Circle Formulas (Area, Circumference, Diameter)

A circle is the set of all points a fixed distance r (the radius) from a center. Its area is A = πr² and its circumference (perimeter) is C = 2πr. The diameter is d = 2r, twice the radius.

Circle formulas

Area: $A = \pi r^2$
Circumference: $C = 2\pi r$
Diameter: $d = 2r$

where r = radius, d = diameter (d = 2r), A = area, C = circumference, π ≈ 3.14159

Area

A = \pi r^2

Worked example

The unit circle (r = 1) has area exactly π ≈ 3.14 square units.

A = \pi r^2 = \pi\cdot 1^2 = \pi \approx 3.14

Circumference

C = 2\pi r

Diameter

d = 2r

Interactive circle explorer

Drag a slider (or type a value, or use the arrow keys) and watch the diagram, the results, and the full step-by-step substitution update live.

Radius r

Area A 78.54units²

Circumference C 31.42units

Diameter d 10units

Step-by-step substitution

A = πr² = π · 5² = 25π ≈ 78.54
C = 2πr = 2 · π · 5 = 10π ≈ 31.42
d = 2r = 2 · 5 = 10

Classroom use: circle formulas

Who it's for: Grades 7–10 · Middle-school students meeting π, area, and circumference, and high-school students arguing informally for where those formulas come from.

Learning objectives

After using this page, students will be able to:

Compute area and circumference from a radius or a diameter.
Convert between radius and diameter correctly before substituting.
Explain why doubling the radius quadruples the area.

Suggested classroom activity

Have students drag the radius and record area at r = 1, 2, 3, 4. The 1 : 4 : 9 : 16 pattern emerges from their own data, which makes the squared term meaningful rather than memorised. The radius-vs-diameter mix-up is the single most common error here, so ask for the diameter each time too.

Standards alignment

Aligned to the Common Core State Standards for Mathematics. Codes are given exactly as published so they can be matched against state crosswalks.
Code	Standard
`7.G.B.4`	Know and use the formulas for the area and circumference of a circle.
`HSG.GMD.A.1`	Give an informal argument for circumference, area, and volume formulas.

Free to use and adapt in class under the terms on our license page — no account, no sign-up, no ads. Content last reviewed August 2026.

Common mistakes

Squaring the diameter instead of the radius in A = πr².
Confusing circumference (2πr) with area (πr²).

Related & next steps

Add circle to your formula sheet →

Frequently asked questions

What is the formula for the area of a circle?

The area of a circle is A = πr², pi times the radius squared. For a circle of radius 7, the area is π × 49 ≈ 153.94 square units.

What is the difference between circumference and diameter?

The diameter d = 2r is the distance across a circle through its center. The circumference C = 2πr = πd is the distance around it — about 3.14 times the diameter.

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