51
Circumference of a circle and area of a disc
Textbook, pp. 222–224
GoalKnow the formulas for the circumference of a circle and the area of a disc; the number π; solve problems.
New words
circumference · aylana uzunligiarea of a disc (circle) · doira yuziradius · radiusthe number π · π soni
Explanation
The ratio of a circle's circumference to its diameter is the same number for every circle; it is denoted by the letter π (pi): C : d = π. So the circumference is C = πd = 2πr, where d is the diameter, r is the radius and d = 2r. The area of a disc is S = πr². The number π is an infinite non-repeating decimal: π = 3.14159265... In problems we usually take π ≈ 3.14. It does not depend on the radius. Circumference is measured in units of length such as cm or m, and the area of a disc in units of area such as cm² or m².
Worked examples
r = 3 cm: C = 2 · 3.14 · 3 = 18.84 cm. Area of the disc S = 3.14 · 3² = 28.26 cm².
If C = 25.12 cm, then r = C : (2π) = 25.12 : 6.28 = 4 cm.
Class activity
“Measure the circle”: groups measure the circumference of different lids with string and divide it by the diameter. In every group the result is close to 3, between 3.1 and 3.2, which is π found by measuring.
Practice
1
Find the circumference of a circle with radius 2.5 cm (π = 3.14).
15.7 cm
2
A circle has circumference 25.12 cm. Find its radius (π = 3.14).
4 cm
3
Find the area of a disc with radius 10 cm (π = 3.14), in cm².
314
4
Why is the ratio C : d the same, π, for big and small circles?
All dimensions of a circle are proportional to its radius: when the radius grows by some factor, both circumference and diameter grow by the same factor, so the ratio stays the same.