Explores the relationship between a circle's diameter and circumference to discover pi. Develops and applies the formulas for circumference (C = pi*d = 2*pi*r) and area (A = pi*r^2). Students solve real-world problems involving circular measurements.
Middle School Geometry & Data • 6-8
Every circle—from a coin to a planet's orbit—shares the same beautiful ratio: the circumference divided by the diameter always equals pi. This lesson explores that constant and the formulas it unlocks.
| Term | Definition |
|---|---|
| Radius (r) | Distance from the center to any point on the circle |
| Diameter (d) | Distance across the circle through the center; d = 2r |
| Circumference (C) | The perimeter — the distance around the circle |
| Pi (pi) | The ratio C / d; approximately 3.14159... (irrational, never-ending) |
If you measure the circumference and diameter of any circular object—a jar lid, a tire, a clock face—and divide C by d, you always get roughly 3.14. This ratio is the same for every circle in the universe. We call it pi. For calculations, use pi ≈ 3.14 or the pi button on your calculator.
Use the first form when you know the diameter. Use the second when you know the radius.
The area depends on the radius squared. This means doubling the radius makes the area four times larger, not just two.
Problems sometimes give the diameter. Always convert to the radius first (divide by 2) before using the area formula. Forgetting this step is the number-one circle mistake.
A circular garden has diameter 14 m. Find its circumference.
A pizza has radius 9 inches. Find the area.
A circular running track has circumference 400 m. Find the radius.
When given diameter = 10 and asked for area, students sometimes compute pi × 10² = 314. But 10 is the diameter, not the radius! The radius is 5, so A = pi × 5² = 78.5. Always check: am I using the radius?
1. Find the circumference of a circle with radius 7 cm. (Use pi ≈ 3.14.)
C = 2 × 3.14 × 7 = 43.96 cm
2. A dinner plate has diameter 26 cm. Find its area.
Radius = 26 / 2 = 13 cm. A = 3.14 × 13² = 3.14 × 169 = 530.66 cm²
3. A wheel has circumference 94.2 cm. Find the diameter.
d = C / pi = 94.2 / 3.14 = 30 cm
4. A sprinkler waters a circular region with radius 12 ft. How much area does it cover?
A = 3.14 × 12² = 3.14 × 144 = 452.16 ft²
5. Circle A has radius 4 and circle B has radius 8. How many times larger is the area of circle B compared to circle A?
A(A) = pi × 16 = 16pi. A(B) = pi × 64 = 64pi. Ratio = 64pi / 16pi = 4 times larger. Doubling the radius quadruples the area because area depends on r².