Pythagoras' Theorem
Pythagoras' theorem states that in any right-angled triangle, the square of the hypotenuse, the longest side, opposite the right angle, equals the sum of the squares of the other two sides: a^2 + b^2 = c^2, where c is the hypotenuse. It finds a missing side of a right-angled triangle from the other two, and can also find the direct distance between two points on a coordinate grid, or the length of a diagonal inside a rectangle.
Before you start
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Method
- Check the triangle has a right angle, and identify the hypotenuse: the longest side, always opposite the right angle.
- Label the two shorter sides a and b, and the hypotenuse c.
- To find the hypotenuse, square both shorter sides, add the results, then take the square root: c = square root of (a^2 + b^2).
- To find a shorter side, square the hypotenuse and the known shorter side, subtract the smaller from the larger, then take the square root: a = square root of (c^2 - b^2).
- Always add to find the hypotenuse and always subtract to find a shorter side; check your answer is sensible by comparing it with the other side lengths.
- For a distance between two coordinates, the horizontal and vertical differences form the two shorter sides of a right-angled triangle, so Pythagoras' theorem gives the direct distance between them.
Worked example
A ladder reaches 4.5 m up a vertical wall. The foot of the ladder is 2.4 m from the base of the wall on level ground. Find the length of the ladder, to 1 decimal place.
- Draw a right-angled triangle: the wall (vertical, 4.5 m), the ground (horizontal, 2.4 m), and the ladder as the hypotenuse.
- Square both shorter sides: 4.5^2 = 20.25 and 2.4^2 = 5.76.
- Add the squares: 20.25 + 5.76 = 26.01.
- Take the square root: square root of 26.01 = 5.1, so the ladder is 5.1 m long.
Practice questions
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Q1A right-angled triangle has shorter sides 3 cm and 4 cm. Find the length of the hypotenuse.Show answer
Answer: square root of (3^2 + 4^2) = square root of 25 = 5 cm
Q2A right-angled triangle has shorter sides 6 cm and 8 cm. Find the length of the hypotenuse.Show answer
Answer: square root of (36 + 64) = square root of 100 = 10 cm
Q3A right-angled triangle has a hypotenuse of 13 cm and one shorter side of 5 cm. Find the other shorter side.Show answer
Answer: square root of (169 - 25) = square root of 144 = 12 cm
Q4A right-angled triangle has a hypotenuse of 17 cm and one shorter side of 8 cm. Find the other shorter side.Show answer
Answer: square root of (289 - 64) = square root of 225 = 15 cm
Q5A rectangle measures 9 cm by 12 cm. Find the length of its diagonal.Show answer
Answer: square root of (9^2 + 12^2) = square root of 225 = 15 cm
Q6Point C is at (-6, 2) and point D is at (14, 23) on a coordinate grid, with each unit representing 1 cm. Find the distance CD.Show answer
Answer: horizontal difference = 20, vertical difference = 21, distance = square root of (400 + 441) = square root of 841 = 29 cm
Q7A right-angled triangle has shorter sides 9 cm and 40 cm. Find its hypotenuse, then find its perimeter.Show answer
Answer: hypotenuse = square root of (81 + 1600) = square root of 1681 = 41 cm; perimeter = 9 + 40 + 41 = 90 cm
Q8A ship sails 24 km due north, then 7 km due east. Find the direct distance from its starting point.Show answer
Answer: square root of (24^2 + 7^2) = square root of 625 = 25 km
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A ladder of length 6.5 m leans against a vertical wall. The foot of the ladder is 2.5 m from the base of the wall. (a) Work out how far up the wall the ladder reaches. (b) The ladder is repositioned so its foot is 1 m closer to the wall. Work out the new height it reaches, to 1 decimal place.
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Triangle ABC has a right angle at B, with AB = 12 cm and AC = 37 cm. (a) Work out the length of BC. (b) The triangle is the cross-section of a triangular prism of length 15 cm. Work out the volume of the prism.
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Free printable worksheet
Want more practice on paper? Download the pythagoras' theorem worksheet pack - 8 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 14 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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