Circles
Wheels, pizzas, clocks, coins, the rim of a mug: circles are everywhere. In this page you'll learn the names of every part of a circle, discover for yourself a strange number called π (pi) that hides inside every circle, and use it to work out the distance round a circle and the space inside it, even for half and quarter circles.
National curriculum KS3: know the formulae for the circumference and area of a circle; calculate perimeters of 2D shapes; identify and name parts of circles, including chord, tangent, arc, sector and segment1 · Meet the circle: its parts
Push a pin into some card, tie a piece of string to it, put a pencil at the other end and walk the pencil round with the string tight. You've drawn a circle: every point on it is exactly the same distance from the pin. A pair of compasses does the same job.
Circles have lots of parts with special names, and you'll need them all. Hover over (or tap) a part of the circle, or press a button, to see what it is called and what it means.
Radius and diameter
The two most important lengths are the radius (centre to edge, the length of the string) and the diameter (edge to edge, straight through the centre). A diameter is two radii joined end to end (radii is the plural of radius), so
d = 2 × r and r = d ÷ 2
Read the question carefully. Questions sometimes give you the radius and sometimes the diameter. Before using any formula, write down both: "r = 5 cm, so d = 10 cm". Mixing them up is the most common circle mistake of all.
2 · Discovering π
Here's a question. If you know how wide a circle is (its diameter), can you work out how far it is round the edge (its circumference)? Let's experiment. Imagine painting the rim of a wheel, then rolling it exactly one full turn along the ground. The paint leaves a line on the ground exactly as long as the circumference. How many diameters long is that line?
One full turn takes you three diameters and a little bit more. Is that just for this wheel? Try measuring other circles. People did exactly this with string and rulers thousands of years ago. Type any radius you like and add it to the table.
Every circle, from a 1p coin to the London Eye, gives the same answer: the circumference is about 3.14 times the diameter. (The tiny wobbles come from rounding the measurement to the nearest mm or 10 cm.) This number is so important that it has its own name and symbol, the Greek letter π, said "pie".
π = C ÷ d = 3.14159265…
The digits of π go on for ever without repeating, so we can never write it down exactly. That's why your calculator has a π button: it holds π to about 10 digits. For rough work, π ≈ 3.14 or even π ≈ 3 is fine.
3 · Circumference: C = πd = 2πr
Since circumference ÷ diameter = π, turn it round: circumference = π × diameter. And because the diameter is 2 radii, you can also write it using the radius:
C = πd or C = 2πr
Both are the same formula. Use whichever matches what you're given. Try it with the calculator below and look at the three ways of handling π:
- π button: the most accurate. Round your final answer, usually to 1 d.p. (one decimal place).
- π ≈ 3.14: fine without a calculator, but it drifts from the true answer as circles get bigger.
- In terms of π: don't multiply by π at all; leave the symbol in the answer, like 10π cm. This is exact, because nothing has been rounded. If a question says "give your answer in terms of π", do this.
Common mistakes with circumference
- Doing π × r and forgetting to double: π × r is only half the circumference.
- Using the diameter in 2πr: that gives twice the right answer. Use πd with d, or 2πr with r.
- Rounding too early. Keep the full calculator value until the very last line, then round.
4 · Area: A = πr²
The area is the amount of flat space inside the circle: how much pizza you get, not how much crust. Area is measured in square units such as cm² (square centimetres).
Here's a beautiful way to see the formula. Cut a circle into equal slices like a pizza (each slice is a sector). Lay them in a row, alternately pointing up and down. Nothing is lost or added, so the area is the same. With lots of thin slices the row turns into an almost perfect rectangle. Slide the controls.
So what are the measurements of that rectangle?
- Height: each slice is a radius long, so the height is r.
- Width: the curved crust is shared out: half of it is along the top and half along the bottom. So the width is half the circumference: ½ × 2πr = πr.
- Area of rectangle = width × height = πr × r = πr².
