factors

4. Factorization in a prime factors tree

For the first 5000 prime numbers, this calculator indicates the index of the prime number. The prime number is denoted as Prime[n], so Prime[1] = 2, Prime[2] = 3, Prime[3] = 5, and so on.

The limit on the input number to factor is less than 10,000,000,000,000 (less than 10 trillion or a maximum of 13 digits).

What is Prime Factorization?

Prime factorization or integer factorization of a number is breaking a number down into the set of prime numbers which multiply together to result in the original number. This is also known as prime decomposition.

How to Find Prime Factorization of a Number

We cover two methods of prime factorization: find primes by trial division, and use primes to create a prime factors tree.

Prime Factorization by Trial Division

Say you want to find the prime factors of 100 using trial division. Start by testing each integer to see if and how often it divides 100 and the subsequent quotients evenly. The resulting set of factors will be prime since, for example, when 2 is exhausted all multiples of 2 are also exhausted.

Find the prime factors of 100:

100 ÷ 2 = 50; save 2

50 ÷ 2 = 25; save 2

25 ÷ 2 = 12.5, not evenly so divide by next highest number, 3

25 ÷ 3 = 8.333, not evenly so divide by next highest number, 4

But, 4 is a multiple of 2 so it has already been checked, so divide by next highest number, 5

25 ÷ 5 = 5; save 5

5 ÷ 5 = 1; save 5

List the resulting prime factors as a sequence of multiples, 2 x 2 x 5 x 5 or as factors with exponents, 2 ² x 5 ².

Examples of Prime Decomposition: Factors and Exponents

Prime factorization of 100 is 2 x 2 x 5 x 5 or 2² x 5²

Prime factorization of 76 is 2 x 2 x 19 or 2² x 19¹

Prime factorization of 50 is 2 x 5 x 5 or 2 x 5²

Prime factorization of 48 is 2 x 2 x 2 x 2 x 3 or 2⁴ x 3¹

Prime factorization of 36 is 2 x 2 x 3 x 3 or 2² x 3²

Prime factorization of 20 is 2 x 2 x 5 or 2² x 5¹

Prime factorization of 10 is 2 x 5 or 2¹ x 5¹

Prime Factors Tree

Using a prime factorization tree to see the work, prime decomposition of 100 = 2 x 2 x 5 x 5 looks like this:

Related Calculators

For a list of the first 1000 prime numbers see our See 1000 Prime Numbers Table.

References and Further Reading

Math is Fun: Prime Factorization.

Weisstein, Eric W. " Prime Number." From MathWorld--A Wolfram Web Resource.

Weisstein, Eric W. " Direct Search Factorization." From MathWorld--A Wolfram Web Resource.

Weisstein, Eric W. " Prime Factorization Algorithms." From MathWorld--A Wolfram Web Resource.

Caldwell, Chris K. The Prime Pages.

Last updated: October 20, 2023

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Prime number calculator to find prime factors. Learn what is a prime number. Perform prime decomposition and create a primes factor tree. Prime factorization of numbers.

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Calculator Use

Do math calculations with mixed numbers (mixed fractions) performing operations on fractions, whole numbers, integers, mixed numbers, mixed fractions and improper fractions. The Mixed Numbers Calculator can add, subtract, multiply and divide mixed numbers and fractions.

Mixed Numbers Calculator (also referred to as Mixed Fractions):

This online calculator handles simple operations on whole numbers, integers, mixed numbers, fractions and improper fractions by adding, subtracting, dividing or multiplying. The answer is provided in a reduced fraction and a mixed number if it exists.

Enter mixed numbers, whole numbers or fractions in the following formats:

Mixed numbers: Enter as 1 1/2 which is one and one half or 25 3/32 which is twenty five and three thirty seconds. Keep exactly one space between the whole number and fraction and use a forward slash to input fractions. You can enter up to 3 digits in length for each whole number, numerator or denominator (123 456/789).

Whole numbers: Up to 3 digits in length.

Fractions: Enter as 3/4 which is three fourths or 3/100 which is three one hundredths. You can enter up to 3 digits in length for each the numerators and denominators (e.g., 456/789).

Adding Mixed Numbers using the Adding Fractions Formula

1. Convert the mixed numbers to improper fractions

2. Use the algebraic formula for addition of fractions:
a/b + c/d = (ad + bc) / bd

3. Reduce fractions and simplify if possible

Adding Fractions Formula

\( \dfrac{a}{b} + \dfrac{c}{d} = \dfrac{(a \times d) + (b \times c)}{b \times d} \)

Example

Add 1 2/6 and 2 1/4

\( 1 \dfrac{2}{6} + 2 \dfrac{1}{4} = \dfrac{8}{6} + \dfrac{9}{4} \)

\( = \dfrac{(8 \times 4) + (9 \times 6)}{6 \times 4} \)

\( = \dfrac{32 + 54}{24} = \dfrac{86}{24} = \dfrac{43}{12} \)

