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What is meant by three dimension shapes?

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What is meant by three dimension shapes?

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In geometry, a three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.

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In mathematics, we study 3-dimensional objects in the concept of solids and try to apply them in real life. Some real-life examples of 3D shapes are shown which are a soccer ball, a cube, a bucket, and a book.

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There are many kinds of 3 dimensional shapes.
Some of them are mentioned below as follows

CUBE

CUBOID

SPHERE

CONE

PRISM

HEMISPHERE

CYLINDER

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In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices

SURFACE AREA OF A CUBE

The surface of the cube is in square shape. Hence, area of each face of the cube is equal to square of edge. Let the length of edge of cube is a.

Therefore, area of one face = a²    [By area of square formula]

There are total 6 faces. Therefore,

TSA of cube = a² + a² + a² + a² + a² +a2

TSA of cube = 6a2

A cube consists of ‘n’ number of square units. Hence the space covered by these square units on the surface of the cube is the surface area. Basically, the surface area is the sum of all the area of all the shapes that cover the surface of the shape or object. In the case of a cube, there are 6 faces. So the surface area will be sum of all the area of six faces.

Problem 1:
If the sidewall of a cubic structure have length 7m, then find the total surface area.

Solution: Given, the length of the sidewall = 7m

As per the formula, we know;

TSA = 6a2

TSA = 6 x 7 x 7 = 294 sq.m

VOLUME OF A CUBE

The volume of a cube defines the number of cubic units, occupied by the cube completely. A cube is a solid three-dimensional figure, which has 6 square faces or sides. To calculate the volume we should know the dimensions of the cube.

Volume of a Cube = Length × Width × Height

V = a × a × a

V = a3

Where ‘a’ is the length the side of cube or edges.

Question 1: Find the volume of the cube, having the sides of length 7 cm.

Solution:

Given, the length of sides of the cube is 7 cm.

We know, Volume of a cube = (length of sides of the cube)3

Therefore, Volume, V = (7 cm)3

V = 343 cm3

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A cuboid is a three-dimensional shape that has 6 faces, 12 edges, and 8 vertices. It is different from a cube since all the faces of a cuboid are rectangular in shape, whereas, a cube has square faces. The three dimensions of a cuboid are its length, width, and height.

Area of Cuboid

The area of a cuboid refers to the surface area as the cuboid is a three dimensional solid. Thus, the area of cuboid can be calculated using the formula of area of rectangle, since the faces of a cuboid are in a rectangular shape.

Cuboid Surface Area

The surface area of the cuboid can be of two types-

(i) Total Surface Area

(ii) Lateral Surface Area or Curved Surface Area

Surface Area of Cuboid Formula

Before going into the concept of area, let us denote the dimensions of a cuboid, which are,

Length, Width, and Height are represented by l, w, h, respectively.

Total Surface Area of a Cuboid

The Total surface area of a cuboid (TSA) is equal to the sum of the areas of it’s 6 rectangular faces, which is given by:

Total Surface Area of a Cuboid (TSA) = 2 (lw + wh + lh) square units

The above formula gives the total surface area of a cuboid having all six faces.

Lateral Surface Area of a Cuboid

The lateral surface area of a cuboid is the sum of 4 planes of a rectangle, leaving the top (upper) and the base (lower) surface. Mathematically, the Lateral Surface Area of a cuboid (LSA) is given as:

Lateral Surface Area of a cuboid (LSA) = 2 (lh + wh) = 2 h (l + w) square units

Example 1:

The dimensions of a cuboid are given as follows:

Length = 4.8 cm

Width = 3.4 cm

Height = 7.2 cm.

Find the Total Surface area and the Lateral Surface area.

Solution:

The total surface area is given as

TSA = 2 (lw + wh + hl)

=2((4.8 ×3.4) + (3.4×7.2) + (7.2×4.8))

= 2(16.32 +24.48 +34.56)

= 2(75.36) cm²

Therefore, TSA of a cuboid= 150.72 cm

Also, the lateral surface area = 2 h (l + w)

= 2×7.2 (4.8 + 3.4)

= 14.4 (8.2) = 118.08

Therefore, LSA of a cuboid = 118.08 cm²

The volume of cuboid: The volume of a cuboid is given by the product of its dimensions.

The volume of a cuboid of length ‘l’, breadth ‘b’, height ‘h’ = l×b×h cubic units

Example
.Find the volume of a cuboid of length 20 cm, breadth 15 cm and height 10 cm.

Solution:

Length of the cuboid = 20 cm

Breadth of the cuboid = 15 cm

Height of the cuboid = 10 cm

Therefore, volume of the cuboid = length × breadth × height

= (20 × 15 × 10) cm³

= 3000 cm³

https://www.math-only-math.com/worksheet-on-volume-of-a-cube-and-cuboid.html

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What is a cuboid in math? Formulas - Volume of Cuboid - Atech Academy

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A sphere is a geometrical object in three-dimensional space that is the surface of a ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk"). ... These are also referred to as the radius and center of the sphere, respectively.

