Circle

Properties

• Every diameter is a line of symmetry.
• The circumference divided by the diameter is the same for every circle. That number is π, about 3.14159.
• A chord is a line joining two points on the circle. The diameter is the longest chord.
• A tangent touches the circle at one point and is at right angles to the radius there.

Why the formulas work

The circumference is π times the diameter by the very definition of π. For the area, cut the circle into many thin slices like a pizza and arrange them head to tail. They form a shape close to a rectangle, with height r and width half the circumference, πr. Its area is πr × r. The thinner the slices, the closer the fit.

Worked example

Take a circle with radius 7 cm.

1. Area. Area = πr². Here: π × 7 × 7 = 153.94 cm² (with π = 22/7: 154 cm²).
2. Circumference. Circumference = 2πr. Here: 2 × π × 7 = 43.98 cm (with π = 22/7: 44 cm).
3. Diameter. Diameter = 2r. Here: 2 × 7 = 14 cm.

Finding a missing measurement

From the circumference, r = C ÷ 2π: a circumference of 44 cm gives r = 7 cm, taking π = 22/7. From the area, r = √(A ÷ π): an area of 616 cm² gives r² = 196, so r = 14 cm.

Mistakes to avoid

• Using the diameter in place of the radius. Halve the diameter first.
• Mixing up 2πr (a length) and πr² (an area).
• Squaring π as well as r: it is π × r × r.

Where you see it

Wheels, coins, rotis, bangles, clock faces and manhole covers are circles. A manhole cover is round because a circle cannot fall through its own hole.

Try these

Work each one out, then tap to check.

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