1 2/6 + 2 1/4 = 8/6 + 9/4 = (8*4 + 9*6) / 6*4 = 86 / 24

So we get 86/24 and simplify to 3 7/12

Subtracting Mixed Numbers using the Subtracting Fractions Formula

1. Convert the mixed numbers to improper fractions

2. Use the algebraic formula for subtraction of fractions: a/b - c/d = (ad - bc) / bd

3. Reduce fractions and simplify if possible

Subtracting Fractions Formula

\( \dfrac{a}{b} - \dfrac{c}{d} = \dfrac{(a \times d) - (b \times c)}{b \times d} \)

Example

Subtract 2 1/4 from 1 2/6

1 2/6 - 2 1/4 = 8/6 - 9/4 = (8*4 - 9*6) / 6*4 = -22 / 24

Reduce the fraction to get -11/12

Multiplying Mixed Numbers using the Multiplying Fractions Formula

1. Convert the mixed numbers to improper fractions

2. Use the algebraic formula for multiplying of fractions: a/b * c/d = ac / bd

3. Reduce fractions and simplify if possible

Multiplying Fractions Formula

\( \dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{a \times c}{b \times d} \)

Example

multiply 1 2/6 by 2 1/4

1 2/6 * 2 1/4 = 8/6 * 9/4 = 8*9 / 6*4 = 72 / 24

Reduce the fraction to get 3/1 and simplify to 3

Dividing Mixed Numbers using the Dividing Fractions Formula

1. Convert the mixed numbers to improper fractions

2. Use the algebraic formula for division of fractions: a/b ÷ c/d = ad / bc

3. Reduce fractions and simplify if possible

Dividing Fractions Formula

\( \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a \times d}{b \times c} \)

Example

divide 1 2/6 by 2 1/4

1 2/6 ÷ 2 1/4 = 8/6 ÷ 9/4 = 8*4 / 9*6 = 32 / 54

Reduce the fraction to get 16/27

Related Calculators

To perform math operations on simple proper or improper fractions use our Fractions Calculator. This calculator simplifies improper fraction answers into mixed numbers.

If you want to simplify an individual fraction into lowest terms use our Simplify Fractions Calculator.

For an explanation of how to factor numbers to find the greatest common factor (GCF) see the Greatest Common Factor Calculator.

If you are simplifying large fractions by hand you can use the Long Division with Remainders Calculator to find whole number and remainder values.

Last updated: October 17, 2023

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Mixed numbers calculator to add, subtract, multiply and divide mixed numbers (mixed fractions), fractions and integers. Do math with mixed numbers and mixed fractions such as 1 1/2 or 3 5/8.

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Calculator Use

Compare and order fractions, integers and mixed numbers in ascending or descending order. Order fractions from least to greatest or from greatest to least.

How to Order Fractions, Integers and Mixed Numbers

To compare and order fractions we must first convert all integers, mixed numbers (mixed fractions) and fractions into values that we can compare. We do this by first converting all terms into fractions, finding the least common denominator ( LCD), then rewriting each term as an equivalent fraction with the LCD. Then we compare the numerators of each fraction and put them in correct order from least to greatest or greatest to least.

Steps to compare and order fractions, integers and mixed numbers

1. Convert integers and mixed numbers to improper fractions

2. Find the least common denominator LCD of all the fractions

3. Rewrite fractions as equivalent fractions using the LCD

4. Order the new fractions by the numerator

5. Match original inputs to the ordered fractions to see the final order

Example using this calculator default inputs

Input: 2, 3/4, 9/12, 3 5/8, -12/16 and order from least to greatest

1. Convert integers and mixed numbers to improper fractions

* 3/4, 9/12 and -12/16 are proper fractions so we can use those as they are written
* 2 in fraction form is 2/1
* Convert 3 5/8 to an improper fraction. Multiply the whole number 3 by the denominator 8 to get 24. Add 24 to the numerator 5 to get 29/8.
So 3 5/8 = 29/8.
* All inputs as fractions: 2/1, 3/4, 9/12, 29/8, -12/16

2. Find the least common denominator: LCD of 1, 4, 8, 12, 16 is 48

3. Rewrite the input fractions as equivalent fractions using the LCD

* 2, 3/4, 9/12, 3 5/8, -12/16 become:

* 96/48, 36/48, 36/48, 174/48, -36/48

4. Order these fractions from least to greatest numerator

* -36/48 < 36/48 = 36/48 < 96/48 < 174/48

5. Place the original fractions in the same order as their equivalents

* -12/16 < 3/4 = 9/12 < 2 < 3 5/8

Related Calculators

To find which fraction is bigger and which is smaller see the Comparing Fractions Calculator.

To add, subtract, multiply and divide fractions see the Fractions Calculator.

For addition, subtraction, multiplication and division on mixed numbers and fractions see the Mixed Numbers Calculator.

To simplify a fraction to lowest terms see the Simplifying Fractions Calculator.

