Pythagoras' Theorem – Grade 9 IGCSE Maths

The Theorem

a² + b² = c² for right-angled triangles

Find Hypotenuse

c = √(a² + b²)

Find Shorter Side

a = √(c² − b²)

Pythagorean Triples

3-4-5, 5-12-13, 8-15-17

Distance Formula

d = √((x₂−x₁)² + (y₂−y₁)²)

3D Pythagoras

Space diagonal of a cuboid

1. Pythagoras' Theorem

In any right-angled triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.

a² + b² = c²
where c is always the hypotenuse (longest side)

The hypotenuse is always opposite the right angle and is always the longest side. Label it c before you start.

The theorem works only in right-angled triangles. Always check for the right angle symbol (□) before applying it.

2. Finding the Hypotenuse

When both shorter sides (legs) a and b are known:

c = √(a² + b²)

Example: Find c when a = 3, b = 4
c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5

Non-integer example: a = 5, b = 7
c² = 25 + 49 = 74
c = √74 ≈ 8.60 (2 d.p.)

3. Finding a Shorter Side

When the hypotenuse c and one leg b are known, rearrange the formula:

a = √(c² − b²)

Example: c = 13, b = 5. Find a.
a² = 13² − 5² = 169 − 25 = 144
a = √144 = 12

Always subtract the leg² from the hypotenuse². If you subtract the wrong way, you get a negative number under the square root — that's impossible.

4. Pythagorean Triples

A Pythagorean triple is a set of three positive whole numbers satisfying a² + b² = c². These exact values appear frequently in IGCSE exams.

Must-know triples:
3, 4, 5 — and multiples: 6,8,10 · 9,12,15 · 15,20,25
5, 12, 13 — and multiples: 10,24,26
8, 15, 17
7, 24, 25

5. Distance Between Two Points

The horizontal and vertical distances between two points form the legs of a right-angled triangle. The straight-line distance is the hypotenuse:

d = √((x₂ − x₁)² + (y₂ − y₁)²)

Example: Distance from A(1, 2) to B(4, 6)
Horizontal: 4 − 1 = 3 Vertical: 6 − 2 = 4
d = √(3² + 4²) = √(9 + 16) = √25 = 5

6. 3D Pythagoras — Cuboid Space Diagonal

For a cuboid with length l, width w and height h, the space diagonal (corner to corner through the interior) is:

d = √(l² + w² + h²)

Two-step method:
Step 1: Base diagonal b = √(l² + w²)
Step 2: Space diagonal d = √(b² + h²) = √(l² + w² + h²)

Example: Cuboid 3 × 4 × 5
d = √(9 + 16 + 25) = √50 ≈ 7.07

7. Word Problems Strategy

1. Read carefully and draw a diagram.
2. Identify the right angle — it is always given or implied.
3. Label the sides: mark c as the hypotenuse (opposite the right angle).
4. Choose the correct formula and substitute.
5. Round to the precision stated in the question.

Common contexts: ladders against walls · ramps · ships' journeys · rectangle diagonals · roof trusses · distances between towns on a grid

Example 1 — Find the Hypotenuse

A right-angled triangle has legs 6 cm and 8 cm. Find the hypotenuse.

Formula: c² = a² + b²

Substitute: c² = 6² + 8² = 36 + 64 = 100

Answer: c = √100 = 10 cm

Example 2 — Find a Shorter Side

A right-angled triangle has hypotenuse 17 cm and one leg 8 cm. Find the other leg.

Formula: a² = c² − b²

Substitute: a² = 17² − 8² = 289 − 64 = 225

Answer: a = √225 = 15 cm

Example 3 — Distance Between Points

Find the distance between A(2, 1) and B(7, 13).

Horizontal change: 7 − 2 = 5

Vertical change: 13 − 1 = 12

Answer: d = √(5² + 12²) = √(25 + 144) = √169 = 13

Example 4 — 3D Space Diagonal

A box is 4 cm × 5 cm × 3 cm. Find the space diagonal.

Formula: d = √(l² + w² + h²)

d = √(4² + 5² + 3²) = √(16 + 25 + 9) = √50

Answer: d = √50 ≈ 7.07 cm

Example 5 — Ladder Word Problem

A 10 m ladder leans against a vertical wall. The foot is 4 m from the wall. How high up the wall does it reach?

The ladder is the hypotenuse (c = 10 m), the floor distance is one leg (b = 4 m).

h² = 10² − 4² = 100 − 16 = 84

Answer: h = √84 ≈ 9.2 m

Example 6 — Verify a Pythagorean Triple

Show that 8, 15, 17 is a Pythagorean triple.

