Geometry · Step-by-step guide

Circle Formulas

Use circumference, area, arc length, and sector area formulas with radians and degrees.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

For radius r, circumference is 2πr\displaystyle 2\pi r and area is πr2\displaystyle \pi r^2. An arc or sector is the same fraction of the whole circle as its central angle is of 360° (or 2π\displaystyle 2\pi radians).

C=2πr,A=πr2\displaystyle C=2\pi r,\quad A=\pi r^2
01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Radius versus diameter

The diameter is twice the radius. Convert before substituting if a formula uses r.

2

Length versus area

Circumference and arc length use linear units; area and sector area use square units.

3

Use the central-angle fraction

Multiply the whole-circle formula by θ/360\displaystyle \theta/360 for degrees or use s=rθ\displaystyle s=r\theta for radian arc length.

02

See the structure

A picture makes the relationships easier to remember.

r
The highlighted segment is a radius, measured from the center to the circle.
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Circumference

A circle has radius 6.

  1. Use C=2πr\displaystyle C=2\pi r.
  2. Substitute r = 6.
  3. Keep π for an exact answer.
Answer12π\displaystyle 12\pi units
Example 2Area from diameter

A circle has diameter 10.

  1. The radius is 5.
  2. Use A=πr2\displaystyle A=\pi r^2.
  3. Square the radius, not the diameter.
Answer25π\displaystyle 25\pi square units
Example 3Arc length

Radius 9, central angle 80°.

  1. Use s=(80/360)(2π⋅9)\displaystyle s=(80/360)(2\pi\cdot9).
  2. Reduce 80/360=2/9\displaystyle 80/360=2/9.
  3. Multiply.
Answer4π\displaystyle 4\pi units
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Substituting diameter for radius.

  • Forgetting to square r in area.

  • Reporting area in linear units.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Circles

Practice circumference, area, arcs, sectors, and circle relationships.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

Find the circumference of a circle with radius 6.

r = 6Find the circumference
Show hint

Use C=2πr\displaystyle C=2\pi r.

Show worked solution
  1. C=2π(6)\displaystyle C=2\pi(6).
  2. Multiply the numerical factors.

Answer: 12π\displaystyle 12\pi units.

2

Find the area of a circle with diameter 10.

d = 10Find the area
Show hint

Convert diameter to radius.

Show worked solution
  1. r=10/2=5\displaystyle r=10/2=5.
  2. A=π(5)2\displaystyle A=\pi(5)^2.

Answer: 25π\displaystyle 25\pi square units.

3

Find the length of a 60° arc in a circle of radius 9.

60°r = 9Find the highlighted arc length
Show hint

Take 60/360 of the circumference.

Show worked solution
  1. C=18π\displaystyle C=18\pi.
  2. Arc =(60/360)(18π)\displaystyle =(60/360)(18\pi).
  3. Simplify the fraction.

Answer: 3π\displaystyle 3\pi units.

4

Find the area of a 90° sector with radius 8.

90°r = 8Find the highlighted sector area
Show hint

Take one fourth of the circle area.

Show worked solution
  1. Whole area =64π\displaystyle =64\pi.
  2. 90/360=1/4\displaystyle 90/360=1/4.
  3. Sector area =16π\displaystyle =16\pi.

Answer: 16π\displaystyle 16\pi square units.

5

A circle has area 49π\displaystyle 49\pi. Find its circumference.

A = 49πr = ?Find the circumference
Show hint

Find r from the area first.

Show worked solution
  1. πr2=49π\displaystyle \pi r^2=49\pi, so r=7\displaystyle r=7.
  2. C=2π(7)\displaystyle C=2\pi(7).

Answer: 14π\displaystyle 14\pi units.

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

For radius r, circumference is 2πr\displaystyle 2\pi r and area is πr2\displaystyle \pi r^2. An arc or sector is the same fraction of the whole circle as its central angle is of 360° (or 2π\displaystyle 2\pi radians).

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

Keep learning

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