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Compare and order fractions, integers and mixed numbers in ascending or descending order. Order fractions from least to greatest or from greatest to least. To compare and order fractions we must first convert all integers, mixed numbers (mixed fractions) and fractions into values that we can compare.

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Use the calculators below to calculate the surface area of several common shapes.

Ball Surface Area

Cone Surface Area

Cube Surface Area

Cylindrical Tank Surface Area

Rectangular Tank Surface Area

Capsule Surface Area

Cap Surface Area

Please provide any two values below to calculate.

Conical Frustum Surface Area

Ellipsoid Surface Area

Square Pyramid Surface Area

Related
Volume Calculator | Area Calculator | Body Surface Area Calculator

The surface area of a solid is a measure of the total area occupied by the surface of an object. All of the objects addressed in this calculator are described in more detail on the Volume Calculator and Area Calculator pages. As such, this calculator will focus on the equations for calculating surface area the objects and the use of these equations. Please refer to the aforementioned calculators for more detail on each individual object.

Sphere

The surface area (SA) of a sphere can be calculated using the equation:

Xael doesn't like sharing her chocolate truffles with anyone. When she receives a box of Lindt truffles, she proceeds to calculate the surface area of each truffle in order to determine the total surface area she has to lick to decrease the probability that anyone will try to eat her truffles. Given that each truffle has a radius of 0.325 inches:

Cone

The surface area of a circular cone can be calculated by summing the surface area of each of its individual components. The "base SA" refers to the circle that comprises the base in a closed circular cone, while the lateral SA refers to the rest of the area of the cone between the base and its apex. The equations to calculate each, as well as the total SA of a closed circular cone are shown below:

Athena has recently taken an interest in southeast Asian culture, and is particularly fascinated by the conical hat, typically referred to as a "rice hat," which is commonly used in a number of southeast Asian countries. She decides to make one of her own, and being a very practical person not mired in sentimentality, retrieves her mother's wedding dress from the dark recesses of the wardrobe in which it resides. She determines the surface area of material she needs to create her hat with a radius of 1 foot and a height of 0.5 feet as follows:

Cube

The surface area of a cube can be calculated by summing the total areas of its six square faces:

Anne wants to give her younger brother a Rubik's cube for his birthday, but knows that her brother has a short attention span and is easily frustrated. She custom orders a Rubik's Cube in which all the faces are black, and has to pay for the customization based on the surface area of the cube with edge length of 4 inches.

Cylindrical Tank

The surface area of a closed cylinder can be calculated by summing the total areas of its base and lateral surface:

base SA = 2πr ²
lateral SA = 2πrh
total SA = 2πr(r + h) where r is radius and h is height

Jeremy has a large cylindrical fish tank that he bathes in because he doesn't like showers or bath tubs. He is curious whether his heated water cools faster than when in a bathtub, and needs to calculate the surface area of his cylindrical tank of height 5.5 feet and radius of 3.5 feet.

total SA = 2π × 3.5(3.5 + 5.5) = 197.920 ft ²

Rectangular Tank

The surface area of a rectangular tank is the sum of the area of each of its faces:

SA = 2lw + 2lh + 2wh
where l is length, w is width, and h is height

Banana, the eldest daughter of a long line of banana farmers, wants to teach her spoiled rotten little sister, Banana-Bread, a lesson about hope and expectations. Banana-Bread has been clamoring all week long about wanting a new set of drawers to house her new Batman action figures. As such, Banana buys her a large Barbie doll house with limited edition kitchen utensils, oven, apron, and realistic rotting bananas for Batman. She packs these into a rectangular box of similar dimensions as the drawer that Banana-Bread wants, and needs to determine the amount of wrapping paper she needs to complete her presentation of the gift of the 3 ft × 4 ft × 5 ft surprise:

SA = (2 × 3 × 4) + (2 × 4 × 5) + (2 × 3 × 5) = 94 ft ²

Capsule

The surface area of a capsule can be determined by combining the surface area equations for a sphere and the lateral surface area of a cylinder. Note that the surface area of the bases of the cylinder is not included since it does not comprise part of the surface area of a capsule. The total surface area is calculated as follows:

Horatio is manufacturing a placebo that purports to hone a person's individuality, critical thinking, and ability to objectively and logically approach different situations. He has already tested the market and has found that a vast majority of the sample population exhibit none of these qualities, and are very ready to purchase his product, further entrenching themselves within the traits they so desperately seek to escape. Horatio needs to determine the surface area of each capsule so that he can coat them with an excessive layer of sugar and appeal to the sugar predisposed tongues of the population in preparation for his next placebo that "cures" all forms of diabetes mellitus. Given each capsule has r of 0.05 inches and h of 0.5 inches:

SA = 4π × 0.05 ² + 2π × 0.05 × 0.5 = 0.188 in ²

Spherical Cap

The surface area of a spherical cap is based on the height of the segment in question. The calculator provided assumes a solid sphere and includes the base of the cap in the calculation of surface area, where the total surface area is the sum of the area of the base and that of the lateral surface of the spherical cap. If using this calculator to compute the surface area of a hollow sphere, subtract the surface area of the base. Given two values of height, cap radius, or base radius, the third value can be calculated using the equations provided on the Volume Calculator. The surface area equations are as follows:

spherical cap SA = 2πRh
base SA = πr ²
Total solid sphere SA = 2πRh + πr ²
where R is spherical cap radius, r is base radius, and h is height

Jennifer is jealous of the globe that her older brother Lawrence received for his birthday. Since Jennifer is two-thirds the age of her brother, she decides that she deserves one-third of her brother's globe. After returning her father's hand saw to the toolshed, she calculates the surface area of her hollow portion of the globe with R of 0.80 feet and h 0.53 feet as shown below:

SA = 2π × 0.80 × 0.53 = 2.664 ft ²

Conical Frustum

The surface area of a solid, right conical frustum is the sum of the areas of its two circular ends and that of its lateral face:

Paul is making a volcano in the shape of a conical frustum for his science fair project. Paul views volcanic eruptions as a violent phenomenon, and being against all forms of violence, decides to make his volcano in the form of a closed conical frustum that does not erupt. Although his volcano is unlikely to impress the science fair judges, Paul must still determine the surface area of material he needs to coat the outer wall of his volcano with R of 1 foot, r of 0.3 feet, and h of 1.5 feet:

Ellipsoid

Calculating the surface area of an ellipsoid does not have a simple, exact formula such as a cube or other simpler shape does. The calculator above uses an approximate formula that assumes a nearly spherical ellipsoid:

Coltaine has always enjoyed cooking and recently won a ceramic knife from a contest. Unfortunately for his family, who almost exclusively eat meat, Coltaine has been practicing his cutting technique on an excessive amount of vegetables. Rather than eating his vegetables, Coltaine's father stares dejectedly at his plate, and estimates the surface area of the elliptical cuts of zucchini with axes 0.1, 0.2, and 0.35 inches:

Square Pyramid

The surface area of a square pyramid is comprised of the area of its square base and the area of each of its four triangular faces. Given height h and edge length a, the surface area can be calculated using the following equations:

Vonquayla's classroom recently completed building a model of the Great Pyramid of Giza. However, she feels that the model does not exude the feeling of architectural wonder that the original does and decides that coating it with "snow" would at least impart an aspect of wonder. She calculates the surface area of melted sugar she would need to fully coat the pyramid with edge length a of 3 feet and height h of 5 feet:

Unlike the Great Pyramid of Giza that has stood for thousands of years, its model, made of graham crackers and coated in sugar, lasted only a matter of days.

Common Area Units

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This free surface area calculator determines the surface area of a number of common shapes, including sphere, cone, cube, cylinder, capsule, cap, conical frustum, ellipsoid, and square pyramid. Explore the area or volume calculator, as well as hundreds of other calculators addressing math, finance, fitness, health, and more.

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The following is a list of volume calculators for several common shapes. Please fill in the corresponding fields and click the "Calculate" button.

Sphere Volume Calculator

Cone Volume Calculator

Cube Volume Calculator

Cylinder Volume Calculator

Rectangular Tank Volume Calculator

Capsule Volume Calculator

Spherical Cap Volume Calculator

Please provide any two values below to calculate.

Conical Frustum Volume Calculator

Ellipsoid Volume Calculator

Square Pyramid Volume Calculator

Tube Volume Calculator

Volume is the quantification of the three-dimensional space a substance occupies. The SI unit for volume is the cubic meter, or m³. By convention, the volume of a container is typically its capacity, and how much fluid it is able to hold, rather than the amount of space that the actual container displaces. Volumes of many shapes can be calculated by using well-defined formulas. In some cases, more complicated shapes can be broken down into simpler aggregate shapes, and the sum of their volumes is used to determine total volume. The volumes of other even more complicated shapes can be calculated using integral calculus if a formula exists for the shape's boundary. Beyond this, shapes that cannot be described by known equations can be estimated using mathematical methods, such as the finite element method. Alternatively, if the density of a substance is known, and is uniform, the volume can be calculated using its weight. This calculator computes volumes for some of the most common simple shapes.

Sphere

A sphere is the three-dimensional counterpart of a two-dimensional circle. It is a perfectly round geometrical object that, mathematically, is the set of points that are equidistant from a given point at its center, where the distance between the center and any point on the sphere is the radius r. Likely the most commonly known spherical object is a perfectly round ball. Within mathematics, there is a distinction between a ball and a sphere, where a ball comprises the space bounded by a sphere. Regardless of this distinction, a ball and a sphere share the same radius, center, and diameter, and the calculation of their volumes is the same. As with a circle, the longest line segment that connects two points of a sphere through its center is called the diameter, d. The equation for calculating the volume of a sphere is provided below:

EX: Claire wants to fill a perfectly spherical water balloon with radius 0.15 ft with vinegar to use in the water balloon fight against her arch-nemesis Hilda this coming weekend. The volume of vinegar necessary can be calculated using the equation provided below:

volume = 4/3 × π × 0.15 ³ = 0.141 ft ³

Cone

A cone is a three-dimensional shape that tapers smoothly from its typically circular base to a common point called the apex (or vertex). Mathematically, a cone is formed similarly to a circle, by a set of line segments connected to a common center point, except that the center point is not included in the plane that contains the circle (or some other base). Only the case of a finite right circular cone is considered on this page. Cones comprised of half-lines, non-circular bases, etc. that extend infinitely will not be addressed. The equation for calculating the volume of a cone is as follows:

where r is the radius and h is the height of the cone

EX: Bea is determined to walk out of the ice cream store with her hard-earned $5 well spent. While she has a preference for regular sugar cones, the waffle cones are indisputably larger. She determines that she has a 15% preference for regular sugar cones over waffle cones and needs to determine whether the potential volume of the waffle cone is ≥ 15% more than that of the sugar cone. The volume of the waffle cone with a circular base with radius 1.5 in and height 5 in can be computed using the equation below:

volume = 1/3 × π × 1.5 ² × 5 = 11.781 in ³

Bea also calculates the volume of the sugar cone and finds that the difference is < 15%, and decides to purchase a sugar cone. Now all she has to do is use her angelic, childlike appeal to manipulate the staff into emptying the containers of ice cream into her cone.

Cube

A cube is the three-dimensional analog of a square, and is an object bounded by six square faces, three of which meet at each of its vertices, and all of which are perpendicular to their respective adjacent faces. The cube is a special case of many classifications of shapes in geometry, including being a square parallelepiped, an equilateral cuboid, and a right rhombohedron. Below is the equation for calculating the volume of a cube:

volume = a ³
where a is the edge length of the cube

EX: Bob, who was born in Wyoming (and has never left the state), recently visited his ancestral homeland of Nebraska. Overwhelmed by the magnificence of Nebraska and the environment unlike any other he had previously experienced, Bob knew that he had to bring some of Nebraska home with him. Bob has a cubic suitcase with edge lengths of 2 feet, and calculates the volume of soil that he can carry home with him as follows:

Cylinder

A cylinder in its simplest form is defined as the surface formed by points at a fixed distance from a given straight line axis. In common use, however, "cylinder" refers to a right circular cylinder, where the bases of the cylinder are circles connected through their centers by an axis perpendicular to the planes of its bases, with given height h and radius r. The equation for calculating the volume of a cylinder is shown below:

volume = πr ²h
where r is the radius and h is the height of the tank

EX: Caelum wants to build a sandcastle in the living room of his house. Because he is a firm advocate of recycling, he has recovered three cylindrical barrels from an illegal dumping site and has cleaned the chemical waste from the barrels using dishwashing detergent and water. The barrels each have a radius of 3 ft and a height of 4 ft, and Caelum determines the volume of sand that each can hold using the equation below:

volume = π × 3 ² × 4 = 113.097 ft ³

He successfully builds a sandcastle in his house, and as an added bonus, manages to save electricity on nighttime lighting, since his sandcastle glows bright green in the dark.

Rectangular Tank

A rectangular tank is a generalized form of a cube, where the sides can have varying lengths. It is bounded by six faces, three of which meet at its vertices, and all of which are perpendicular to their respective adjacent faces. The equation for calculating the volume of a rectangle is shown below:

volume= length × width × height

EX: Darby likes cake. She goes to the gym for 4 hours a day, every day, to compensate for her love of cake. She plans to hike the Kalalau Trail in Kauai and though extremely fit, Darby worries about her ability to complete the trail due to her lack of cake. She decides to pack only the essentials and wants to stuff her perfectly rectangular pack of length, width, and height 4 ft, 3 ft and 2 ft respectively, with cake. The exact volume of cake she can fit into her pack is calculated below:

volume = 2 × 3 × 4 = 24 ft ³

Capsule

A capsule is a three-dimensional geometric shape comprised of a cylinder and two hemispherical ends, where a hemisphere is half a sphere. It follows that the volume of a capsule can be calculated by combining the volume equations for a sphere and a right circular cylinder:

where r is the radius and h is the height of the cylindrical portion

EX: Given a capsule with a radius of 1.5 ft and a height of 3 ft, determine the volume of melted milk chocolate m&m's that Joe can carry in the time capsule he wants to bury for future generations on his journey of self-discovery through the Himalayas:

Spherical Cap

A spherical cap is a portion of a sphere that is separated from the rest of the sphere by a plane. If the plane passes through the center of the sphere, the spherical cap is referred to as a hemisphere. Other distinctions exist, including a spherical segment, where a sphere is segmented with two parallel planes and two different radii where the planes pass through the sphere. The equation for calculating the volume of a spherical cap is derived from that of a spherical segment, where the second radius is 0. In reference to the spherical cap shown in the calculator:

