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This website gives a different perspective from the classroom while still accomplishing the same task of learning the Pythagorean Theorem.

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Pythagoras

Over 2000 years ago there was an amazing discovery about triangles:
When a triangle has a right angle (90°) ...... and squares are made on each of the three sides, ...... then the biggest square has the exact same area as the other two squares put together!

It is called "Pythagoras' Theorem" and can be written in one short equation:

Note:

Definition

The longest side of the triangle is called the "hypotenuse", so the formal definition is:

In a right angled triangle:
the square of the hypotenuse is equal to
the sum of the squares of the other two sides.

Sure ... ?

Let's see if it really works using an example.

Let's check if the areas are the same:

3 ² + 4 ² = 5 ²

Calculating this becomes:

9 + 16 = 25
It works ... like Magic!

Why Is This Useful?

If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!)

How Do I Use it?

Write it down as an equation:

a ² + b ² = c ²

Then we use algebra to find any missing value, as in these examples:

Example: Solve this triangle

Read Builder's Mathematics to see practical uses for this.

Also read about Squares and Square Roots to find out why √169 = 13

Example: Solve this triangle.

Example: What is the diagonal distance across a square of size 1?

It works the other way around, too: when the three sides of a triangle make , then the triangle is right angled.

Example: Does this triangle have a Right Angle?

They are equal, so ...

Yes, it does have a Right Angle!

Example: Does an 8, 15, 16 triangle have a Right Angle?

So, NO, it does not have a Right Angle

Example: Does this triangle have a Right Angle?

Yes, it does!

So this is a right-angled triangle

And You Can Prove The Theorem Yourself !

Get paper pen and scissors, then using the following animation as a guide:

Draw a right angled triangle on the paper, leaving plenty of space.

Draw a square along the hypotenuse (the longest side)

Draw the same sized square on the other side of the hypotenuse

Draw lines as shown on the animation, like this:

Cut out the shapes

Arrange them so that you can prove that the big square has the same area as the two squares on the other sides

Another, Amazingly Simple, Proof

Here is one of the oldest proofs that the square on the long side has the same area as the other squares.

Watch the animation, and pay attention when the triangles start sliding around.

You may want to watch the animation a few times to understand what is happening.

The purple triangle is the important one.

becomes

We also have a proof by adding up the areas.

Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived.

511,512,617,618, 1145, 1146, 1147, 2359, 2360, 2361

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Pythagoras Over 2000 years ago there was an amazing discovery about triangles: When a triangle has a right angle (90°) ... ... and squares are made on each of the three sides, ... ... then the biggest square has the exact same area as the other two squares put together!

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This website gives a more in-depth focus on Pythagorean Theorem as students will learn about the common Pythagorean Triples.

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A "Pythagorean Triple" is a set of positive integers, a, b and c that fits the rule:

Example: The smallest Pythagorean Triple is 3, 4 and 5.

Let's check it:

Calculating this becomes:

9 + 16 = 25

Yes, it is a Pythagorean Triple!

Triangles

When a triangle's sides are a Pythagorean Triple it is a right angled triangle.

See Pythagoras' Theorem for more details.

Example: The Pythagorean Triple of 3, 4 and 5 makes a Right Angled Triangle:

Here are two more Pythagorean Triples:

5, 12, 13

9, 40, 41

5 ² + 12 ² = 13 ²

9 ² + 40 ² = 41 ²

25 + 144 = 169

(try it yourself)

And each triangle has a right angle!

List of the First Few

Here is a list of the first few Pythagorean Triples ( not including "scaled up" versions mentioned below):

Scale Them Up

The simplest way to create further Pythagorean Triples is to scale up a set of triples.

Example: scale 3, 4, 5 by 2 gives 6, 8, 10

36 + 64 = 100

If you want to know more about them read Pythagorean Triples - Advanced

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A "Pythagorean Triple" is a set of positive integers, a, b and c that fits the rule: Let's check it: Calculating this becomes: 9 + 16 = 25 Yes, it is a Pythagorean Triple! When a triangle's sides are a Pythagorean Triple it is a right angled triangle.

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This activity allows students to practice right triangle content and see their knowledge overall on Right Triangles. 

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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This review allows students to review all their right triangle content to prepare for any future assessments. 

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This activity allows students to practice solving sides and angles of right triangles and check their overall knowledge of the content. 

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No mobile devices in your classroom? No worries! You can enjoy Nearpod from any web browser :) Create, engage, and assess your students in every lesson!

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This formula sheet is for students to refer back to the commonly used formulas for right triangles (Pythagorean Theorem and Special Right Triangle properties).

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This formula sheet for students to refer back to commonly used Trigonometry Ratios and their inverse.

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These flashcards help students memorize their vocabulary and formulas that will be used frequently with Right Triangles.

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This video helps students memorize the Pythagorean Theorem. 

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Colin Dodds - Pythagorean Theorem (Math Song)

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This video is a quick overview of Special Right Triangles, specifically 45-45-90 Triangles.

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0:00 / 1:00•Watch full video

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This video is a quick overview of Special Right Triangles, specifically 30-60-90 Triangles.

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0:00 / 0:30•Watch full video

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This vocabulary activity helps students match their vocabulary with the correct definition/formulas that are commonly used with Right Triangles.

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No mobile devices in your classroom? No worries! You can enjoy Nearpod from any web browser :) Create, engage, and assess your students in every lesson!

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This activity allows students to practice solving sides and angles of right triangles and check their overall knowledge of the content at their own pace. 

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This website allows students to make the required calculations Pythagorean Theorem and Right Triangle Trig.

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A beautiful, free online scientific calculator with advanced features for evaluating percentages, fractions, exponential functions, logarithms, trigonometry, statistics, and more.

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This video is a quick overview of Right Triangle Trigonometry and their ratios.