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Rule

Circumference of a Circle

The circumference of a circle is calculated by multiplying its diameter by π.

C=π d

This can be visualized in the following diagram.

Since the diameter is twice the radius, the circumference of a circle can also be calculated by multiplying 2r by π.

C=2π r

Proof

Consider two circles and their respective diameters and circumferences.

By the Similar Circles Theorem, all circles are similar. Therefore, their corresponding parts are proportional. C_B/C_A=d_B/d_A This proportion can be rearranged.

C_B/C_A=d_B/d_A

▼

Rearrange equation

MultEqn

LHS * C_A=RHS* C_A

C_B=d_B/d_A(C_A)

DivEqn

.LHS /d_B.=.RHS /d_B.

C_B/d_B=1/d_A(C_A)

MoveRightFacToNumOne

1/b* a = a/b

C_B/d_B=C_A/d_A

Therefore, for any two circles, the ratio of the circumference to the diameter is always the same. This means that this ratio is constant. This constant is defined as π. With this information, it can be shown that the circumference of a circle is the product between its diameter and π.

C/d=π ⇒ C=π d

Exercises

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