⭕ ⭕ Circle Calculator: Radius, Diameter, Area, and Circumference

The circle is one of the most fundamental shapes in geometry, and its properties appear throughout daily life — from wheels and pipes to pizza, coins, and satellite dishes. Four measurements define a circle completely: radius, diameter, circumference, and area. Knowing any one of them lets you calculate all the others, connected through the mathematical constant π (pi).

The Four Circle Measurements

• Radius (r): The distance from the center to any point on the circle
• Diameter (d): The distance across the circle through the center = 2 × radius
• Circumference (C): The perimeter — the total distance around the circle
• Area (A): The space enclosed within the circle

The Circle Formulas

What is π (Pi)?

Pi (π) is the ratio of any circle's circumference to its diameter: π = C ÷ d. This ratio is the same for every circle, no matter how large or small — it is a universal constant approximately equal to 3.14159265358979...

Pi is an irrational number (its decimal expansion never ends or repeats) and a transcendental number (it cannot be the root of any polynomial equation with rational coefficients). For most calculations, using π ≈ 3.14159 or 22/7 as an approximation is sufficient.

Worked Examples

Example 1: Circle with radius 5 cm

• Diameter = 2 × 5 = 10 cm
• Circumference = 2π × 5 = 10π ≈ 31.42 cm
• Area = π × 5² = 25π ≈ 78.54 cm²

Example 2: Find the radius of a circle with area 154 m²

• r = √(A ÷ π) = √(154 ÷ 3.14159) = √49.01 ≈ 7 m
• Circumference = 2π × 7 ≈ 43.98 m

Example 3: Find the area of a circle with circumference 62.83 cm

• r = C ÷ (2π) = 62.83 ÷ 6.2832 = 10 cm
• Area = π × 10² = 100π ≈ 314.16 cm²

Arc Length and Sector Area

A sector is a "pie slice" of a circle — bounded by two radii and an arc. An arc is a portion of the circumference.

Arc length = (θ/360°) × 2πr (where θ is the central angle in degrees)

Sector area = (θ/360°) × πr²

Example: A 60° sector of a circle with radius 9 cm:

• Arc length = (60/360) × 2π × 9 = (1/6) × 56.55 = 9.42 cm
• Sector area = (60/360) × π × 81 = (1/6) × 254.47 = 42.41 cm²

Real-World Circle Calculations

Pipe and Cylinder Cross-Sections

A pipe with an outer diameter of 6 inches and an inner diameter of 5.5 inches (wall thickness 0.25 inches):

• Outer area = π × 3² = 28.27 in²
• Inner area = π × 2.75² = 23.76 in²
• Cross-sectional area of pipe wall = 28.27 − 23.76 = 4.51 in²

Circular Lawn Irrigation

A sprinkler covers a circular area of radius 12 m. What area does it water?

Area = π × 12² = 144π ≈ 452.4 m²

Pizza Size Comparison

Is a 16-inch pizza more than twice the size of an 8-inch pizza?

• 8-inch pizza area = π × 4² ≈ 50.27 in²
• 16-inch pizza area = π × 8² ≈ 201.06 in²
• Ratio = 201.06 ÷ 50.27 = 4× bigger — not 2×!

This is why doubling the diameter quadruples the area — area scales with r², so doubling r multiplies area by 4. This is one of the most counterintuitive facts about circles.

Circles and Other Shapes

• The circle has the maximum area for a given perimeter of any 2D shape (isoperimetric inequality)
• A circle inscribed in a square with side length s has radius = s/2; area = π(s/2)²
• A circle circumscribed around a square with side s has radius = s√2/2; area = πs²/2