courses › Geometry

Circles

level 22 course

Area, circumference, and slices of a circle, in terms of π.

In your head, jot if needed · no calculator why?

Opens at level 22.

Builds on: Area & perimeter (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

Every circle measurement comes from its radius r, the distance from the centre to the edge. The diameter d goes all the way across, so it is twice the radius.

The circumference, the distance around, is 2πr, the same as πd. The area is πr². Answers here are written as a number times π, like 25π, so you never need a decimal for π.

A sector is a slice, like a slice of pizza. Its angle says what fraction of the full 360° it takes, and it gets that fraction of the area and of the circumference.

Techniques

Radius first

Any circle area or circumference.

Given the diameter? Halve it to get the radius.
Area: square the radius. The number times π is the area.
Circumference: double the radius, which is the diameter. That number times π is the circumference.

worked example

Example: A circle has diameter 14. Its area is kπ. What is k?

Radius: 14 ÷ 2 = 7.
Square it: 7 × 7 = 49, so the area is 49π.

Answer: 49

Take the slice's fraction

A sector's area or an arc's length, given the central angle.

Write the angle over 360 and simplify: 90° is 1/4, 120° is 1/3.
Sector area: that fraction of πr².
Arc length: that fraction of 2πr. A fraction answer is fine.

worked example

Example: A sector of a circle has radius 6 and a central angle of 60°. Its area is kπ. What is k?

60/360 = 1/6, so the sector is one sixth of the circle.
Whole circle: 6 × 6 = 36, so its area is 36π.
One sixth of 36π is 6π.

Answer: 6

Tips by skill

TipCircle area: Halve a diameter to get the radius. Then k is the radius squared.
TipCircumference: k is twice the radius, which is the same as the diameter.
TipSectors & arcs: The angle over 360 is the fraction. For area, k is that fraction of r²; for arc length, of 2r.

Watch out for

Using the diameter as the radius. Halve it first, or the area comes out four times too big.
Mixing the formulas: 2πr is the distance around, and πr² is the area inside.
Giving the arc length when the question asks for the sector's area, or the reverse.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

Circle area par 13 s not tried yet
worked example
A circle has diameter 2. Its area is kπ. What is k?

Answer: 1
1. The radius is half the diameter: 1.
2. Area = πr² = 1² × π = π, so k = 1.
Circumference par 9 s not tried yet
worked example
A circle has diameter 15. Its circumference is kπ. What is k?

Answer: 15
1. Circumference = πd = 15π, so k = 15.
Sectors & arcs par 20 s not tried yet
worked example
A sector of a circle has radius 2 and a central angle of 45°. Its area is kπ. What is k? Give k as a whole number or a fraction.

Answer: 1/2
1. 45° is 1/8 of a full turn.
2. The whole circle's area is 2² × π = 4π.
3. The sector is 1/8 × 4π = (1/2)π, so k = 1/2.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

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