Circle circumference and area
Use radius and diameter correctly, retain exact pi answers and round only at the end.
Core and Extended.
Before you begin
- Square numbers and use square roots.
- Distinguish radius from diameter.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
A diameter is 14 cm. What is the radius?
What you will learn
- Find circumference and area from radius or diameter.
- Recover a radius from a stated area or circumference.
The boundary uses one length; the area uses its square
A circle's circumference is 2πr or πd, where d = 2r. Its area is πr². If diameter is 14 cm, first use radius 7 cm for the area, giving 49π cm², while circumference is 14π cm. Squaring the diameter would make the area four times too large.
An answer in terms of π is exact within the mathematical model. If a decimal is requested, use your calculator's π value and round the final answer to the specified accuracy. Inverse problems follow the formula: r = C/(2π) for circumference and r = √(A/π) for area. A radius must be positive in a physical circle.
Match the measure to the context
A fence around a circular garden needs circumference; turf covering it needs area. A 4 m radius garden requires 8π m of fence and 16π m² of turf before any gate opening or waste allowance. Read those extra conditions separately rather than rounding the radius or adding an invented allowance.
For a circular ring, subtract the inner circle's area from the outer circle's area: π(R² − r²). Its boundary includes two circumferences if both edges are requested. A half-circle's curved arc is half a circumference, but its full perimeter also includes the diameter; cutting a shape changes which boundaries count.
Pause and explain
A circle has radius 5 cm. What is its exact area?
Worked example
A circle has diameter 14 cm. Give its circumference and area exactly.
Show the worked solution
- The radius is 14/2 = 7 cm.
- Circumference is πd = 14π cm.
- Area is πr² = 49π cm².
Answer 14π cm; 49π cm².
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find exact circumference and area for radius 3 cm.
Give me a hint
Use the radius in both formulas.
Compare my reasoning
- Circumference is 2π × 3 = 6π cm.
- Area is π × 3² = 9π cm².
- Check that the radius is squared only for area.
6π cm; 9π cm².
Look for these in your work
- Kept pi exact.
- Used the correct dimensions.
A circular disc has area 81π cm². Find its radius and diameter.
Give me a hint
Divide by pi, then take a positive square root.
Compare my reasoning
- r² = 81, so r = 9 cm.
- Diameter is twice this radius, 18 cm.
- Check π × 9² = 81π cm².
Radius 9 cm; diameter 18 cm.
Look for these in your work
- Took a square root rather than dividing by two.
- Distinguished radius and diameter.
A circular garden has radius 4 m and a 2 m gate opening. Give exact fence length and turf area, excluding the gate from the fence.
Give me a hint
Subtract the opening from circumference, not area.
Compare my reasoning
- The complete circumference is 8π m.
- Fence length is 8π − 2 m.
- Turf covers π × 4² = 16π m²; the gate does not remove garden area.
Fence (8π − 2) m; turf 16π m².
Look for these in your work
- Applied the opening to the correct measure.
- Kept the remaining answer exact.
Common mistakes
- Squaring the diameter for area.
- Using square units for circumference.
- Rounding pi or the radius too early.
What can you explain now?
A circle has circumference 12π cm. Find its radius.
Compare with the explanation
6 cm
2πr = 12π, so dividing by 2π gives r = 6 cm.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, start with an exact area and recover radius, diameter and circumference.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources