BananaLearnLearning LibraryGamesA circle has radius 7 cm. Find circumference and area using π ≈ 22/7.
- Circumference = 2 × 22/7 × 7 = 44 cm.
- Area = 22/7 × 7 × 7 = 154 cm².
- The answers have different units because one measures around and the other inside.
Two measurements answer different practical questions
To put ribbon around a circular card, measure its boundary: circumference. To cover its face with coloured paper, measure the inside region: area. A diagram can show both, but they are not interchangeable. A 44 cm length of ribbon cannot be described as 44 square centimetres.
The radius is a line segment from the centre to the circle. A diameter passes through the centre and joins two points on the circle, so d=2r. A line that crosses the circle away from the centre is a chord; it is not necessarily a diameter.
| Known measure | Circumference | Area |
|---|---|---|
| Radius r | 2πr | πr² |
| Diameter d | πd | π(d/2)² |
| r = 7 cm, π ≈ 22/7 | 44 cm | 154 cm² |
| d = 10 cm, exact | 10π cm | 25π cm² |
Why the formulas have different powers
For every circle, circumference divided by diameter is the same constant, π. That relationship gives C=πd=2πr. The area formula πr² can be understood by dividing a circle into many narrow sectors and arranging them alternately. Their approximate base is half the circumference, πr, and their height approaches r; base times height approaches πr².
This rearrangement explains why squaring the radius matters. Doubling all lengths doubles the circumference, but covers four times the area. Think of scaling a small circular mat into a larger one: both horizontal and vertical dimensions grow.
Keep exact values until the question asks for an approximation
For radius 3 cm, 6π cm is an exact circumference. Using π≈3.14 gives 18.84 cm; using a calculator’s π gives a slightly different decimal. Follow the question’s instruction and round at the end. The fraction 22/7 is an approximation to π, even when its use makes the arithmetic convenient.
If given area 49π cm², solve r²=49 and choose r=7 cm because radius is positive. If given circumference 18π cm, solve 2r=18 to get r=9 cm. Do not take a square root for circumference; the two formulas have different structures.
Parts of circles need an extra boundary check
A semicircle has half the area of the full circle. Its curved edge is half the circumference, but its complete perimeter also includes the straight diameter. For radius 7 cm and π≈22/7, the curve is 22 cm and the diameter is 14 cm, so the full boundary is 36 cm.
For a circular path around a garden, the path’s area is the area of the outer circle minus that of the inner circle. Draw and label both radii first. The width of the path is added to the radius, not to the diameter.
Worked examples: follow each decision
A circular sign has diameter 10 cm. Find its area in terms of π.
- Radius = 10 ÷ 2 = 5 cm.
- Area = π × 5² = 25π cm².
- Using π ≈ 3.14 would give 78.5 cm².
A circular garden has radius 7 m. A 1 m wide path goes around it. Find the path area.
- Outer radius=8 m, not 15 m.
- Area difference=π×8²−π×7²=15π m².
- Using π≈3.14 gives 47.1 m². Subtracting the radii first and squaring would describe a different circle.
13 questions with explained answers
Attempt a short group of questions before opening the solutions. Use a drawing or written method where helpful. Explain the answer, then try a similar question with a detail changed.
Diameter 8 m: radius?
Show answer and reasoning
4 m
Radius is half the diameter.
Radius 3 cm: exact circumference?
Show answer and reasoning
6π cm
Use 2πr.
Radius 4 cm: exact area?
Show answer and reasoning
16π cm²
Square the radius, then multiply by π.
Radius doubles: what happens to area?
Show answer and reasoning
It becomes four times as large
(2r)² = 4r².
Diameter 6 m: edging using π ≈ 3.14?
Show answer and reasoning
18.84 m
Edging is circumference: πd.
Diameter 14 cm: exact circumference?
Show answer and reasoning
14π cm
C=πd.
Diameter 14 cm: exact area?
Show answer and reasoning
49π cm²
Radius is 7 cm, so A=π×7².
C=20π m. Find radius.
Show answer and reasoning
10 m
2πr=20π.
A=81π cm². Find diameter.
Show answer and reasoning
18 cm
Radius is 9 cm; double it.
A circle’s radius triples. Area factor?
Show answer and reasoning
9
(3r)²=9r².
Semicircle radius 4 cm: complete perimeter?
Show answer and reasoning
4π+8 cm
Half the circumference plus diameter.
Ring with outer radius 5 cm and inner radius 3 cm: area?
Show answer and reasoning
16π cm²
25π−9π=16π.
Which needs square metres: fencing or covering a pond?
Show answer and reasoning
Covering
Fencing uses length; covering uses area.
Check what you can now explain
- Are you measuring around, inside or a partial boundary?
- Did you convert diameter to radius for area?
- Are your units and approximation instructions correct?
Keep these distinctions clear
- Do not square the entire πr expression; πr² means π × r × r.
- A diameter is not a radius. Label your diagram first.
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Keep the ideas connected
- Area and perimeter: the difference, examples and practice
- Grade 8 Cylinder Surface Area and Volume
- Choose a Maths practice game by skill
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The examples teach the stated concept; check your school’s terminology and sequence when using the guide alongside a course.