Given two values, the calculator provided computes the third value and the volume. The equations for converting between the height and the radii are shown below:

Given R and h: r = √
where r is the radius of the base, R is the radius of the sphere, and h is the height of the spherical cap

EX: Jack really wants to beat his friend James in a game of golf to impress Jill, and rather than practicing, he decides to sabotage James' golf ball. He cuts off a perfect spherical cap from the top of James' golf ball, and needs to calculate the volume of the material necessary to replace the spherical cap and skew the weight of James' golf ball. Given James' golf ball has a radius of 1.68 inches, and the height of the spherical cap that Jack cut off is 0.3 inches, the volume can be calculated as follows:

volume = 1/3 × π × 0.3 ² (3 × 1.68 - 0.3) = 0.447 in ³

Unfortunately for Jack, James happened to receive a new shipment of balls the day before their game, and all of Jack's efforts were in vain.

Conical Frustum

A conical frustum is the portion of a solid that remains when a cone is cut by two parallel planes. This calculator calculates the volume for a right circular cone specifically. Typical conical frustums found in everyday life include lampshades, buckets, and some drinking glasses. The volume of a right conical frustum is calculated using the following equation:

where r and R are the radii of the bases, h is the height of the frustum

EX: Bea has successfully acquired some ice cream in a sugar cone, and has just eaten it in a way that leaves the ice cream packed within the cone, and the ice cream surface level and parallel to the plane of the cone's opening. She is about to start eating her cone and the remaining ice cream when her brother grabs her cone and bites off a section of the bottom of her cone that is perfectly parallel to the previously sole opening. Bea is now left with a right conical frustum leaking ice cream, and has to calculate the volume of ice cream she must quickly consume given a frustum height of 4 inches, with radii 1.5 inches and 0.2 inches:

volume=1/3 × π × 4(0.2 ² + 0.2 × 1.5 + 1.5 ²) = 10.849 in ³

Ellipsoid

An ellipsoid is the three-dimensional counterpart of an ellipse, and is a surface that can be described as the deformation of a sphere through scaling of directional elements. The center of an ellipsoid is the point at which three pairwise perpendicular axes of symmetry intersect, and the line segments delimiting these axes of symmetry are called the principal axes. If all three have different lengths, the ellipsoid is commonly described as tri-axial. The equation for calculating the volume of an ellipsoid is as follows:

where a, b, and c are the lengths of the axes

EX: Xabat only likes eating meat, but his mother insists that he consumes too much, and only allows him to eat as much meat as he can fit within an ellipsoid shaped bun. As such, Xabat hollows out the bun to maximize the volume of meat that he can fit in his sandwich. Given that his bun has axis lengths of 1.5 inches, 2 inches, and 5 inches, Xabat calculates the volume of meat he can fit in each hollowed bun as follows:

volume = 4/3 × π × 1.5 × 2 × 5 = 62.832 in ³

Square Pyramid

A pyramid in geometry is a three-dimensional solid formed by connecting a polygonal base to a point called its apex, where a polygon is a shape in a plane bounded by a finite number of straight line segments. There are many possible polygonal bases for a pyramid, but a square pyramid is a pyramid in which the base is a square. Another distinction involving pyramids involves the location of the apex. A right pyramid has an apex that is directly above the centroid of its base. Regardless of where the apex of the pyramid is, as long as its height is measured as the perpendicular distance from the plane containing the base to its apex, the volume of the pyramid can be written as:

Generalized pyramid volume:

where b is the area of the base and h is the height

Square pyramid volume:

where a is the length of the base's edge

EX: Wan is fascinated by ancient Egypt and particularly enjoys anything related to the pyramids. Being the eldest of his siblings Too, Tree and Fore, he is able to easily corral and deploy them at his will. Taking advantage of this, Wan decides to re-enact ancient Egyptian times and have his siblings act as workers building him a pyramid of mud with edge length 5 feet and height 12 feet, the volume of which can be calculated using the equation for a square pyramid:

volume = 1/3 × 5 ² × 12 = 100 ft ³

Tube Pyramid

A tube, often also referred to as a pipe, is a hollow cylinder that is often used to transfer fluids or gas. Calculating the volume of a tube essentially involves the same formula as a cylinder ( volume=pr²h), except that in this case, the diameter is used rather than the radius, and length is used rather than height. The formula, therefore, involves measuring the diameters of the inner and outer cylinder, as shown in the figure above, calculating each of their volumes, and subtracting the volume of the inner cylinder from that of the outer one. Considering the use of length and diameter mentioned above, the formula for calculating the volume of a tube is shown below:

where is the outer diameter, is the inner diameter, and l is the length of the tube

EX: Beulah is dedicated to environmental conservation. Her construction company uses only the most environmentally friendly of materials. She also prides herself on meeting customer needs. One of her customers has a vacation home built in the woods, across a creek. He wants easier access to his house, and requests that Beulah build him a road, while ensuring that the creek can flow freely so as not to disrupt his favorite fishing spot. She decides that the pesky beaver dams would be a good point to build a pipe through the creek. The volume of patented low-impact concrete required to build a pipe of outer diameter 3 feet, inner diameter 2.5 feet, and length of 10 feet, can be calculated as follows:

Common Volume Units

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The following is a list of volume calculators for several common shapes. Please fill in the corresponding fields and click the "Calculate" button. Please provide any two values below to calculate. Volume is the quantification of the three-dimensional space a substance occupies. The SI unit for volume is the cubic meter, or m3 .

