Practice Maths

Pythagoras’ Theorem: Finding a Shorter Side

Key Ideas

Key Terms

a2 + b2 = c2
Where c is the hypotenuse (longest side, opposite the right angle).
shorter side (leg)
Rearrange: a2 = c2 − b2, then take the square root.
hypotenuse first
It is the side opposite the right angle and is always the longest side.
Pythagorean triples
3–4–5,   5–12–13,   8–15–17,   7–24–25, and their multiples (e.g. 6–8–10).
exact surds
Unless a decimal or rounded value is specifically requested.
FormulaWhen to useRearrangement
a2 = c2 − b2Finding a leg when hypotenuse is knowna = √(c2 − b2)
Hot Tip When finding a shorter side, you SUBTRACT the smaller square from the hypotenuse square. When finding the hypotenuse, you ADD. A common mistake is adding when you should subtract — always ask: “Do I know the hypotenuse?” If yes, subtract.

Worked Example

Question: A right triangle has hypotenuse 20 cm and one leg 12 cm. Find the length of the other leg.

Step 1 — Identify the hypotenuse.
c = 20, b = 12. We want a.

Step 2 — Rearrange Pythagoras’ theorem.
a2 = c2 − b2 = 202 − 122 = 400 − 144 = 256

Step 3 — Take the square root.
a = √256 = 16 cm

Check: 122 + 162 = 144 + 256 = 400 = 202. ✓   (This is a 3–4–5 triple scaled by 4.)

Rearranging for a Shorter Side

When the unknown is one of the shorter sides (a or b), rearrange Pythagoras' Theorem before substituting. Starting from c2 = a2 + b2, subtract b2 from both sides to get: a2 = c2 − b2, so a = √(c2 − b2). The key difference from finding the hypotenuse is that you subtract, not add. This is the most common error students make in this type of question.

Worked Example

Find the unknown shorter side in a right-angled triangle with hypotenuse 13 cm and one shorter side 5 cm. Here c = 13 and b = 5. Substitute: a2 = 132 − 52 = 169 − 25 = 144. So a = √144 = 12 cm. This is the 5–12–13 Pythagorean triple. A second example with non-exact answer: hypotenuse = 10 m, one side = 6 m. a2 = 100 − 36 = 64. a = √64 = 8 m (the 6–8–10 triple, which is double 3–4–5).

The Most Common Error

The single most frequent mistake is adding the squares instead of subtracting. This happens when students apply c2 = a2 + b2 without first rearranging, treating the unknown shorter side as if it were the hypotenuse. Always ask yourself first: "Is the unknown side the hypotenuse (longest side, opposite right angle) or a shorter side?" If it is a shorter side, you subtract.

Key tip: After finding a shorter side, do a quick reasonableness check: the answer must be less than the hypotenuse. If you get a value larger than the hypotenuse, you have added instead of subtracted. Also, the answer must be positive — if you get a negative number under the square root, it means the given measurements are impossible (no right-angled triangle can have those dimensions).

When the Answer Is Not a Whole Number

Most exam questions use numbers that produce neat answers (Pythagorean triples or perfect squares), but sometimes you will get a surd. For example, hypotenuse = 9 cm, one side = 4 cm: a2 = 81 − 16 = 65. So a = √65 ≈ 8.06 cm. Leave the answer as √65 if an exact form is required, or round as directed. As with finding the hypotenuse, always check what the question specifies before deciding whether to leave a surd or calculate a decimal.

Identifying Which Side Is Unknown

In word problems or diagrams, confirm which side you are looking for before writing any equations. Label the three sides: c is always the hypotenuse; a and b are the two shorter sides (it does not matter which you call a or which you call b). If both shorter sides are given and the hypotenuse is unknown, add. If the hypotenuse and one shorter side are given and the other shorter side is unknown, subtract.

