Math 7 · Circles and area
How far round, and how much inside
Every circle, from a coin to a Ferris wheel, has the same ratio of its distance round to its distance across: a little more than 3, called π. This unit uses π for circumference and area — and before that, finds the area of parallelograms and triangles by turning them into rectangles.
- 1. The words, first
- 2. Radius, diameter and circumference
- 3. Area of parallelograms and triangles
- 4. Area of a circle
- 5. What costs marks
The words, first
The idea: Radius and diameter are the pair to keep straight. Most wrong answers on this page use one for the other.
| Word | What it means |
|---|---|
| Radius, r | A straight line from the centre of a circle to the circle, or its length. |
| Diameter, d | A straight line across the circle through the centre. It is two radii long: d = 2 × r. |
| Circumference, C | The distance all the way round a circle: its perimeter. |
| π (pi) | Any circle's circumference divided by its diameter: 3.14159…, never ending. Grade 7 uses 3.14 where a number is needed. |
| Central angle | An angle whose corner is at the centre of a circle. All the way round, the central angles add to 360°. |
| Area | The flat surface inside a shape, in square units such as cm² or m². |
| Base, height | The base is the side you measure from. The height is the distance from the base to the opposite side or corner, measured at 90° to the base — not along a slanted side. |
Radius, diameter and circumference
The idea: The circumference is always π times the diameter. Measure across, multiply by 3.14, and you know the distance round.
d = 2 × r · C = π × d, which is the same as 2 × π × r
Why π is the same for every circle. A circle twice as wide is the same shape scaled up, so it is exactly twice as far round too. Circumference ÷ diameter never changes.
The circle in the figure. r = 5 cm, so d = 10 cm, and C = 3.14 × 10 = 31.4 cm.
Worked example. A bike wheel is 60 cm across. How far does the bike roll in one turn of the wheel? In 100 turns?
- The distance across is the diameter: d = 60 cm.
- One turn rolls one circumference: C = 3.14 × 60 = 188.4 cm.
- 100 turns: 188.4 × 100 = 18 840 cm, which is 188.4 m.
Central angles. All the way round the centre is 360°. A pizza cut into 8 equal slices has 360° ÷ 8 = 45° at the tip of each.
Area of parallelograms and triangles
The idea: A parallelogram can be rearranged into a rectangle, so its area is base × height. A triangle is half of a parallelogram, so its area is base × height ÷ 2.
Parallelogram: A = b × h · Triangle: A = b × h ÷ 2
Why. Cut the triangle off one slanted end of a parallelogram and slide it to the other end: it becomes a rectangle with the same base and height. And two copies of any triangle fit together into a parallelogram, so one triangle is half of b × h.
Worked example. A parallelogram has a base of 9 cm, slanted sides of 6 cm and a height of 5 cm. Its area is 9 × 5 = 45 cm². Using the slanted side gives 54 cm², too big, because a slanted side is always longer than the height.
Worked example. A triangular garden bed has a base of 12 m and a height of 7 m. Its area is 12 × 7 ÷ 2 = 42 m². Forgetting to halve gives 84 m², the whole parallelogram.
Area of a circle
The idea: A = π × r × r. The sliced-up circle is a parallelogram whose base is half the circumference and whose height is the radius.
A = π × r², which means π × r × r
Why. In the figure, the base is half the circumference — half of 2 × π × r, which is π × r — and the height is r. Base × height gives π × r × r.
The circle in the figure. A = 3.14 × 5 × 5 = 3.14 × 25 = 78.5 cm², the same as 15.7 × 5.
Worked example. A pizza is 30 cm across. What is its area?
- Radius first: 30 cm across is the diameter, so r = 15 cm.
- Square the radius: 15 × 15 = 225.
- Multiply by π: 3.14 × 225 = 706.5 cm².
The two classic slips. Using the diameter gives 3.14 × 30 × 30 = 2826 cm², four times too big. Reading r² as r × 2 gives 3.14 × 30 = 94.2, which is the circumference.
What costs marks
The idea: Half of these are radius and diameter mixed up. The rest are heights and units.
- Diameter in the area formula. Halve it first.
- Reading r² as 2 × r. 5² = 5 × 5 = 25, not 10.
- A slanted side as the height. The height meets the base at 90°.
- Forgetting to halve for a triangle.
- Mixing up C and A. Circumference is a length, in cm. Area is in cm².