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Circle Area &
Circumference

Master the formulas for area and circumference of a circle, understand how \(\pi\) connects them, and solve real-world problems with confidence.

Mathematics

What Is a Circle?

center
radius \( (r) \)
diameter \( (d = 2r) \)

A circle is the set of all points that are the same distance from a fixed point called the center. The radius \(r\) is the distance from the center to any point on the circle. The diameter \(d\) stretches all the way across and equals \(2r\).

The Magic Number: \(\pi\)

If you wrap a string around any circle and then lay it flat across the diameter, it always fits a little more than 3 times. That ratio is \(\pi\) (pi):

This means for every circle, no matter how big or small, dividing the circumference by the diameter always gives \(\pi\).

Why Do Circles Matter?

Engineering
Wheels, gears, pipes, and turbines all rely on circular measurements.
Nature
Tree rings, ripples in water, and planetary orbits are circular.
Everyday Life
Pizzas, clocks, coins, and sports fields use area and circumference.

The Circumference Formula

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

The Area Formula

\(r\)
\(A\)
\(A = \pi r^2\)
\( \text{Area} = \pi \times r \times r \)
measured in square units

Why Does \(A = \pi r^2\) Work?

Slice into wedges → ≈ rectangle
\(\frac{1}{2} C = \pi r\)
\(r\)

Circumference vs. Area

Circumference
Distance around the circle
Formula: \(C = 2\pi r\)
Units: cm, m, ft (linear)
Think: fence around a garden
Area
Space inside the circle
Formula: \(A = \pi r^2\)
Units: cm², m², ft² (square)
Think: grass filling the garden

How to Solve Circle Problems

1
Identify what you need: Are you finding circumference (around) or area (inside)?
2
Find the radius: If given the diameter, divide by 2. If given circumference, solve backwards.
3
Plug into the formula: Use \(C = 2\pi r\) for circumference or \(A = \pi r^2\) for area.
4
Compute and label units: Use linear units for circumference, square units for area.

Example 1: Find the Circumference

7 cm
\(r = 7 \text{ cm}\)
Find \(C\)

Example 2: Find the Area

10 m
\(d = 10 \text{ m}\)
so \(r = 5 \text{ m}\)
Find \(A\)

Example 3: Working Backwards

A circular track has a circumference of 62.8 meters. What is the radius of the track?

Example 4: Semicircle Area

6 in
\(r = 6 \text{ in}\)
Find area of
the semicircle

Common Mistakes to Avoid

✕ Using Diameter as Radius
If given \(d = 12\), remember to halve it: \(r = 6\). Plugging 12 into \(\pi r^2\) gives 4 times the correct area.
✕ Forgetting to Square
For area, you must square the radius: \(\pi r^2\), not \(\pi r\). The exponent makes a huge difference.
✕ Wrong Units
Circumference uses linear units (cm, m). Area uses square units (cm², m²). Always check your label.

Formula Summary

Property Circumference Area
What it measures Distance around Space inside
Formula (with \(r\)) \(C = 2\pi r\) \(A = \pi r^2\)
Formula (with \(d\)) \(C = \pi d\) \(A = \pi \left(\frac{d}{2}\right)^2\)
Units cm, m, ft cm², m², ft²
Solve for \(r\) \(r = \frac{C}{2\pi}\) \(r = \sqrt{\frac{A}{\pi}}\)

Question 1 of 5

A circle has a radius of 9 cm. What is its circumference? (Use \(\pi \approx 3.14\))

Question 2 of 5

A circular pond has a diameter of 14 meters. What is the area of the pond? (Use \(\pi \approx \frac{22}{7}\))

Question 3 of 5

A bike wheel has a circumference of 75.36 inches. What is the diameter of the wheel? (Use \(\pi \approx 3.14\))

Question 4 of 5

Which formula correctly gives the area of a circle with radius \(r\)?

Question 5 of 5

A pizza has an area of \(113.04\) square inches. What is its radius? (Use \(\pi \approx 3.14\))