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Examples: 1+2, 1/3+1/4, 2^3 * 2^2
(x+1)(x+2) (Simplify Example), 2x^2+2y @ x=5, y=3 (Evaluate Example)
y=x^2+1 (Graph Example), 4x+2=2(x+6) (Solve Example)

Algebra Calculator is a calculator that gives step-by-step help on algebra problems.

See More Examples "

Disclaimer: This calculator is not perfect. Please use at your own risk, and please alert us if something isn't working. Thank you.

How to Use the Calculator

Type your algebra problem into the text box.

For example, enter 3x+2=14 into the text box to get a step-by-step explanation of how to solve 3x+2=14.

More Examples

Trying the examples on the Examples page is the quickest way to learn how to use the calculator.

Math Symbols

If you would like to create your own math expressions, here are some symbols that the calculator understands:

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Algebra Calculator shows you the step-by-step solutions! Solves algebra problems and walks you through them.

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Number Line

Graph

Frequently Asked Questions (FAQ)

How do you solve algebraic expressions?

To solve an algebraic expression, simplify the expression by combining like terms, isolate the variable on one side of the equation by using inverse operations. Then, solve the equation by finding the value of the variable that makes the equation true.

What are the basics of algebra?

The basics of algebra are the commutative, associative, and distributive laws.

What are the 3 rules of algebra?

The basic rules of algebra are the commutative, associative, and distributive laws.

What is the golden rule of algebra?

The golden rule of algebra states Do unto one side of the equation what you do to others. Meaning, whatever operation is being used on one side of equation, the same will be used on the other side too.

What are the 5 basic laws of algebra?

The basic laws of algebra are the Commutative Law For Addition, Commutative Law For Multiplication, Associative Law For Addition, Associative Law For Multiplication, and the Distributive Law.

algebra-calculator

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Graph How do you solve algebraic expressions? To solve an algebraic expression, simplify the expression by combining like terms, isolate the variable on one side of the equation by using inverse operations. Then, solve the equation by finding the value of the variable that makes the equation true.

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iCalculator - Ratio Division Calculator | Ratio Calculator | Math Calculators

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This Ratio Division Calculator allows you to divide an amount by a specific ratio. A ratio is a numerical comparison of 2 or more quantities and indicate the relative size of each quantity. Ratios are used in a number of daily activities and business markets, the Ratio Division Calculator can be used to divide a specific ratio, you can also access additional Ratio Calculators for application in math, engineering and other every day scenarios.

Notes: How to Calculate Ratios using the Ratio Division Calculator

The following notes refer to the number adjacent the ratio calculation property:

1. Amount ¹ Enter total amount to be shared out. As an example, it could be several people sharing their lottery winning. So, enter the jackpot amount.

2. Ratio ² Enter ratio, numbers separated by: (decimal numbers are allowed). In our lottery example, the ratio is the share that each person gets perhaps based on the amount they contribute to the lottery tickets. so, 5 people, putting different amounts into the lottery fund could be 1:5:2:3:9

3. Number Count ³ Count of numbers in ratio. In our lottery example, the number count is the number of people in the group who are entitled to a share of the lottery winnings.

4. Total number of shares ⁴ Total number of shares (total of ratio figures). In our lottery example, the total amount of shares is the sum of each person's ratio, so 1:5:2:3:9 = 1 + 5 + 2 + 3 +9 = 20 shares.

5. Amount per share ⁵ Amount per share (amount divided by number of shares). This is self-explanatory, in our lottery example we have 20 shares, the figure here is the value of one share = the total lottery win divided by the number in the group (20).

6. Decimal places ⁶ You can refine the calculations by controlling the number of decimal places that the calculator calculates to. The default is 2 decimal places. In our lottery example, we think that if a group of you are lucky enough to win the lottery, decimal places won't be a problem. For the rest of us, this is useful for helping to calculate share index split and more refined answers for math homework etc.

Instructions: How to use the Ratio Division Calculator

Ratio division is as simple as calculating the sum of all shares, then dividing the amount by the total of all shares to define the value of one share. The single share amount can then by multiplied by each individual ratio split to calculate the value of each amount. The sum of all amounts calculated in the ratio should, when summed up, equal the total original amount. Completing the final sum of ratio amounts is a useful way to check that you have your ratio calculations correct, alliteratively you can use the ratio calculators on icalculator to check the math for you, lets look at how to use the ratio division calculator.

