Pythagoras' Theorem, Area and Volume: Year 8 Study Guide
Pythagoras connects the sides of a right-angled triangle, and measurement formulas let us calculate real objects.
In a right-angled triangle, the square of the hypotenuse (the longest side) is the sum of the squares of the other two sides: c² = a² + b². A 6 cm by 8 cm right-angled triangle has a hypotenuse of √(36 + 64) = 10 cm. The theorem also finds a shorter side: b = √(c² − a²).
A circle has circumference C = 2πr = πd and area A = πr². The volume of a prism or cylinder = base area × height. Cylinder: V = πr²h. Surface area is the total area of all faces. Capacity: 1 cm³ = 1 mL; 1000 cm³ = 1 L; 1 m³ = 1000 L.
Using Pythagoras
A ladder 5 m long leans on a wall with its base 1.5 m from the wall. Height = √(25 − 2.25) = √22.75 ≈ 4.77 m. To test for a right angle, check if a² + b² = c². Pythagorean triples: (3, 4, 5), (5, 12, 13), (8, 15, 17).
Identify the hypotenuse.
c² = a² + b².
Square, add, root.
Check it is reasonable.
Circles and cylinders
For a circle of radius 6 cm: C = 2π × 6 ≈ 37.7 cm and A = π × 36 ≈ 113.1 cm². A cylindrical water tank with radius 1 m and height 2 m has a volume of π × 1 × 2 ≈ 6.28 m³, which is 6283 L. Its curved surface area is 2πrh. Use your calculator's π key rather than 3.14 for accuracy.
| Shape | Formula | Example |
|---|---|---|
| Circle circumference | 2πr | r = 5 → 31.4 |
| Circle area | πr² | r = 5 → 78.5 |
| Prism volume | base area × height | 12 × 5 = 60 |
| Cylinder volume | πr²h | r = 2, h = 7 → 87.96 |
Composite figures
Break a complex shape into simple ones: a rectangle with a semicircle on top, for instance. Add the areas, or subtract a hole. Keep units consistent: convert cm to m (or the reverse) before multiplying.
Worked examples and common mistakes
Example: A TV screen is 90 cm wide and 50 cm high. Its diagonal is √(8100 + 2500) ≈ 103 cm. Common mistakes: using the diameter instead of the radius in πr², applying Pythagoras to a triangle that is not right-angled and forgetting to square-root. Estimate: is the hypotenuse the longest side?
Check your understanding: model answers
1. 15. 2. 16π ≈ 50.3 square units. 3. 90 cm³.
✨ Key Takeaways
- c² = a² + b² for right-angled triangles.
- A = πr², C = 2πr.
- Volume = base area × height.
- Keep units consistent.
💬 Questions for Discussion & Reflection
- Find the hypotenuse for sides 9 and 12.
- Find the area of a circle with radius 4 (use π).
- Find the volume of a prism with base area 18 cm² and height 5 cm.