Check: 8² + 15² = 64 + 225 = 289

17² = 289 ✓

Conclusion: Since a² + b² = c², it is a valid Pythagorean triple.

📐 Right Triangle Calculator

Select which side to find, enter the other two values, and see the triangle drawn with labels.

Find: Side a: Side b:

Enter two sides and click Calculate.

Exercise 1 — Find the Hypotenuse (2 d.p.)

1. a = 3, b = 4. Find c.

2. a = 5, b = 12. Find c.

3. a = 6, b = 8. Find c.

4. a = 9, b = 12. Find c.

5. a = 5, b = 5. Find c.

6. a = 4, b = 7. Find c.

7. a = 7, b = 9. Find c.

8. a = 8, b = 11. Find c.

9. a = 4, b = 5. Find c.

10. a = 11, b = 14. Find c.

Exercise 2 — Find a Shorter Side (2 d.p.)

1. c = 5, b = 3. Find a.

2. c = 13, b = 5. Find a.

3. c = 10, b = 8. Find a.

4. c = 17, b = 15. Find a.

5. c = 9, b = 7. Find a.

6. c = 11, b = 8. Find a.

7. c = 15, b = 11. Find a.

8. c = 18, b = 13. Find a.

9. c = 8, b = 6.5. Find a.

10. c = 22, b = 15.5. Find a.

Exercise 3 — Distance Between Two Points (2 d.p.)

1. Distance from (0,0) to (3,4).

2. Distance from (1,1) to (4,5).

3. Distance from (2,4) to (8,12).

4. Distance from (0,0) to (5,12).

5. Distance from (−1,−1) to (4,4).

6. Distance from (1,2) to (8,8). [Δx=7, Δy=6]

7. Distance from (−2,1) to (6,7).

8. Distance from (3,−2) to (6,3). [Δx=3, Δy=5]

9. Distance from (0,2) to (5,7).

10. Distance from (−3,−2) to (7,12).

Exercise 4 — 3D Space Diagonal of a Cuboid (2 d.p.)

1. Cuboid 3 × 4 × 5. Find the space diagonal.

2. Cuboid 4 × 8 × 8. Find the space diagonal.

3. Cuboid 6 × 6 × 5. Find the space diagonal.

4. Cuboid 7 × 9 × 8. Find the space diagonal.

5. Cuboid 3 × 4 × 3. Find the space diagonal.

6. Cuboid 5 × 7 × 6. Find the space diagonal.

7. Cuboid 5 × 5 × 5. Find the space diagonal.

8. Cuboid 9 × 10 × 8. Find the space diagonal.

9. Cuboid 3 × 3 × 2. Find the space diagonal.

10. Cuboid 10 × 12 × 11. Find the space diagonal.

Exercise 5 — Word Problems (1 d.p.)

1. A ladder 8.5 m leans against a wall. Its foot is 4 m from the base. How high does it reach?

2. A rectangle is 5 m by 11 m. Find its diagonal length.

3. A boat sails 7 km east then 8 km north. How far is it from the start?

4. A square has diagonal 22 cm. Find the side length. (side = diag ÷ √2)

5. Two legs of a right triangle are both 5.5 m. Find the hypotenuse.

6. A pole is 8 m tall. A wire runs from its top to a point 4.5 m from its base. Find the wire length.

7. A field is 7 m × 8 m. Find the diagonal path length.

8. A ramp rises 5 m over a horizontal run of 12 m. Find the ramp length.

9. A kite string is 10 m. Horizontal distance is 8 m. How high is the kite?

10. A room is 9 m × 14 m. Find the diagonal of the floor.

🏋️ Practice — 20 Questions

1. a=3, b=4. Find c.

2. a=5, b=12. Find c.

3. a=6, b=8. Find c.

4. a=9, b=12. Find c.

5. a=5, b=5. Find c (2 d.p.).

6. a=4, b=7. Find c (2 d.p.).

7. a=7, b=9. Find c (2 d.p.).

8. a=8, b=11. Find c (2 d.p.).

9. a=4, b=5. Find c (2 d.p.).

10. a=11, b=14. Find c (2 d.p.).

11. c=5, b=3. Find a.

12. c=13, b=5. Find a.

13. c=10, b=8. Find a.

14. c=17, b=15. Find a.

15. c=9, b=7. Find a (2 d.p.).

16. c=11, b=8. Find a (2 d.p.).

17. c=15, b=11. Find a (2 d.p.).

18. c=18, b=13. Find a (2 d.p.).

19. c=8, b=6.5. Find a (2 d.p.).

20. c=22, b=15.5. Find a (2 d.p.).

🏆 Challenge — 8 Questions