## What You Will Learn

After this lesson you will be able to calculate the area and circumference of any circle when given its radius or diameter. You will understand the role of π, know when to use each formula, and confidently solve problems involving semicircles, quarter-circles, and real-world circular shapes.

## Theory

### Key vocabulary

Every circle problem starts with three measurements:

- **Radius (r)** — the distance from the center of the circle to any point on its edge.
- **Diameter (d)** — the distance across the circle through the center. It is always exactly twice the radius: d=2r.
- **Pi (π)** — a mathematical constant approximately equal to 3.14159. It is the ratio of any circle's circumference to its diameter.

### Circumference — the distance around the circle

The circumference is the circle's "perimeter." There are two equivalent forms of the formula:

C = 2πr  
or  
C = πd

Use the first form when you know the radius. Use the second when you know the diameter. They give the same result because d=2r.

For example, a circle with radius 5 cm has circumference:

C = 2π(5) = 10π ≈ 31.42 cm

### Area — the space inside the circle

The area formula tells you how much flat surface the circle covers:

A = πr²

This means: square the radius, then multiply by π. Always remember to **square the radius first**, not multiply π by r and then try to square.

For the same circle with r=5 cm:

A = π(5)² = 25π ≈ 78.54 cm²

### Semicircles and quarter-circles

For a **semicircle** (half a circle):

Aₘₑₘᵢ = 1/2 πr²  
Cₘₑₘᵢ = πr + 2r

For a **quarter-circle**:

Aₖₑₐᵝ = 1/4 πr²  
Cₖₑₐ = 1/2 πr + 2r

## Worked Examples

### Example 1: Finding circumference and area from the radius (easy)

**Problem:** A circle has a radius of 8 cm. Find its circumference and area. Use π ≈ 3.14.

**Step 1:** Calculate the circumference.
C = 2πr = 2 × 3.14 × 8 = 50.24 cm

**Step 2:** Calculate the area.
A = πr² = 3.14 × 8² = 3.14 × 64 = 200.96 cm²

**Answer:** Circumference = **50.24 cm**, Area = **200.96 cm²**

### Example 2: Given the diameter instead of the radius (easy)

**Problem:** A circular table has a diameter of 1.2 m. Find the area of the tabletop. Use π ≈ 3.14.

**Step 1:** Find the radius.
r = d/2 = 1.2/2 = 0.6 m

**Step 2:** Calculate the area.
A = πr² = 3.14 × (0.6)² = 3.14 × 0.36 = 1.1304 m²

**Answer:** ≈ **1.13 m²**

### Example 3: Finding the radius from the circumference (medium)

**Problem:** A bicycle wheel has a circumference of 2.2 m. What is its radius? Use π ≈ 3.14.

**Step 1:** Start from the circumference formula and solve for r.
C = 2πr ⟹ r = C/(2π)

**Step 2:** Substitute.
r = 2.2/(2 × 3.14) ≈ 0.35 m

**Answer:** ≈ **0.35 m** (or about 35 cm)

### Example 4: Area of a semicircular window (medium)

**Problem:** A window is shaped as a semicircle with a diameter of 80 cm. Find the area of the glass. Use π ≈ 3.14.

**Step 1:** Find the radius.
r = 80/2 = 40 cm

**Step 2:** Calculate the area of the full circle and take half.
A = 1/2 πr² = 1/2 × 3.14 × 40² = 2512 cm²

**Answer:** **2512 cm²**

### Example 5: Composite shape — rectangle with semicircles (challenging)

**Problem:** A running track is shaped like a rectangle (100 m by 60 m) with a semicircle on each short end. Find the total area enclosed by the track. Use π ≈ 3.14.

**Step 1:** The two semicircles together form one full circle. The diameter of each semicircle equals the short side: d=60 m, so r=30 m.

**Step 2:** Calculate the area of the rectangle.
A = 100 × 60 = 6000 m²

**Step 3:** Calculate the area of the full circle.
A = πr² = 3.14 × 30² = 2826 m²

**Step 4:** Add the two areas.
A_total = 6000 + 2826 = 8826 m²

**Answer:** **8826 m²**

## Common Mistakes

**Mistake 1: Forgetting to square the radius in the area formula**

Always write r² to remind yourself to square first.

**Mistake 2: Using the diameter directly in the area formula**

The area formula requires the radius, not the diameter. Always check whether the problem gives you r or d.

**Mistake 3: Confusing circumference and area formulas**

Circumference is a length measured in units (cm, m). Area is a surface measured in square units (cm², m²). A quick unit check reveals which formula you need.

## Practice Problems

Try these on your own before checking the answers:
1. A circle has a radius of 6 cm. Find its circumference and area. (Use π ≈ 3.14.)
2. A circular pond has a diameter of 14 m. Find its area.
3. A pizza has a circumference of 40 cm. What is its radius?
4. Find the area of a quarter-circle with radius 10 cm.
5. A rectangular yard (20 m by 12 m) has a circular fountain (diameter 4 m) in the center. Find the area of the yard that is NOT covered by the fountain.

## Summary

- The circumference of a circle is C = 2πr = πd — it measures the distance around the circle.
- The area of a circle is A = πr² — always use the radius, and remember to square it.
- If you are given the diameter, divide by 2 to get the radius before using the area formula.
- For semicircles and quarter-circles, take the corresponding fraction of the full circle's area, and add the straight edges for the perimeter.
- Check your units: circumference is in linear units (cm, m), area is in square units (cm², m²).