SURFACE AREA OF A SPHERE

The surface area of a sphere is defined as the region covered by its outer surface in three-dimensional space.  A Sphere is a three-dimensional solid having a round shape, just like a circle. The formula of total surface area of a sphere in terms of pi (π) is given by:

Surface area =  4 π r 2 square units

Example 1-
Calculate the curved surface area of a sphere having radius equals to 3.5 cm(Take π= 22/7)

Solution–

We know,

Curved surface area = Total surface area = 4 π r2 square units

= 4 × (22/7) × 3.5 × 3.5

Therefore, the curved surface area of a sphere= 154 cm2

VOLUME OF SPHERE

The volume of sphere is the capacity it has. The shape of the sphere is round and three -dimensional. It has three axes such as x-axis, y-axis and z-axis which defines its shape. All the things like football and basketball are examples of the sphere which have volume.

The volume here depends on the diameter of radius of the sphere since if we take the cross-section of the sphere, it is a circle. The surface area of sphere is the area or region of its outer surface. To calculate the sphere volume, whose radius is ‘r’ we have the below formula:

Volume of a sphere = 4/3 πr 3

Example 1

Find the volume of a sphere whose radius is 3 cm?

Solution :

Given: Radius, r = cm

Volume of a sphere = 4/3 πr³ cubic units

V = 4/3 x 3.14 x 33

V = 4/3 x 3.14 x 3 x 3 x 3

V = 113.04 cm3

https://www.storyofmathematics.com/volume-of-a-sphere

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A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines connecting a common point, the apex, to all of the points on a base that is in a plane that does not contain the apex

Surface Area of a cone Formula

The surface area of a cone is the total area occupied by its surface in a 3D plane. The total surface area will be equal to the sum of its curved surface area and circular base area.

Surface area of cone = πr(r+√(h²+r²))
where r is the radius of the circular base
h is the height of coneOrSurface area of cone =πr(r+L)
where L is the slant height of the coneAndCurved Surface area of cone = πrl

Example 1

1. Determine the curved surface area of a cone whose base radius is 7 cm and slant height is 15 cm.

Solution: Curved surface area of a cone = πrl

= (22/7)× 7 ×15

=  330 cm2

VOLUME OF CONE

The volume of a cone defines the space or the capacity of the cone
The volume of a cone formula is given as

The volume of a cone = (1/3) πr ² h cubic units

Where,

‘r’ is the base radius of the cone

‘l’ is the slant height of a cone

‘h’ is the height  of the cone

Example 1

Calculate the volume if r= 2 cm and h= 5 cm.

Solution:

Given:

r = 2

h= 5

Using the Volume of Cone formula

The volume of a cone = (1/3) πr ² h cubic units

V= (1/3) × 3.14 × 22 ×5

V= (1/3) × 3.14 × 4 ×5

V= (1/3) × 3.14 × 20

V = 20.93 cm3
https://www.ask-math.com/volume-of-cone.html

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In math, a hemisphere is defined as a three-dimensional shape that's half of a sphere with one flat, circular side. A sphere, on the other hand, is a nearly perfectly round three-dimensional shape.

Curved Surface Area:
The area of the outer surface of the hemisphere is called the curved surface area.

Total Surface Area:
The area of the curved surface and the area of the circle (base) is called the total surface area

From the surface area of a sphere, we can easily calculate the surface area of the hemisphere.

Since hemisphere is half of the sphere

CSA of hemisphere = (1/2)surface area of the sphere

CSA = (1/2)4πr2

CSA = 2πr2

The curved surface area of a hemisphere = 2πr 2 square units.

The total surface area of the sphere = Curved surface area of sphere + base area

We know that the base of the hemisphere is circular in shape, use the area of the circle.

TSA = 2πr2 + πr2 = 3πr2

Therefore,

The total surface area of a hemisphere = 3πr 2 square units

Where π is a constant whose value is equal to 3.14 approximately.

“r” is the radius of the hemisphere.

Question:

Find the surface area of a hemisphere whose radius is 4 cm?

Solution:

Given:

Radius, r = 4 cm

The curved surface area = 2πr2 square units.

The total surface area = 3πr2 square units

Substitute the value of r in the formula.

(i) CSA of the hemisphere= 2 × 3.14 × 4 × 4

CSA = 3.14 × 32

CSA = 100.48 cm2

(ii) TSA of the hemisphere = 3 × 3.14 × 4 × 4

TSA = 3.14 × 48

TSA = 150.72 cm2

Therefore, the curved and the total surface area of the hemisphere are 100.48 and 150.72 cm2, respectively.

Volume of hemisphere

We can easily find the volume of the hemisphere since the base of the sphere is circular. The volume of the hemisphere is derived by Archimedes.

The volume of a hemisphere = (2/3)πr 3 cubic units.

Where π is a constant whose value is equal to 3.14 approximately.

“r” is the radius of the hemisphere.

Question:

Find the volume of the hemisphere whose radius is 6 cm.

