Area and volume
Area of plane shapes
Rectangle
A = l · b
Square
A = s²
Triangle
A = g · h / 2
Parallelogram
A = g · h
Trapezium
A = (a + b)·h / 2
Circle
A = π r², O = 2πr
Volume of solids
Right prism / cuboid
V = l · b · h
Cube
V = s³
Cylinder
V = π r² h
Cone
V = π r² h / 3
Pyramid
V = G · h / 3
Sphere
V = ⁴⁄₃ π r³, O = 4πr²
Example
Triangle: Base 10 cm, height 6 cm.
A = (10 · 6) / 2 = 30 cm²
Circle: Radius 5 cm.
A = π · 5² = 25π ≈ 78.5 cm²
Cylinder: Radius 3 cm, height 10 cm.
V = π · 3² · 10 = 90π ≈ 282.7 cm³
Sphere: Radius 4 cm.
V = (4/3) · π · 4³ = (256/3)π ≈ 268.1 cm³
Pythagoras' theorem
In a right-angled triangle we have:
a2 + b2 = c2
Where c is the hypotenuse (the longest side, opposite the right angle), and a and b are the two shorter sides.
Using Pythagoras' theorem
Find the hypotenuse: c = √(a2 + b2)
a = 3, b = 4: c = √(9 + 16) = √25 = 5
Find a shorter side: a = √(c2 − b2)
c = 13, b = 5: a = √(169 − 25) = √144 = 12
Check for a right-angled triangle: Does a2 + b2 = c2?
3, 4, 5: 9 + 16 = 25 ✓ → Right-angled!
Proof idea (geometric)
Draw a square with side length (a + b). Place four copies of the right-angled triangle in the corners. A small square remains in the middle with side length c.
The total area can be calculated in two ways:
- Outer square: (a + b)²
- 4 triangles + inner square: 4 · (ab/2) + c²
Setting these equal gives the proof:
(a + b)2 = 4 · ab2 + c2 ⇒ a2 + 2ab + b2 = 2ab + c2 ⇒ a2 + b2 = c2
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