🏰 Geometry Kingdom · Geometry

Circles

Use radius, diameter, circumference and area, and work with arcs and sectors.

In short

The diameter is twice the radius, and pi is circumference divided by diameter.
$C = 2 pi r$ is a length; $A = pi r 2$ is an area, and $r 2$ means $r x r$ .
Answers are either approximate $(pi = 3.14)$ or exact (left in terms of pi) — the question says which.
A sector with angle t is the fraction $t/360$ of the circle, and both its arc and its area scale by that fraction.

Radius, diameter and pi

A circle is the set of all points at the same distance from a centre. That distance is the radius r; all the way across through the centre is the diameter d, and $d = 2r$ .

Measure the distance around any circle (its circumference) and divide by its diameter, and you always get the same number, a little more than 3. That number is pi, written with the Greek letter and roughly 3.14159...

Pi never ends and never repeats, so answers involving it come in two styles:

Approximate: replace pi by 3.14 and give a decimal.
Exact: leave pi as a symbol, so a circumference is written $10*pi$ rather than 31.4.

Read every question carefully — it will say which style it wants. An exact answer is never "wrong by rounding", which is why it is preferred in later mathematics.

Circumference and area

Because pi is defined as circumference divided by diameter,

$C = pi x d = 2 x pi x r$

For area, imagine slicing the circle into very thin wedges and laying them alternately point-up and point-down. They form a shape that gets closer and closer to a rectangle of height r and width half the circumference, that is $(1/2)(2 pi r) = pi r$ . Multiplying gives

$A = pi x r 2$

The two formulas are easy to confuse. The check is the units: circumference is a distance and uses r once; area is a space and uses r squared.

Note also that $r 2$ means $r x r$ , not 2r. For $r = 6$ , the area uses 36, not 12.

Working backwards

Both formulas can be run in reverse.

Given the circumference, divide by pi to get the diameter, then halve it if you want the radius:

$C = 31.4$ with $pi = 3.14 -> d = 31.4 / 3.14 = 10$ , so $r = 5$

Given the area, divide by pi and then take a square root — undoing a square needs a root, not a halving:

$A = 49*pi -> r 2 = 49$ , so $r = 7$

Watch which one the question asks for. Radius and diameter differ by a factor of 2, and it is easy to hand back the wrong one.

Arcs and sectors

A sector is a slice of the circle bounded by two radii; the curved edge of that slice is an arc.

A sector with central angle t degrees is the fraction $t/360$ of the whole circle, and both the arc and the sector area scale by exactly that fraction:

arc length $= (t/360) x 2 pi r$ sector area $= (t/360) x pi r 2$

$A 90°$ sector of a circle of radius 8 is a quarter of the circle: arc $= (1/4)(16 pi) = 4*pi$ , area $= (1/4)(64 pi) = 16*pi$ .

Keep the two apart the same way as before: an arc is a length (built from the circumference formula) and a sector is an area (built from the $r 2$ formula).

Worked examples

Example 1

A circle has radius 6 cm. Find its area, leaving the answer in terms of pi.

Use $A = pi x r 2$ .
Square the radius first: $62 = 36$ .
So $A = 36 x pi$ .
The exact area is $36*pi cm 2$ — no rounding needed.

Example 2

A circle has circumference 47.1 cm. Using $pi = 3.14$ , find its radius.

$C = pi x d$ , so $47.1 = 3.14 x d$ .
Divide both sides by $3.14: d = 47.1 / 3.14 = 15$ cm.
That is the diameter, and the radius is half of it.
$r = 15 / 2 = 7.5$ cm.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

A circle has radius 4 cm. Find its circumference. Use $pi = 3.14$ and round to 2 decimal places.

Answer: 25.12 cm

$d = 2r = 2 x 4 = 8$ cm
$C = pi x d = 3.14 x 8$
$C = 25.12$ cm

Problem 2

Difficulty 3 of 5

A circle has diameter 22 cm. Find its area. Use $pi = 3.14$ and round to 2 decimal places.

Answer: $379.94 cm 2$

$r = d / 2 = 22 / 2 = 11$ cm
$A = pi x r 2 = 3.14 x 11 2$
$112 = 121$
$A = 3.14 x 121 = 379.94 cm 2$

Problem 3

Difficulty 4 of 5

A circle has radius 16 cm. Find its circumference, leaving the answer in terms of pi (for example $25*pi)$ .

Answer: $32*pi$

$C = 2 x pi x r = 2 x pi x 16$
$2 x 16 = 32$
Answer: $32*pi$ cm

Common mistakes

Using the area formula when the circumference was wanted, or the reverse.
Doubling the radius instead of squaring it.
Squaring the diameter instead of the radius when only the diameter is given.
Giving a decimal when the question asked for an answer in terms of pi.

What you should be able to do

Convert between radius and diameter.
Compute circumference and area of a circle.
Find an arc length and a sector area.
Work backwards from circumference or area to the radius.

Where this fits in the curriculum

Common Core

7.G.B.4
Grade 7 — Know and use the formulas for the area and circumference of a circle, and relate them to each other.
HSG-C.B.5
High school — Derive the fact that arc length and sector area are proportional to the radius, and define radian measure.

Ontario

G7.E2.3
Grade 7 — Use the relationships between the radius, diameter and circumference of a circle to solve problems.
G7.E2.5
Grade 7 — Solve problems involving the area of a circle, and the relationship between its area and its circumference.

SAT

Additional Topics in Math
Circumference, area, arcs, sectors and circle equations.