Practice Maths

Pythagoras’ Theorem: Finding the Hypotenuse

Key Ideas

Key Terms

right-angled triangle
Has one angle equal to 90°.
hypotenuse
(c) is the longest side and is always opposite the right angle.
legs
(a and b).
Pythagoras’ theorem
a2 + b2 = c2
exact
(as a surd, e.g. √50) or decimal approximations (rounded to a given number of decimal places).
Pythagorean triple
A set of three whole numbers that satisfy the theorem (e.g. 3, 4, 5 or 5, 12, 13).
FormulaWhen to useRearrangement
c2 = a2 + b2Finding the hypotenusec = √(a2 + b2)
Hot Tip Always identify the hypotenuse first — it is the side opposite the right angle, not necessarily the side on the right or at the top. Label it c before you set up your equation.

Worked Example

Question: A right-angled triangle has legs of length 3 cm and 4 cm. Find the length of the hypotenuse.

Step 1 — Write the formula.
c2 = a2 + b2

Step 2 — Substitute the known values.
c2 = 32 + 42 = 9 + 16 = 25

Step 3 — Solve for c.
c = √25 = 5 cm

Solution: The hypotenuse is 5 cm. (3, 4, 5 is a Pythagorean triple.)

Pythagoras' Theorem and the Hypotenuse

Pythagoras' Theorem applies to any right-angled triangle. It states: c2 = a2 + b2, where c is the length of the hypotenuse and a and b are the lengths of the two shorter sides. The hypotenuse is always the side directly opposite the right angle — it is always the longest side in a right-angled triangle. To find c, square both shorter sides, add the results, then take the square root: c = √(a2 + b2).

Worked Example

Find the hypotenuse of a right-angled triangle with shorter sides 6 cm and 8 cm. Substitute into the formula: c2 = 62 + 82 = 36 + 64 = 100. So c = √100 = 10 cm. This is a Pythagorean triple (6, 8, 10, which is double the 3-4-5 triple). A second example: sides 5 cm and 7 cm. c2 = 25 + 49 = 74. c = √74 ≈ 8.60 cm (to 2 decimal places).

Exact vs Decimal Answers

When the number under the square root is a perfect square (like 25, 49, 100), the answer is exact and whole. When it is not a perfect square (like 74), you can leave the answer as an exact surd (√74) or calculate a decimal approximation. Always check what the question asks: "exact answer" means leave as a surd; "correct to 2 decimal places" means use your calculator. Unless told otherwise, a surd is the more precise form.

Pythagorean Triples

A Pythagorean triple is a set of three whole numbers that satisfy a2 + b2 = c2. The most important ones to recognise are: 3–4–5 (9 + 16 = 25), 5–12–13 (25 + 144 = 169), and 8–15–17 (64 + 225 = 289). Any multiple of a triple is also a triple: 6–8–10, 9–12–15, 10–24–26 are all valid. Recognising these saves time in exams because you can state the hypotenuse without calculating.

Key tip: Always identify the hypotenuse before setting up the formula. The hypotenuse is opposite the right angle — look for the square corner symbol in the diagram. A common error is applying the formula with the hypotenuse in the wrong position. If no diagram is given, draw one first and mark the right angle clearly.

Identifying the Hypotenuse in a Diagram

In most Pythagoras problems, the right angle is marked with a small square symbol. The side opposite this mark is always the hypotenuse — it does not matter which way the triangle is oriented on the page. If the triangle is tilted, rotated, or appears unusual, the hypotenuse is still simply the side that is not touching the right angle. Never assume the longest-looking side in a rough sketch is the hypotenuse; always confirm using the right-angle marker.

