Circle Calculator
Circle Formulas
Quick Reference
Related Calculators
Understanding Circles
Circle Definitions
A circle is a shape where all points are equidistant from a center point. This distance is called the radius. Circles appear everywhere in nature and human design, from wheels to planetary orbits.
Radius (r)
The distance from the center to any point on the circle. The fundamental measurement from which all other properties are derived.
Diameter (d)
A line passing through the center connecting two points on the circle. Always twice the radius: d = 2r.
Circumference (C)
The distance around the circle (perimeter). Calculated as C = 2πr or C = πd.
Area (A)
The space enclosed by the circle. Calculated as A = πr² (pi times radius squared).
The Constant π (Pi)
Pi (π) is the ratio of a circle's circumference to its diameter. It's an irrational number that never ends or repeats:
Common approximations: 3.14, 22/7, or 355/113
Circle Formulas
| To Find | Given | Formula |
|---|---|---|
| Circumference | Radius | C = 2πr |
| Circumference | Diameter | C = πd |
| Area | Radius | A = πr² |
| Area | Diameter | A = πd²/4 |
| Radius | Circumference | r = C/(2π) |
| Radius | Area | r = √(A/π) |
Example Calculations
Circle with r = 5
- Diameter: 2 × 5 = 10
- Circumference: 2π × 5 ≈ 31.42
- Area: π × 5² ≈ 78.54
Circle with d = 10
- Radius: 10 ÷ 2 = 5
- Circumference: π × 10 ≈ 31.42
- Area: π × 5² ≈ 78.54
Real-World Applications
- Calculate fence length for a circular garden (circumference)
- Determine how much paint for a circular wall (area)
- Find pizza size comparison (area scales with r²)
- Calculate tire circumference for odometer calibration
- Design circular pools, ponds, or fountains
Frequently Asked Questions
How do you find the area of a circle?
The area of a circle is calculated using A = pi x r squared, where r is the radius. For example, a circle with radius 5 has area = pi x 25 = approximately 78.54 square units. If you know the diameter, use A = pi x (d/2) squared or A = pi x d squared / 4.
What is the relationship between circumference and diameter?
The circumference of any circle divided by its diameter always equals pi (approximately 3.14159). This means C = pi x d. This ratio is the definition of pi and holds true for all circles regardless of size.
How do you find the radius from the area?
To find radius from area, rearrange the area formula: r = square root of (A / pi). For example, if area = 50 square units, radius = square root of (50 / 3.14159) = square root of 15.92 = approximately 3.99 units.
Why do pizza sizes seem to grow faster than their diameter suggests?
Pizza area grows with the square of the radius, not linearly. A 16-inch pizza has about 201 square inches, while a 12-inch pizza has only about 113 square inches. So the 16-inch pizza is 78% larger in area, not just 33% larger. This is why larger pizzas are often better value.