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Explainer Video

Tutor: Bilal

Summary

Pythagoras' theorem

In a nutshell

Pythagoras' theorem (or the Pythagorean theorem) is a formula that involves 3 sides of a right-angled triangle.

Hypotenuse

The hypotenuse of a triangle is the longest side. For a right-angled triangle, the hypotenuse will be opposite the right angle.

The formula for Pythagoras' theorem

Pythagoras' theorem states that

$a^2\,+b^2\,=c^2$

Where $c$ is the hypotenuse of the triangle, and $a$ and $b$ are the other 2 sides.

Maths; Trigonometry; KS3 Year 7; Pythagoras’ theorem

Example 1

A right-angled triangle has the 2 shorter sides with length $5cm$ and $12cm$ . Find the length of the longest side.

Sketch the triangle:

Label the sides:

Use Pythagoras' theorem:

$a^2\,+b^2\,=c^2$

Replace $a$ and $b$ with your values, and solve:

$12^2+5^2=c^2$

$144+25=c^2$

$c^2=169$

$c=\sqrt{169}=13$

$\underline{c=13cm}$

Note: Don't forget to write the correct unit for your length!

Example 2

A right-angled triangle has a hypotenuse of length $8cm$ , and a base of length $2cm$ . What is the area of the triangle to two decimal places?

First, work out the perpendicular height of the triangle using Pythagoras' theorem. Let $a=2$ and $c=8$ :

$a^2+b^2=c^2$ 

$2^2+b^2=8^2$ 

$b^2=8^2-2^2=64-4=60$ 

$b=\sqrt{60}cm$ 

Then, work out the area of the triangle using the area formula:

$\text{Area}=\dfrac{1}{2}\times\text{base}\times\text{perpendicular\,height}$ 

$\text{Area}=\dfrac{1}{2}\times2\times\sqrt{60}$ 

$\text{Area}=7.74596669...$ 

The area is $\underline{7.75 \ cm^2 \space (2\thinspace d.p)}$ .

Learn with Basics

Length:

Unit 1

Geometric shapes: Types and properties

Unit 2

Triangles and regular polygons: Types and properties

Jump Ahead

Optional

Unit 3

Pythagoras’ theorem

Final Test

Create an account to complete the exercises

FAQs - Frequently Asked Questions

How do you use Pythagoras' theorem to find the area of a right-angled triangle?

Use Pythagoras' theorem to find the length of the base or perpendicular height - whichever isn't given. Then, use the formula for the area of a triangle which is 1/2 x base x perpendicular height.

What is the hypotenuse of a triangle?

The hypotenuse of a triangle is the longest side.

When can I use Pythagoras' Theorem?

You can only use Pythagoras' Theorem when you are given 2 sides of a right-angled triangle and want to find the third side.

