Circles: Circumference and Area
Derive the formulas C = πd and A = πr² using the relationship between circumference and diameter. Solve problems involving exact answers (in terms of π) and decimal approximations.
A lesson of Pre-Algebra, in Geometry Foundations, about 10 minutes.
By the end of this lesson, you will be able to calculate the circumference and area of circles, and solve problems using these formulas.
Circle Vocabulary
Radius ($r$): distance from center to any point on the circle.
Diameter ($d$): distance across the circle through the center; $d = 2r$.
Circumference ($C$): the distance around the circle (perimeter).
$\pi \approx 3.14159$ — the ratio of circumference to diameter for any circle.
Formulas
$$C = 2\pi r = \pi d \qquad A = \pi r^2$$
Use $\pi \approx 3.14$ for approximations, or leave answers in terms of $\pi$ for exact answers.
Examples
Find $C$ and $A$ of a circle with radius 7 cm:
$C = 2\pi(7) = 14\pi \approx 43.96$ cm
$A = \pi(7)^2 = 49\pi \approx 153.94$ cm²
Find radius given circumference = $30\pi$ inches:
$30\pi = 2\pi r \to r = 15$ inches
Find diameter given area = $64\pi$ m²:
$64\pi = \pi r^2 \to r^2 = 64 \to r = 8$ m $\to d = 16$ m
Semicircles and Quarter Circles
A semicircle is half a circle:
- Circumference = $\frac{1}{2}(2\pi r) + 2r = \pi r + 2r$ (half circumference + diameter)
- Area = $\frac{1}{2}\pi r^2$
A quarter circle: divide by 4.
Summary
$C = 2\pi r = \pi d$; $A = \pi r^2$
Exact answer: leave in terms of $\pi$ (e.g., $49\pi$ cm²).
Approximate answer: multiply by 3.14 or use a calculator.