Pythagorean Theorem

Welcome!

Welcome to the Pythagoras page! It has been designed so you can work through it in the order presented, but you can also jump to any section you want whenever you want!

Pythagoras was a Greek mathematician who lived thousands of years ago. He discovered a rule (called the Pythagorean Theorem) that allows us to calculate the length of the third side of a right-angled triangle if we have measured the other two sides. Doesn't sound like much, but it is one of the most important discoveries in all of mathematics. Read below to find out why.

What is a right-angled triangle?

A right-angle triangle is a particular triangle which has two really important features:

One of its interior angles is 90°, also known as a right-angle. The right-angle is often shown as a small square box in the corner of the triangle
The side opposite the right-angle is always the longest side of the triangle and is called the hypotenuse

Because of their special properties, right-angled triangles are really useful helping us solve different problems. Here are a few examples:

Finding the lengths of lines on a grid

Finding the heights of isosceles triangles

Finding the heights of large objects in real life

Squares and square roots

We're almost ready to start solving problems using the Pythagorean Theorem, but before that we need to make sure we understand how to square and square root numbers using a calculator.

Squaring a number means we multiply it by itself. For example 3 squared is written as 3² and it means 3 x 3, therefore 3² = 9, because 3 x 3 = 9.

The square root of a number is generally a smaller number that you can square (multiply by itself) to get the bigger number. For example the square root of 64 is written like this: √64. What number when squared gives us 64? Well 8 x 8 = 64, so √64 = 8.

Below you can find these buttons on some of the more common calculators:

Try the following worksheets to make sure you feel comfortable using the square and square root buttons on your calculator. Each worksheet has answers at the end to check you're on the right track! Click the "pop out" button in the top right corner to view the worksheets in full size and complete the work in your book.

Squaring Numbers Worksheet.pdf

Square Root Worksheet.pdf

You can also use the square and square root button at the same time. But just make sure you use brackets to tell the calculator all the things you want to be squared or square rooted. Something like this:

Pythagorean Theorem

You made it - now we can finally start learning about the Pythagorean Theorem and how we can use it to solve problems! The Pythagorean Theorem explore the special relationship between the three side lengths of a right-angled triangle. Check out this video which demonstrates this visually.

The following video has three important sections about the Pythagorean Theorem:

Calculate the length of the hypotenuse [0:00 - 2:25] and [9:40-11:00]: After watching these sections, complete the worksheet below on finding the hypotenuse of a right-angled triangle
Calculate the length of a short side [2:25-4:30]: After watching this section, complete the worksheet below on finding a short side of a right-angled triangle
Using the Pythagorean Theorem in 3D situations [4:30-9:40]: Feel free to watch this now, but we won't worry too much about 3D situations just yet. We'll get back to this on the "Extension: 3D Pythagoras" page

Calculate the length of the hypotenuse

If you have watched the above video from the beginning to 2:36, then try this worksheet. There are worked answers at the end so you can check to see if you're on the right track!

(For more practice, check out Beta Textbook page 258-259)

Finding the Hypotenuse - Pythagoras.pdf

Calculate the length of a short side

If you have watched the above video from 2:37-4:45 then try this worksheet. There are worked answers at the end so you can check to see if you're on the right track!

(For more practice, check out Beta Textbook page 261)

Finding a shorter side - Pythagoras.pdf

Mix it up!

If you're feeling confident you can solve questions about finding both the hypotenuse and a shorter side of a right-angled triangle, then try this next worksheet. It has a mixture of the two types of questions so you will need to decide which method you use. There are worked answers at the end so you can check to see if you're on the right track!

(For more practice, check out Beta Textbook page 262-267)

Mixed Problems - Pythagoras.pdf

I've done all that, now what?

If you've gone through the video, tried the worksheets and MyMaths quizzes then there's a couple things you can do:

3D Pythagoras

If you haven't already, now would be a great time to go back to the main video and watch the 3D Pythagoras section from 4:46-9:57. After that you can try this worksheet. There are answers at the end so you can check to see if you're on the right track

3D Pythagoras Worksheet.pdf

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