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Level 2 · Topic 3

Measures, Shape & Space

Circles, cylinders, Pythagoras' theorem, complex compound shapes and density

Circles — circumference and area

Circles use the special number π (pi) ≈ 3.14159… The radius is the distance from the centre to the edge. The diameter is twice the radius.

Circumference = 2πr = πd
Area = πr²
π ≈ 3.14159 (use the π button on your calculator)
💡 Remember: Area uses r² (radius squared). Circumference uses r (radius to the power 1). A common mistake is using diameter instead of radius in the area formula.

Volume of cylinders and prisms

A cylinder is a prism with a circular cross-section. A prism is any 3D shape with a uniform cross-section along its length.

Volume of cylinder = πr²h
Volume of prism = cross-sectional area × length
1 litre = 1,000 cm³  |  1 m³ = 1,000,000 cm³
πr²h
Cylinder volume
A × l
Prism volume
cm³
Volume units

Pythagoras' theorem

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

a² + b² = c²
c = √(a² + b²) ← finding hypotenuse
a = √(c² − b²) ← finding a shorter side
💡 Identify the hypotenuse: It is always the side opposite the right angle — the longest side. Label it c. The other two sides are a and b.

Density, pressure and speed

These are all examples of compound measures — rates combining two different units.

Density = Mass ÷ Volume   (g/cm³ or kg/m³)
Pressure = Force ÷ Area   (N/m² or Pa)
Speed = Distance ÷ Time   (m/s or km/h)
💡 Triangle method: Cover the quantity you want to find — the remaining two show you whether to multiply or divide.
1Circumference and area of a circle
▼
A circular pond has a radius of 3.5 m. Find (a) the circumference and (b) the area. Give answers to 2 d.p.
  1. 1(a) Circumference: C = 2πr = 2 × π × 3.5 = 21.99 m
  2. 2(b) Area: A = πr² = π × 3.5² = π × 12.25 = 38.48 m²
✅ Circumference = 21.99 m  |  Area = 38.48 m²
2Volume of a cylinder
▼
A cylindrical water tank has radius 0.8 m and height 1.5 m. What is its volume in litres?
  1. 1Volume: V = πr²h = π × 0.8² × 1.5 = π × 0.64 × 1.5 = 3.016 m³
  2. 2Convert to litres: 3.016 m³ × 1,000 = 3,016 litres
✅ Volume ≈ 3,016 litres
3Pythagoras — finding the hypotenuse
▼
A right-angled triangle has shorter sides 9 cm and 12 cm. Find the hypotenuse.
  1. 1c² = a² + b² = 9² + 12² = 81 + 144 = 225
  2. 2c = √225 = 15 cm
✅ Hypotenuse = 15 cm
4Pythagoras — finding a shorter side
▼
A right-angled triangle has hypotenuse 13 m and one side 5 m. Find the other side.
  1. 1Rearrange: a² = c² − b² = 13² − 5² = 169 − 25 = 144
  2. 2a = √144 = 12 m
✅ Missing side = 12 m
5Calculating density
▼
A block of metal has a mass of 540 g and a volume of 60 cm³. What is its density?
  1. 1Density = Mass ÷ Volume
  2. 2Density = 540 ÷ 60 = 9 g/cm³
✅ Density = 9 g/cm³ (similar to copper)
6Area of a compound shape including a semicircle
▼
A shape consists of a rectangle 10 m × 4 m with a semicircle of diameter 4 m on top. Find the total area.
  1. 1Rectangle area: 10 × 4 = 40 m²
  2. 2Semicircle: radius = 2 m. Area = ½ × π × 2² = ½ × π × 4 = 6.28 m²
  3. 3Total: 40 + 6.28 = 46.28 m²
✅ Total area ≈ 46.28 m²
Question 1
A circle has diameter 14 cm. What is its area? (Use π ≈ 3.14)
Question 1 (corrected)
A circle has radius 6 cm. What is its circumference to 1 d.p.? (π ≈ 3.14159)
Question 2
A right-angled triangle has legs 6 cm and 8 cm. What is the hypotenuse?
Question 2 (corrected)
A right-angled triangle has hypotenuse 17 cm and one leg 8 cm. Find the other leg.
Question 3
A cylinder has radius 4 cm and height 10 cm. What is its volume to 1 d.p.?
Question 4
A substance has mass 350 g and volume 50 cm³. What is its density?
Question 5
A shape is made of a rectangle (8m × 5m) with a triangle (base 8m, height 3m) on top. What is the total area?
Question 5 (corrected)
A ladder 10 m long leans against a wall. Its base is 6 m from the wall. How high up the wall does it reach?
Question 6
A metal block has density 8 g/cm³ and volume 25 cm³. What is its mass?
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