Here we see a circle with radius r units. What is its perimeter? How do we find out?
Answer: Perimeter (circumference) = 2πr units. We find it from the fact that circumference ÷ diameter is the same number π ≈ 3.14 for every circle, so C = π × 2r.
Step-by-step solution
Idea: Every circle is an enlarged or shrunk copy of every other circle, so the ratio circumference : diameter is the same for all circles. This fixed ratio is called π.
- Measure many circles (wrap a thread around a round object, then measure the thread and the diameter). Circumference ÷ diameter always comes to about 3.14. This constant is called π.1 mark
- So C ÷ D = π, which gives C = πD. The diameter is twice the radius, D = 2r, so C = 2πr units.1 mark
The perimeter of a circle of radius r is 2πr units. It follows from circumference ÷ diameter = π for every circle.
Check: A circle of radius 7 cm: 2 × 227 × 7 = 44 cm. A thread around such a circle would measure about 44 cm ✓.
Answer to write in the exam
C ÷ D = π (same for all circles)
C = πD = π × 2r
∴ Perimeter = 2πr units
Common mistakes that cost marks
- Writing πr2 for the perimeter. That is the area.
- Using πr (half the circle) or πD with D taken as the radius.
How this can come in the exam
MCQ (1 mark)
The perimeter of a circle of radius 3.5 cm is (use π = 227)
- 11 cm
- 22 cm
- 38.5 cm
- 44 cm
Show answer
(B) 22 cm
2 × 227 × 3.5 = 22 cm.
Try one yourself
Find the perimeter of a circle of radius 21 cm. (Use π = 227.)
Show answer
2 × 227 × 21 = 132 cm.
More questions like this
- What is the connection between this question and the one about the 400 m athletics track?
- What happens to the perimeter of a square if we double its side?
- What about a circle? What is its perimeter (usually called the circumference) in terms of its diameter? Is the ratio of circumference (C) to diameter (D) the same for circles of all sizes (see the figure)? What do you think?
- What is the value of the C/D ratio? How would you estimate this ratio?
- You can do a simple measurement at home to estimate the C/D ratio. Take a cotton reel with thin thread around it. Measure the diameter D of the reel as accurately as possible. Unwrap and then tightly wrap the thread around the reel 20 times. Unwrap it again; measure its length L, and calculate L20D. This is the ratio we want. For accuracy, the thread should be very thin. Please do the experiment! Do you get a ratio between 3 and 4? Between 3.1 and 3.2? It is also possible to estimate the C/D ratio using pure geometry, i.e., without any measurements at all! Can you imagine how?