Mastery Practice

  1. Find the hypotenuse. Give exact answers where possible; otherwise round to 2 decimal places. Fluency

     Leg aLeg bHypotenuse c (2 d.p.)
    (a)3 cm4 cm 
    (b)5 cm12 cm 
    (c)6 cm8 cm 
    (d)8 cm15 cm 
    (e)7 cm24 cm 
    (f)9 cm40 cm 
    (g)4 cm7 cm 
    (h)5 cm9 cm 
    (i)6 cm6 cm 
    (j)10 cm10 cm 

  2. Find the hypotenuse for triangles with decimal side lengths. Round to 2 decimal places. Fluency

     Leg aLeg bHypotenuse c (2 d.p.)
    (a)2.5 m3.5 m 
    (b)1.5 m2.0 m 
    (c)4.2 cm5.6 cm 
    (d)3.6 m4.8 m 
    (e)7.5 cm10.0 cm 
    (f)0.6 m0.8 m 
    (g)5.5 m7.2 m 
    (h)9.1 cm12.3 cm 

  3. Find the exact length of the hypotenuse. Leave answers as simplified surds where the result is not a whole number. Fluency

     Leg aLeg bExact answerDecimal (2 d.p.)
    (a)1 cm1 cm  
    (b)2 cm3 cm  
    (c)4 cm6 cm  
    (d)5 cm5 cm  
    (e)3 cm7 cm  
    (f)6 cm10 cm  

  4. True or False? Write T or F and give a brief reason or counter-example for each. Fluency

    1. The hypotenuse is always the longest side of a right-angled triangle.
    2. If a = 6 and b = 8, then c = 14.
    3. The formula c2 = a2 + b2 only works when the triangle has a right angle.
    4. For the triangle with legs 5 and 12, the hypotenuse is 13.
    5. The hypotenuse is always opposite the largest angle in the triangle.
    6. If you double both legs of a right triangle, the hypotenuse also doubles.

  5. Ladder and distance problems. Understanding

    Ladder against a wall. A 5 m ladder leans against a vertical wall. The base of the ladder is 3 m from the base of the wall. The ladder, wall, and ground form a right-angled triangle where the ladder is the hypotenuse.
    1. Draw and label this right triangle.
    2. Find how high the ladder reaches up the wall.
    3. If the base is moved to 4 m from the wall, recalculate how high the ladder reaches. Is it higher or lower?

  6. Diagonal of a rectangle. Understanding

    Rectangular garden. A rectangular garden is 9 m wide and 12 m long. The owner wants to put a diagonal path from one corner to the opposite corner.
    1. Draw the rectangle and mark the diagonal. What right triangle is formed?
    2. Find the exact length of the diagonal path.
    3. Paving costs $18.50 per metre. How much does the diagonal path cost?

  7. Screen diagonal. Understanding

    Television screen. A television screen has a width of 80 cm and a height of 45 cm. Television screens are advertised by their diagonal measurement.
    1. Find the diagonal size of this television, rounded to the nearest centimetre.
    2. A second TV is described as “55 inches diagonal” with a height of 27 inches. Find its width, to the nearest inch.

  8. Pythagorean triples check. For each set of three numbers, test whether a2 + b2 = c2. Write Yes or No and show your working. Understanding

     Three sidesTest: a2 + b2 = c2?Right-angled?
    (a)6, 8, 10  
    (b)5, 12, 13  
    (c)4, 5, 6  
    (d)8, 15, 17  
    (e)9, 40, 41  
    (f)7, 24, 26  

  9. 3D diagonal of a box. Problem Solving

    Moving a pole. A removal company needs to carry a long pole through a rectangular room. The room is 8 m long, 6 m wide, and 3 m high. To find the longest pole that will fit diagonally across the room (corner to corner), apply Pythagoras’ theorem twice.
    1. First, find the diagonal of the floor (using length and width).
    2. Then use that diagonal and the room height to find the full space diagonal.
    3. Will a 10.5 m pole fit? Justify your answer.
    4. What is the longest whole-number pole (in metres) that will fit? Round the space diagonal down.

  10. Multi-step problem: building and cable. Problem Solving

    Antenna cables. A vertical antenna pole stands 12 m tall on flat ground. Two support cables are attached to the top of the pole and anchored to the ground. Cable A is anchored 5 m from the base of the pole. Cable B is anchored 9 m from the base on the opposite side.
    1. Find the length of Cable A. Give an exact answer and a decimal approximation to 2 d.p.
    2. Find the length of Cable B. Give an exact answer and a decimal approximation to 2 d.p.
    3. Cable costs $12.40 per metre. Find the total cost of both cables.
    4. A third cable is attached halfway up the pole (6 m) and anchored 4 m from the base. How long is this cable?
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