Learning Objectives
- identify radius, diameter and circumference;
- use C = 2πr and C = πd;
- calculate area using A = πr²;
- solve semicircle and sector-style perimeter problems;
- find missing circle measurements.
Introduction
This lesson develops Circumference and Area of Circles from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Visual Exploration
Use the diagrams, tables or examples in this lesson to identify what changes, what stays fixed and which relationship connects the values.
Parts of a Circle
Circumference
Find the circumference of a circle with radius 7 cm.
C = 2π(7) = 14π ≈ 44.0 cm.
Area of a Circle
Find the area of a circle with diameter 10 cm.
Radius = 5 cm.
A = π(5²) = 25π ≈ 78.5 cm².
Semicircles
The area is half the circle, but the perimeter includes the curved half plus the diameter.
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Worked Examples
Reason through the method
- Identify the known information and what must be found.
- Choose the definition, property or formula that connects them.
- Substitute carefully and show each step.
- Check the notation, units and whether the result is sensible.
Practice Questions
Foundation- Find the circumference of a circle with radius 5 cm.
- Find the area of a circle with radius 6 m.
- A circle has diameter 14 cm. Find its circumference.
- A circle has circumference 31.4 cm. Find its radius, using π≈3.14.
- Find the perimeter of a semicircle of radius 8 cm.
- Find the area of a semicircle with diameter 20 cm.
- A circular path has outer radius 10 m and inner radius 7 m. Find the area of the path.
✅ Check Your Work
Show practice answers
- 10π ≈ 31.4 cm
- 36π ≈ 113.1 m²
- 14π ≈ 44.0 cm
- 31.4 = 2×3.14×r, so r=5 cm.
- 8π+16 ≈ 41.1 cm.
- Radius=10 cm; area=½π(10²)=50π≈157.1 cm².
- π(10²−7²)=51π≈160.2 m².
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- d = 2r.
- C = 2πr = πd.
- A = πr².
- Semicircle perimeter includes the diameter.