Circumference and Area of Circles

Unit 10: Area of Polygons and Circles · ~15–30 min

Radius, diameter, and pi

Every circle's circumference (the distance around it) is a fixed multiple of its diameter — that constant multiple is π (pi), approximately 3.14159. The radius r is the distance from center to edge, and the diameter d is twice the radius.

• Circumference: C = πd, or equivalently C = 2πr.
• Area: A = πr².

The area formula also traces back to the regular-polygon idea from the last lesson: think of a circle as a regular polygon with an enormous number of sides. As the number of sides grows, the apothem approaches the radius r and the perimeter approaches the circumference 2πr, so A = ½aP becomes A = ½r(2πr) = πr².

Guided practice

• 1.Find the circumference of a circle with radius 6 m, in terms of π.
  Show answer

  C = 2π(6) = 12π m.
• 2.Find the area of a circle with diameter 14 ft, in terms of π.
  Show answer

  r = 7. A = π(7)² = 49π ft².
• 3.A circle has circumference 31.4 cm. Find its radius (use π ≈ 3.14).
  Show answer

  31.4 = 2πr ≈ 6.28r, so r = 31.4 ÷ 6.28 = 5 cm.
• 4.Explain why doubling a circle's radius more than doubles its area.
  Show answer

  Area depends on r², not r, so doubling r multiplies the area by 2² = 4 — area grows with the square of the radius, not in direct proportion to it.
• 5.A circular pool has area 154 m² (use π ≈ 22/7). Find its radius and diameter.
  Show answer

  154 = (22/7)r², so r² = 154 × 7/22 = 49, giving r = 7 m. Diameter = 14 m.