Mastery Practice

  1. Find the unknown shorter side. Give exact answers (as surds where necessary). Fluency

     Hypotenuse cKnown leg bUnknown leg a
    (a)13 cm5 cm 
    (b)17 cm8 cm 
    (c)25 cm7 cm 
    (d)10 cm6 cm 
    (e)15 cm9 cm 
    (f)26 cm10 cm 
    (g)√50 m5 m 
    (h)10 m√19 m 
    (i)29 cm20 cm 
    (j)41 cm9 cm 

  2. Find the unknown leg. Round each answer to 2 decimal places. Fluency

     Hypotenuse cKnown leg bUnknown leg a (2 d.p.)
    (a)11 cm6 cm 
    (b)20 cm11 cm 
    (c)7.5 m4.5 m 
    (d)9 cm5 cm 
    (e)30 m18 m 
    (f)14.5 cm8.7 cm 
    (g)6.5 m2.5 m 
    (h)18 cm13 cm 

  3. Each set is a Pythagorean triple or a multiple of one. Find the missing side without a calculator and name the original triple. Fluency

     HypotenuseKnown legMissing legBase triple
    (a)108  
    (b)1512  
    (c)6525  
    (d)3416  
    (e)5030  

  4. True or False? Write T or F and justify each answer. Fluency

    1. To find a shorter side, the formula is: a2 = c2 + b2.
    2. In a right triangle with hypotenuse 10 and one leg 6, the other leg is 8.
    3. A leg of a right triangle can never be longer than the hypotenuse.
    4. If both legs are equal and the hypotenuse is 10, then each leg is 5.
    5. The formula a = c − b can be used to find a shorter side.
    6. You can always check your answer by verifying a2 + b2 = c2.

  5. Ladder problems. Understanding

    Ladder against a wall. A 5 m ladder leans against a vertical wall. The base of the ladder is 2 m from the wall. The wall is vertical and the ground is horizontal.
    1. How high up the wall does the ladder reach? Give an exact answer and a decimal approximation to 2 d.p.
    2. For safety, the base should be at least 1.5 m from the wall. Is this ladder positioned safely?
    3. If the ladder were 6 m long instead, how high would it reach with the same base distance of 2 m? (Round to 2 d.p.)

  6. Gate diagonal brace. Understanding

    Rectangular gate. A rectangular gate is 3.6 m wide and has a diagonal brace of 4.5 m. The brace is the hypotenuse of the right triangle formed by the gate’s width and height.
    1. Find the height of the gate, to 2 decimal places.
    2. How does the height compare to the width? Which is longer?
    3. A second gate is 2.4 m wide with a 3.0 m diagonal brace. Find the height of the second gate.

  7. Height of an isosceles triangle. Understanding

    Isosceles triangle. An isosceles triangle has two equal sides of 13 cm each and a base of 10 cm. The perpendicular height from the apex to the base bisects the base, creating two right triangles with hypotenuse 13 cm and one leg of 5 cm.
    1. Find the exact height of the triangle.
    2. Find the area of the isosceles triangle.
    3. A second isosceles triangle has equal sides of 10 cm and base 12 cm. Find its height and area.

  8. Identify and correct the errors. Understanding

    1. Student A says: “Hypotenuse = 13, one leg = 5. So the other leg = √(132 + 52) = √194 ≈ 13.9.” What is the error? Write the correct working.
    2. Student B says: “Hypotenuse = 10, one leg = 6. Other leg = 10 − 6 = 4.” What is wrong with this approach? What is the correct answer?
    3. Student C says: “Hypotenuse = 9, one leg = 4. Other leg = 9 − 4 = 5.” Why is this incorrect? Find the actual shorter leg.

  9. Rectangular field and tent. Problem Solving

    Park field. A park has a rectangular field 80 m long. The diagonal path across the field is 100 m.
    1. Find the width of the field. Show full working.
    2. Calculate the area of the field.
    3. A fence is needed along the perimeter. How much fencing (in metres) is required?
    4. Fencing costs $45 per metre and the diagonal path costs $12 per metre. Find the total cost of fencing the perimeter and building both diagonal paths across the field.

  10. Coordinate geometry and tent. Problem Solving

    Tent and coordinates. A tent has a triangular cross-section. The slant height (hypotenuse) of each side is 2.5 m and the base of the tent is 4 m wide. On a coordinate grid, a right triangle has its right angle at the origin O(0, 0), with one vertex at A(0, k) and another at B(8, 0).
    1. Find the exact height of the tent (the perpendicular height from the base midpoint to the apex).
    2. Find the area of the triangular cross-section of the tent.
    3. If the hypotenuse AB of the coordinate triangle = 10 units, find the value of k.
    4. A second tent has slant height 3 m and base 4 m. Find its height and compare it to the first tent.
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