Solution:

Given:

Radius, r = 6 cm

The volume of a hemisphere = (2/3)πr3 cubic units.

Substitute the value of r in the formula.

V = (2/3) × 3.14 × 6 × 6 × 6

V = 2× 3.14 × 2 × 6 × 6

V = 452.16

Therefore, the volume of the hemisphere is 452.16 cubic units.

https://www.vedantu.com/maths/volume-of-hemisphere
https://www.google.com/amp/s/www.embibe.com/exams/volume-of-hemisphere/

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Hemisphere | Fundamental properties and formulas | [ GEOMETRY ]

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In geometry, a prism is a polyhedron comprising an n-sided polygon base, a second base which is a translated copy of the first, and n other faces, necessarily all parallelograms, joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases

Surface Area of Prism
The Surface Area of a Prism Formula is given as,
Surface Area Of A Rectangular Prism is A = 2 (wl + lh + hw)
Surface Area Of A Triangular Prism is A = bh + L (s1 + s2 + s3)

Where,
a – apothem length of the prism.
b – base length of the prism.
l – base width of the rectangular prism.
h – height of the prism.

Question 1:
What will be the surface area of a triangular prism if the apothem length, base length and height are 7 cm, 10 cm and 18 cm respectively ?
Solution:

Given,
a = 7 cm
b = 10 cm
h = 18 cm
Surface area of a triangular prism
= ab + 3bh
= (7 cm $\times$ 10 cm) + (3 $\times$ 10 cm $\times$ 18 cm)
= 70 cm² + 540 cm2
= 610 cm2

Volume of Prism
The volume of a prism is defined as the total space occupied by the three-dimensional object. Mathematically, it is defined as the product of the area of the base and the length.

Therefore,

The volume of a Prism = Base Area × Length

The measurement unit used to represent the volume of a three-dimensional object is cubic units.
https://www.onlinemathlearning.com/volume-prism-1.html

https://www.varsitytutors.com/high_school_math-help/how-to-find-the-volume-of-a-prismath-help/how-to-find-the-volume-of-a-prism

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A cylinder is a solid composed of two congruent circles in parallel planes, their interiors and all the line segments parallel to the segment containing the centers of both circles with endpoints on the circular regions. The circles and their interiors are the bases . The radius of the cylinder is the radius of a base.

Area of a Cylinder Formula

The total area of a cylinder is based on:

Curved Surface Area (CSA)

Base Area

Curved Surface Area

The curved surface area of a cylinder (CSA) is defined as the area of the curved surface of any given cylinder having base radius ‘r’, and height ‘h’, It is also termed as Lateral Surface Area (LSA). The formula for a curved area or lateral area is given by:

CSA or LSA = 2π × r × h Square units

Base Area of Cylinder

The base of the cylinder is in a circular shape. Hence, by the formula of area of the circle, we know,

Area of the circular bases of cylinder = 2 (πr²)  [Since the cylinder has two circular bases]

Total Surface Area of Cylinder

The total surface area of a cylinder is equal to the sum of areas of all its faces. The total surface area with radius ‘r’, and height ‘h’ is equal to the sum of the curved area and circular areas of the cylinder.

TSA = 2π × r × h + 2πr ²= 2πr (h + r) Square units

Problem
Find the total surface area of a container in cylindrical shape whose diameter is 28 cm and height is 15 cm.

Solution, Given, diameter = 28 cm, so radius = 28/2 = 14 cm

and height = 15 cm

By the formula of total surface are, we know;

TSA = 2πr (h + r) = 2x 22/7 x 14 x (15 + 14)

TSA = 2 x 22 x 2 x 29

TSA = 2552 sq.cm

Hence, the total surface area of container is 2552 sq.cm.

Volume of a cylinder

Volume of a Cylinder Formula

A cylinder can be seen as a collection of multiple congruent disks, stacked one above the other. In order to calculate the space occupied by a cylinder, we calculate the space occupied by each disk and then add them up. Thus, the volume of the cylinder can be given by the product of the area of base and height.

For any cylinder with base radius ‘r’, and height ‘h’, the volume will be base times the height.

Therefore, the cylinder’s volume of base radius ‘r’, and height ‘h’ = (area of base) × height of the cylinder

Since the  base is the circle, it can be written as

Volume =  πr2 × h

Therefore, the volume of a cylinder = πr ² h cubic units.

Problem

Calculate the volume of a given cylinder having height 20 cm and base radius of 14 cm. (Take pi = 22/7)

Solution:

Given:

Height  = 20 cm

radius = 14 cm

we know that;

Volume, V = πr²h  cubic units

V=(22/7) × 14  × 14  × 20

V= 12320 cm3

Therefore, the volume of a cylinder = 12320 cm3

https://courses.lumenlearning.com/prealgebra/chapter/finding-the-volume-and-surface-area-of-a-cylinder/

https://www.mathwarehouse.com/solid-geometry/cylinder/formula-volume-of-cylinder.php