Circle Basics

TipLearning Objectives
  • Identify and compute radius, diameter, circumference, and area of circles.
  • Convert between radius and diameter.
  • Interpret circle measurements in real-world problems.

Key Ideas

  • Radius: distance from center to any point on the circle
  • Diameter: full width across the circle through the center
    \[ d = 2r \]
  • Circumference: distance around the circle
    \[ C = 2\pi r \]
  • Area:
    \[ A = \pi r^2 \]

Circle labeled with center, radius, and diameter.

Common Problem Types

Converting Radius ↔︎ Diameter

If \(r=5\), then \(d=10\).

Finding Circumference

Use \(C=2\pi r\) or \(C=\pi d\).

Finding Area

Square the radius first, then multiply by \(\pi\).

Solving Real-World Problems

Using tires, clocks, wheels, satellite dishes, etc.

Using \(\pi\)-Exact vs Approximate

Some questions expect answers in terms of \(\pi\).

Strategies

  • Always check whether the given length is \(r\) or \(d\).
  • Keep answers in terms of \(\pi\) unless told otherwise.
  • Draw a quick circle for visualization.

Worked Examples

Example 1

Find circumference of a circle with radius 7.
\[ C = 2\pi(7) = 14\pi. \]

Example 2

Find area if diameter is 10.
Radius = 5.
\[ A = \pi(5^2) = 25\pi. \]

WarningCommon Mistakes
  • Forgetting to square radius in area formula.
  • Using \(2\pi r\) for area or \(\pi r^2\) for circumference.
  • Using radius when diameter was given (or vice versa).

Practice Problems

  1. Radius = 3 → find area.
  2. Diameter = 16 → find circumference.
  3. A wheel has circumference \(18\pi\). Find its radius.
  1. \(A = 9\pi.\)
  2. \(C = 16\pi.\)
  3. \(2\pi r = 18\pi \Rightarrow r=9.\)

Summary

  • \(d=2r\); \(C=2\pi r\); \(A=\pi r^2.\)
  • Keep \(\pi\) exact unless told otherwise.
  • If you see \(18\pi\) or similar, divide by \(2\pi\) to get radius.
  • Square first, then multiply, for area.