You know the radius is 5. But what's the area? The circumference? The diameter? Or you know the circumference and need the radius. Every circle problem starts from one known value — here's how to get all the rest.
The Four Core Formulas
All circle properties derive from the radius (r):
| Property | Formula | From Radius | |----------|---------|-------------| | Diameter | d = 2r | Double the radius | | Circumference | C = 2πr | 2 × π × radius | | Area | A = πr² | π × radius squared |
π ≈ 3.14159 (or use 22/7 for quick estimates)
Calculating from Each Known Value
From Radius
Given r = 7:
- d = 2(7) = 14
- C = 2π(7) = 43.98
- A = π(49) = 153.94
From Diameter
Given d = 10:
- r = 10/2 = 5
- C = πd = 10π = 31.42
- A = π(5²) = 25π = 78.54
From Circumference
Given C = 31.42:
- r = C/(2π) = 31.42/6.283 = 5
- d = 2r = 10
- A = π(5²) = 78.54
From Area
Given A = 78.54:
- r = √(A/π) = √(78.54/3.14159) = √25 = 5
- d = 10, C = 31.42
Worked Examples
Example 1: Pizza Size
A 12-inch pizza has diameter 12 inches:
- r = 6 inches
- A = π(36) = 113.1 sq inches
A 14-inch pizza:
- r = 7 inches
- A = π(49) = 153.9 sq inches
The 14-inch pizza is 36% larger than the 12-inch. Always buy the larger size — you get more pizza per rupee.
Example 2: Running Track
A circular track has circumference 400m:
- r = 400/(2π) = 63.66 m
- Area enclosed = π(63.66²) = 12,732 sq m
Example 3: Garden Sprinkler
A sprinkler sprays water in a circle with radius 8m:
- Area covered = π(64) = 201.06 sq m
- Diameter of coverage = 16 m
The Trench Truth: The most common circle mistake: confusing radius and diameter. If the problem says "diameter = 10," you MUST divide by 2 to get radius = 5 before using the area or circumference formula. Using d instead of r gives you 4× the correct area.
Try our circle calculator — enter any one known value and get all properties instantly.
Related: Area Converter · Square Root Calculator · Derivative Calculator
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