Circumference and Area of a Circle
From the wheels on your bike to the pizza you share with friends, circles are everywhere. This guide will show you how to calculate the distance around a circle (circumference) and the space inside it (area) — essential skills for your GCSE Maths exam and everyday life.
Anatomy of a Circle
Key Definitions
- Circumference: The distance around the outside edge of a circle.
- Area: The amount of flat space a circle covers.
- Radius (r): The distance from the centre to the circumference.
- Diameter (d): The distance across the circle through the centre. Always twice the radius ($d = 2r$).
- Pi ($\pi$): A special constant (≈ 3.14159…) that links a circle’s diameter to its circumference.
The Key Formulas
Circumference (C)
To find the distance around a circle:
$C = \pi d$ or $C = 2\pi r$
Area (A)
To find the space inside a circle (always use the radius):
$A = \pi r^2$
⚠️ If you’re given the diameter, halve it to find the radius before using the area formula!
Worked Examples
Example 1: Using the Radius
A circular pond has a radius of 5 m. Calculate its circumference and area.
- Circumference:
$C = 2\pi r = 2 \times \pi \times 5 = 10\pi \approx 31.4 \text{ m}$. - Area:
$A = \pi r^2 = \pi \times 5^2 = 25\pi \approx 78.5 \text{ m}^2$.
Example 2: Using the Diameter (Semicircle)
A semicircular window has a diameter of 80 cm. Calculate its perimeter and area.
- Perimeter (curved part + straight edge):
$P = \frac{1}{2}\pi d + d = \frac{1}{2}\pi \times 80 + 80 = 40\pi + 80 \approx 205.7 \text{ cm}$. - Area: Radius $= 80 \div 2 = 40$ cm.
$A = \frac{1}{2}\pi r^2 = \frac{1}{2} \times \pi \times 40^2 = 800\pi \approx 2513.3 \text{ cm}^2$.
Tutor Insights
🤔 Common Misunderstandings
- Confusing Area and Circumference formulas. Tip: Area is in square units (cm²), so its formula has $r^2$.
- Forgetting the diameter in a semicircle’s perimeter. The straight edge ($d$) must be added to the curved half-circumference.
📝 Common Exam Mistakes
- Using the diameter instead of the radius in the area formula.
- Rounding too early. Keep the full calculator value until the very last step.
- Forgetting units (cm, m²) or using the wrong ones.
Practice Questions
- A circular garden has a radius of 3 metres. Calculate its circumference and area to 1 decimal place.
- A bicycle wheel has a diameter of 65 cm. Calculate its circumference to 1 decimal place.
- The circumference of a circular clock face is 94.2 cm. What is its diameter to 1 decimal place?
- A running track is shaped like a rectangle (100 m long, 60 m wide) with a semicircle at each end. Calculate the total perimeter of the track to 1 decimal place.
Show Answers
- Circumference: $2 \times \pi \times 3 \approx 18.8 \text{ m}$. Area: $\pi \times 3^2 \approx 28.3 \text{ m}^2$.
- Working: $C = \pi \times 65 \approx 204.2$.
Answer: 204.2 cm. - Working: $d = C \div \pi = 94.2 \div \pi \approx 30.0$.
Answer: 30.0 cm. - Working: The two semicircles form a full circle with diameter 60 m. Perimeter $= (2 \times 100) + (\pi \times 60) = 200 + 188.5 = 388.5$.
Answer: 388.5 m.
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