Home

Maths

Trigonometry

Pythagoras’ theorem

Videos

Summary

Exercises

Select Lesson

Statistics

Qualitative and quantitative data

Representing data: Graphs and charts

Mean, median, mode and range

Frequency tables

Averages from frequency tables

Averages from grouped frequency tables

Scatter graphs

Probability

Calculating probability

Equal and unequal probabilities

Relative frequency

Listing outcomes

Venn diagrams

Trigonometry

Pythagoras’ theorem

Trigonometric ratios: SOH CAH TOA

Finding missing lengths in a triangle

Finding missing angles in a triangle

Drawing shapes

Transformations

Constructions

Constructing triangles

Lines and angles

Types of angles

Angle rules

Interior and exterior angles

Parallel lines

Measures

Area and perimeter: Triangles and quadrilaterals

Area of compound shapes

Volume of prisms

Properties of shapes

Symmetrical shapes and rotational symmetry

Congruent shapes

Similar shapes

Nets and surface area

Shapes

Quadrilaterals: Types and properties

Triangles and regular polygons: Types and properties

Circles: Area and circumference

3D shapes: Identification and properties

Rates of change

Percentage change

Conversion factors

Imperial units

Solving best buy problems

Reading timetables

Maps and scale drawings

Speed: Formula and units

Density: Formula and units

Proportion

Ratio

Other graphs

Interpreting graphs

Solving simultaneous equations using graphs

Quadratic graphs

Travel graphs: Distance-time graphs

Conversion graphs

Straight line graphs

X and Y coordinates

Straight line graphs

Drawing straight line graphs

Finding the gradient

Equation of a straight line: y = mx + c

Drawing a graph of a linear equation

Estimating values using linear graphs

Formulae and equations

Substituting numbers into an expression

Writing and using formulae

Solving equations

Rearranging formulae

Number patterns and sequences

The nth term

Inequalities: Greater than or less than

Simultaneous equations

Algebraic manipulation

Algebraic notation

Algebraic terms

Simplifying algebraic expressions

Multiplying and dividing algebraic expressions

Single brackets: Expanding and factorising

Expanding double brackets: FOIL and multiplication grid

Fractions, decimals and percentages

Converting between fractions, decimals and percentages

Fractions

Fraction operations: Add, subtract, multiply, divide

Percentages

Rounding: Decimal places and significant figures

Rounding errors and estimating

Number calculations

Ordering numbers: Whole numbers and decimals

Addition and subtraction using the column method

Multiplying and dividing by powers of 10

Adding and subtracting decimals

Methods for multiplication and division

Order of operations: BIDMAS

Types of numbers

Understanding place value

Negative numbers: Add, subtract, multiply, divide

Special types of numbers

Multiples, factors and prime factors

Lowest common multiple and highest common factor

Square roots and cube roots

Writing indices and index laws

Standard form

Explainer Video

Tutor: Bilal

Summary

Pythagoras' theorem

In a nutshell

Pythagoras' theorem (or the Pythagorean theorem) is a formula that involves 3 sides of a right-angled triangle.

Hypotenuse

The hypotenuse of a triangle is the longest side. For a right-angled triangle, the hypotenuse will be opposite the right angle.

The formula for Pythagoras' theorem

Pythagoras' theorem states that

$a^2\,+b^2\,=c^2$

Where $c$ is the hypotenuse of the triangle, and $a$ and $b$ are the other 2 sides.

Example 1

A right-angled triangle has the 2 shorter sides with length $5cm$ and $12cm$ . Find the length of the longest side.

Sketch the triangle:

Label the sides:

Use Pythagoras' theorem:

$a^2\,+b^2\,=c^2$

Replace $a$ and $b$ with your values, and solve:

$12^2+5^2=c^2$

$144+25=c^2$

$c^2=169$

$c=\sqrt{169}=13$

$\underline{c=13cm}$

Note: Don't forget to write the correct unit for your length!

Example 2

A right-angled triangle has a hypotenuse of length $8cm$ , and a base of length $2cm$ . What is the area of the triangle to two decimal places?

First, work out the perpendicular height of the triangle using Pythagoras' theorem. Let $a=2$ and $c=8$ :

$a^2+b^2=c^2$ 

$2^2+b^2=8^2$ 

$b^2=8^2-2^2=64-4=60$ 

$b=\sqrt{60}cm$ 

Then, work out the area of the triangle using the area formula:

$\text{Area}=\dfrac{1}{2}\times\text{base}\times\text{perpendicular\,height}$ 

$\text{Area}=\dfrac{1}{2}\times2\times\sqrt{60}$ 

$\text{Area}=7.74596669...$ 

The area is $\underline{7.75 \ cm^2 \space (2\thinspace d.p)}$ .

Learn with Basics

Length:

Unit 1

Geometric shapes: Types and properties

Unit 2

Triangles and regular polygons: Types and properties

Jump Ahead

Optional

Unit 3

Pythagoras’ theorem

Final Test

Create an account to complete the exercises

FAQs - Frequently Asked Questions

How do you use Pythagoras' theorem to find the area of a right-angled triangle?

Use Pythagoras' theorem to find the length of the base or perpendicular height - whichever isn't given. Then, use the formula for the area of a triangle which is 1/2 x base x perpendicular height.

What is the hypotenuse of a triangle?

The hypotenuse of a triangle is the longest side.

When can I use Pythagoras' Theorem?

You can only use Pythagoras' Theorem when you are given 2 sides of a right-angled triangle and want to find the third side.