Year 8
Circles
Learn how to name the parts of a circle and use pi to work out the circumference and area of circles, parts of circles and shapes with curved edges.
Definition
A circle is the set of points that are all the same distance from a centre. That distance is the radius, the diameter runs right across through the centre, and the circumference is the distance all the way around. Whatever the size of the circle, the circumference is always just over 3 times the diameter, and that fixed number is pi (π), roughly 3.14159. Because π links the measurements of every circle, two short formulae give the circumference and the area.
Why Do I Have to Learn This?
Circles are behind wheels, clocks, coins and orbits. Knowing their parts helps you measure and make round things.
Key Rules:
- 1. The radius runs from the centre to the edge and the diameter runs right across through the centre, so the diameter is twice the radius.
- 2. Pi (π) is the number of times the diameter fits around the circumference, about 3.142, and it is the same for every circle.
- 3. Work out the circumference with C = πd, or C = 2πr when you are given the radius.
- 4. Work out the area with A = πr², squaring the radius before multiplying by π, and halving a diameter first if you need the radius.
- 5. For a part of a circle, take the matching fraction of the full circle: a semicircle uses 12 and a quarter circle uses 14.
- 6. Perimeter means the whole distance around the outside, so add any straight edges to the curved parts, and split a shape with curves into circle parts and straight-sided pieces before you start.
Remember:
- •C = πd and A = πr²: multiply the diameter for the distance around, square the radius for the space inside.
Don’t Forget...
- •Using the diameter in the area formula and working out π × d². A = πr² uses the radius, so a circle with diameter 10 has area π × 5² = 25π, and π × 10² gives an answer 4 times too big.
- •Forgetting the straight edges when finding the perimeter of a semicircle or quarter circle. Halving the circumference only gives the curved edge; the perimeter is the whole way around, so the diameter, or the two radii, must be added on.
- •Squaring the radius by doubling it, writing 5² = 10. Squaring means multiplying a number by itself, so 5² = 5 × 5 = 25, and doubling instead makes every area answer far too small.
Circle
A flat shape where every point on the edge is the same distance from the centre. Circumference = π × diameter and area = π × radius².
A coin, a clock face and a wheel are all circles.
Worked Example:
A circle has a radius of 5 cm. Work out its circumference and its area, giving both answers to 1 decimal place.
The diameter is twice the radius: d = 2 × 5 = 10 cm.
Circumference: C = πd = π × 10 = 31.4159..., which rounds to 31.4 cm.
For the area, square the radius first: r² = 5 × 5 = 25.
Area: A = πr² = π × 25 = 78.5398..., which rounds to 78.5 cm².
Answer: Circumference 31.4 cm and area 78.5 cm²
Why It Works:
Every circle is an enlargement of every other circle, so the circumference divided by the diameter always gives the same value, and that fixed ratio is what π is. The area formula comes from cutting a circle into thin slices and laying them out almost like a rectangle with height r and width πr, half the circumference, which multiplies to πr².
Another Example:
A semicircle has a diameter of 12 cm. Work out its perimeter to 1 decimal place.
The curved edge is half the circumference of the full circle: 12 × π × 12 = 18.8495...
The perimeter also includes the straight edge, which is the diameter: 12 cm.
Add the curved edge and the straight edge: 18.8495... + 12 = 30.8495...
Answer: 30.8 cm
Animated Example:
Free Circles Worksheet
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Let’s Practise the Concept
Step 1 of 3
Put them in order
Tap the numbers from smallest to largest.
In order so far: nothing yet