Circle Calculator
Calculate circle area, circumference, and diameter instantly from radius.
Formula
Area = πr² | C = 2πr
Area depends on the square of the radius. Circumference is the linear distance around the edge.
Worked Examples
Example 1: Everything from the Radius
Problem:A circle has a radius of 5 cm. Find its diameter, circumference and area.
Solution:Diameter d = 2r = 10 cm Circumference C = 2πr = 2 × 3.141593 × 5 = 31.416 cm Area A = πr² = 3.141593 × 25 = 78.540 cm² Check the defining ratio: C/d = 31.416/10 = 3.1416, which is π, as it is for every circle regardless of size.
Result:d = 10 cm | C = 31.416 cm | A = 78.540 cm²
Example 2: Working Backwards from the Area
Problem:A circular tabletop covers 200 cm². What are its radius, diameter and circumference?
Solution:Rearrange A = πr² to r = the square root of (A/π). r² = 200 / 3.141593 = 63.662 r = 7.979 cm Diameter = 2r = 15.958 cm Circumference = 2πr = 2 × 3.141593 × 7.979 = 50.133 cm Check: π × 7.979² = 3.141593 × 63.662 = 200.0 cm². Correct.
Result:r = 7.979 cm | d = 15.958 cm | C = 50.133 cm
Example 3: Why Doubling the Radius Quadruples the Area
Problem:A 12 inch pizza and a 16 inch pizza are priced at $12 and $18. Which is better value per square inch?
Solution:Pizza sizes are diameters, so the radii are 6 and 8 inches. 12 inch: A = π(6²) = 113.10 in², at $12 that is $0.1061 per in² 16 inch: A = π(8²) = 201.06 in², at $18 that is $0.0895 per in² The 16 inch pizza is 77.8% larger in area for only 50% more money, so it is the better buy. The general rule: area scales with the square of the radius, so a 33% larger diameter buys 78% more pizza.
Result:16 inch = $0.0895/in² vs 12 inch = $0.1061/in² - the larger pizza wins
Frequently Asked Questions
What is Pi (π)?
Pi is the mathematical constant representing the ratio of a circle's circumference to its diameter. It is an irrational number approx 3.14159.
What is the radius?
The radius is the distance from the center of the circle to any point on its edge. It is half of the diameter.
What is the diameter?
The diameter is the longest straight line that can be drawn inside a circle, passing through the center. It is twice the radius.
What is the circumference?
The circumference is the distance around the outside of the circle (its perimeter). Formula: C = 2πr or C = πd.
How do I find the area of a circle?
Use the formula A = πr². If you have the diameter, divide it by 2 to get the radius first, then square it and multiply by π.
What is a Chord?
A chord is a straight line segment whose endpoints both lie on the circle. The diameter is the longest possible chord.
What is a Tangent?
A tangent is a straight line that touches the circle at exactly one point. It is always perpendicular to the radius at that point.
What is a Sector?
A sector is a portion of a circle enclosed by two radii and an arc (like a slice of pizza). Its area is proportional to the central angle.
What is an Arc?
An arc is a portion of the circumference of a circle. Its length depends on the radius and the angle it subtends.
What are concentric circles?
Circles that share the same center point but have different radii (like a target or bullseye).
Background & Theory
Circle Properties:
- Radius (r): Distance from center to edge.
- Diameter (d): Distance across circle through center (d = 2r).
- Circumference (C): Distance around the circle (Perimeter).
- Area (A): Space inside the circle.
Formulas:
- Diameter: d = 2r
- Circumference: C = 2πr = πd
- Area: A = πr²
Pi (π): An irrational number approximately equal to 3.14159...