1. The ratio should be entered as numbers separated by a colon, e.g. 1:2:3

2. There is no limit to the amount of numbers in the ratio, and decimal numbers are allowed

3. There are example buttons below to demonstrate different inputs: Example 1 uses integer(whole number) values in the ratio, Example 2 includes decimal numbers. Ratio Division examples 2 and 4 are similar to Ratio Division example 2, just a little more complex.

4. Calculations can be saved to a table by clicking the "Add to Table" button. (The table appears the first time the button is clicked)

What Is a Ratio?

In mathematical terms, a ratio is the quantitative relation between two amounts, a ratio defines the number of times one value contains or is contained within the other related value. A ratio can also be defined as the relationship between two or more things when compared in quantity, amount or size, for example the ratio of cost between previous metals, strength of different construction materials and their tolerances.

Examples of Ratios

Utility costs - the chargeable rate per unit - see the Electrical Consumption Calculator for examples where costs are calculated as a direct relation (ratio) to units consumed.

Staff Costs - where individuals are paid or charge a specific amount per hour - see the Time Sheet Calculator for examples of how staffing costs are calculated on the number of hours completed at a fixed hourly rate.

Relative Payroll Calculations - the ratio of amounts paid toward income tax, medical and pension commitments can be expressed as a ratio of income received or salary package - see the United Stated $55k Salary Example

Gambling - Where ratios are referred to as odds of winning.

How can Ratios be expressed?

Ratios can be expressed in a numbers of ways, as a decimal, a fraction as a related comparison or as is more commonly seen a percentage. for example a ratio of 1 to 2 can be expressed as:

1. 1-2 is a ratio expressed in using a hyphen, whilst this is quite commonly used it should be avoided as the hyphen also represents the minus sign in math and can be confusing in long algorithms or if using computer code or computers to process a math calculation

2. 1:2 is a ratio expressed in typical math format using a colon

3. 1 to 2 is a ratio expressed in written words

4. 1 per 2 is a ratio expressed in written words

5. 0.5 is a ratio expressed as a decimal number

6. 50% is a ratio expressed as a percentage

7. ½ is a ratio expressed as a fraction

Note: The colon is the preferred mathematical written means of communicating a ratio where as verbal communication normally uses a percentage as it is typically easer to visualise and comprehend a percentage figure when comparing relative quantities.

Simplifying Ratios in Math

Simplification in math is always easier for communicating and resolving a problem, whether complex or simple math. Simplifying ratios is no exception and the simplification of ratios makes for simpler communication as a percentage when communicating relative values verbally. As an example, lets say we have a ratio of 12:16, this can be simplified to 3:4 or the ratio expressed verbally as 75%. You can use the Ratio Calculator to compute related ratios and, the smallest ratio in figurative terms and identify the lowest common denominator with a ratio.

Is a Percentage a Ratio ?

Yes, a percentage figure is a a ratio. A percentage is a number expressed as a fraction of 100, so a percentage is a ratio of a number of unit of 100. A Percentage is a special ratio with a single number expressed and followed by a percent sign (%), or the abbreviations 'pct ' / 'pc ' though the use of 'pc ' is less common that 'pct '.

Math Calculators

You may also find the following Math calculators useful.

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Ratio Division Calculator | Ratio Calculator | Math Calculators

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This Ratio Division Calculator allows you to divide an amount by a specific ratio. A ratio is a numerical comparison of 2 or more quantities and indicate the relative size of each quantity.

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What do you want to calculate?

Example: 2x^2-5x-3=0

Step-By-Step Example

Learn step-by-step how to use the quadratic formula!

Example (Click to try)

About the quadratic formula

Solve an equation of the form a x2 +b x+c=0 by using the quadratic formula:

Step-By-Step Video Lesson

Learn all about the quadratic formula with this step-by-step video lesson:

Need more problem types? Try MathPapa Algebra Calculator

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What do you want to calculate? Example: 2x^2-5x-3=0 Learn step-by-step how to use the quadratic formula! Solve an equation of the form a x2 + b x + c = 0 by using the quadratic formula: Learn all about the quadratic formula with this step-by-step video lesson: Need more problem types?

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The main purpose of this calculator is to find expression for the n th term of a given sequence. Also, it can identify if the sequence is arithmetic or geometric. The calculator will generate all the work with detailed explanation.

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Find nth term in arithmetic or geometric sequence

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The main purpose of this calculator is to find expression for the n th term of a given sequence. Also, it can identify if the sequence is arithmetic or geometric. The calculator will generate all the work with detailed explanation.

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Get the free "Simultaneous Equations Solver" widget for your website, blog, Wordpress, Blogger, or iGoogle. Find more Mathematics widgets in Wolfram|Alpha.