Area & Perimeter Calculator
Calculate area and perimeter for circles, rectangles, triangles, squares, trapezoids, parallelograms, and hexagons.
Area & Perimeter
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The Formula
A circle has the highest area-to-perimeter ratio of any 2D shape. Area grows with the square of the radius (doubling the radius quadruples the area). The circumference is the perimeter — the distance around the circle.
Variable Definitions
Radius
Distance from center to the outer edge. All points on the circle are exactly radius r from center.
Diameter
Twice the radius: d = 2r. The longest distance across the circle.
Pi
A mathematical constant ~3.14159. The ratio of any circle's circumference to its diameter.
How to Use This Calculator
- 1
Select "Circle" from the shape tabs above.
- 2
Enter the radius of the circle.
- 3
View the area, circumference, and diameter.
A circle has the highest area-to-perimeter ratio of any 2D shape
Understanding the Concept
The circle is the most efficient 2D shape — for a given perimeter, it encloses the maximum possible area. This is why bubbles form spheres, why pizza pans are round, and why manhole covers are circular. The formula A = πr² was known to ancient mathematicians: Archimedes proved it by approximating circles with inscribed and circumscribed polygons with 96 sides each. The constant π (~3.14159) appears in both the area and circumference formulas because the circle is the same shape scaled — all circles are similar to each other.
Frequently Asked Questions
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