Pythagoras Theorem Questions

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Welcome to our Pythagoras' Theorem Questions area.

Here you will find help, support and questions to help you master Pythagoras' Theorem and apply it.

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Pythagoras' Theorem Questions Page

Here you will find our support page to help you learn to use and apply Pythagoras' theorem.

Please note: Pythagoras' Theorem is also called 'The Pythagorean Theorem'

There are a range of sheets involving finding missing sides of right triangles, testing right triangles and solving word problems using Pythagoras' theorem.

Using these sheets will help your child to:

  • learn Pythagoras' right triangle theorem;
  • use and apply the theorem in a range of contexts to solve problems.

Pythagoras' Theorem

Right triangle labelled

Pythagoras' theorem

a2 + b2 = c2

where a,b and c are the sides of a right triangle. Side c is the hypotenuse (longest side).

Pythagoras' Theorem - in more detail

Pythagoras' theorem states that in a right triangle (or right-angled triangle) the sum of the squares of the two smaller sides of the triangle is equal to the square of the hypotenuse.

Right triangle labelled

In other words, a2 + b2 = c2.

where c is the hypotenuse (the longest side) and a and b are the other sides of the right triangle.

What does this mean?

This means that for any right triangle, the orange square (which is the square made using the longest side) has the same area as the other two blue squares added together.

Right triangle pythagoras

Other formulae

As a result of the formula a2 + b2 = c2, we can also deduce that:

  • b2 = c2 - a2 and a2 = c2 - b2
  • c = √ (a2 + b2)
  • b = √ (c2 - a2) and a = √ (c2 - b2)

Examples

Example 1

Pythagoras theorem example 1

In this example, we need to find the hypotenuse (longest side of a right triangle).

So using pythagoras, the sum of the two smaller squares is equal to the square of the hypotenuse.

This gives us 42 + 62 = ?2

So ?2 = 16 + 36 = 52

So ? must be equal to √52 = 7.21cm to 2dp.

Example 2

Pythagoras example 2

In this example, we need to find the length of the base of the triangle, given the other two sides.

So using pythagoras, the sum of the two smaller squares is equal to the square of the hypotenuse.

This gives us ?2 + 52 = 82

So ?2 = 82 - 52 = 64 - 25 = 39

So ? = √39 = 6.25cm to 2dp.

Pythagoras' Theorem Question Worksheets

The following questions involve using Pythagoras' theorem to find the missing side of a right triangle.

The first sheet involves finding the hypotenuse only.

A range of different measurement units have been used in the triangles, which are not drawn to scale.

Pythagoras' Theorem Questions - Testing Right Triangles

The following questions involve using Pythagoras' theorem to find out whether or not a triangle is a right triangle, (whether the triangle has a right angle).

If Pythagoras' theorem is true for the triangle, and c2 = a2 + b2 then the triangle is a right triangle.

If Pythagoras' theorem is false for the triangle, and c2 = a2 + b2 then the triangle is not a right triangle.

A range of different measurement units have been used in the triangles, which are not drawn to scale.

Pythagoras' Theorem Questions - Word Problems

The following questions involve using Pythagoras' theorem to solve a range of word problems involving 'real-life' type questions.

On the first sheet, only the hypotenuse needs to be found, given the measurements of the other sides.

Illustrations have been provided to support students solving these word problems.

Geometry Formulas

Geometry Formula Sheet

Here you will find a support page packed with a range of geometric formula.

Included in this page are formula for:

  • areas and volumes of 2d and 3d shapes
  • interior angles of polygons
  • angles of 2d shapes
  • triangle formulas and theorems

This page will provide a useful reference for anyone needing a geometric formula.

Triangle Formulas

Here you will find a support page to help you understand some of the special features that triangles have, particularly right triangles.

Using this support page will help you to:

  • understand the different types and properties of triangles;
  • understand how to find the area of a triangle;
  • know and use Pythagoras' Theorem.

All the free printable geometry worksheets in this section support the Elementary Math Benchmarks.

Geometry Cheat Sheets

Here you will find a range of geometry cheat sheets to help you answer a range of geometry questions.

The sheets contain information about angles, types and properties of 2d and 3d shapes, and also common formulas associated with 2d and 3d shapes.

Included in this page are:

  • images of common 2d and 3d shapes;
  • properties of 2d and 3d shapes;
  • formulas involving 2d shapes, such as area and perimeter, pythagoras' theorem, trigonometry laws, etc;
  • formulas involving 3d shapes about volume and surface area.

Using the sheets in this section will help you understand and answer a range of geometry questions.

 

 

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