A = πr²
Read πr² as "π times r squared": square the radius first, then multiply by π (BIDMAS: indices before multiplication). Another way to picture it: r² is a square with sides r, and the circle's area is just over 3 of those squares (π of them, in fact).
Common mistakes with area
- Using the diameter. The formula needs the radius. If you're given d, halve it first.
- Squaring π × r. πr² means π × (r × r), not (π × r)². On a calculator type π × 5 x², or work out r² first.
- Mixing up the formulas. 2πr is a length (circumference, cm). πr² is an area (cm²). The ² in the answer's unit matches the ² on the r.
- Wrong units. Areas are in cm², m², mm²; lengths are in cm, m, mm.
5 · Working backwards
Sometimes you know the circumference or the area and need the radius. Write down the formula, put in what you know, and undo each step with the inverse operation (see the rearranging formulae page).
- From the circumference: C = 2πr, so r = C ÷ (2π). (Or d = C ÷ π, then halve.)
- From the area: A = πr², so r² = A ÷ π, then r = √(A ÷ π). Don't forget the square root!
6 · Semicircles, quarter circles and composite shapes
A semicircle is half a circle; a quarter circle is a quarter. A composite shape is made by joining simple shapes, like a rectangle with a semicircle on top (an arched window) or a running track.
Area is easy: take the right fraction of πr² and add on any other pieces. Perimeter (the total distance round the outside) needs more care: the curved part is a fraction of the circumference, but you must also add every straight edge. Forgetting them is the classic mistake. In the picture, curved edges are orange and straight edges are plum: the perimeter is all of them added together.
7 · Extension: arcs and sectors
An arc is part of the circumference and a sector is a slice. A full turn is 360°, so a sector with angle θ (the Greek letter theta) is the fraction θ/360 of the whole circle. Take that fraction of the circumference for the arc, and that fraction of the area for the sector.
arc = θ/360 × 2πr sector = θ/360 × πr²
Check: θ = 180° gives ½, a semicircle; θ = 90° gives ¼, a quarter circle; θ = 360° is the whole circle.
8 · Practice: mixed circle questions
Time to try it yourself. Each question is new. Use the π button and round to 1 d.p. unless the question says "in terms of π" (then just type the number in front of π, e.g. 12 for 12π). If you go wrong, the feedback tries to spot which mistake you made.
Worked examples
A bicycle wheel has radius 33 cm. Find its circumference to 1 d.p.
A pizza has diameter 30 cm. Find its area to 1 d.p.
A circle has radius 6 cm. Give its circumference and area in terms of π.
A circular pond has area 50 m². Find its radius to 1 d.p.
A garden is a 10 m × 6 m rectangle with a semicircle on one 6 m end. Find its perimeter and area to 1 d.p.
Extension: a sector has angle 60° and radius 9 cm. Find its arc length, perimeter and area.
Quick check
What you've learned
- The parts of a circle: centre, radius (centre to edge), diameter (edge to edge through the centre), circumference (the distance round), chord (a straight line joining two points on the circle), tangent (a straight line touching the circle at one point), arc (part of the circumference), sector (a "pizza slice" between two radii) and segment (the piece cut off by a chord).
- d = 2r and r = d ÷ 2. Always write both down before you start.
- For every circle, circumference ÷ diameter is the same number, π = 3.14159…
- Circumference: C = πd = 2πr (a length, in cm). Area: A = πr² (in cm²), which you can see by rearranging sectors into a rectangle πr wide and r tall.
- Use the π button and round at the end (e.g. to 1 d.p.), or give an exact answer in terms of π, like 12π cm.
- Work backwards with inverse operations: r = C ÷ 2π, and r = √(A ÷ π).
- For semicircles, quarter circles and composite shapes, the perimeter is the curved part plus all the straight edges; the area is the fraction of πr² plus any other pieces.
- Extension: arc length = θ/360 × 2πr and sector area = θ/360 × πr².