Circle Calculator

This circle calculator works as a circumference calculator, an area of a circle calculator, a radius calculator, and a diameter to circumference calculator all at once. Enter any one property of a circle and the tool returns the other three. Use it for homework, fabrication, landscaping layout, or any time you need to go from a tape measurement to a clean number.

1. Pick which input you have. The four buttons are Radius, Diameter, Circumference, and Area. Only one is needed.
2. Radius mode (r): enter the distance from the center of the circle to the edge. If r = 5, the tool returns d = 10, C ≈ 31.4159, and A ≈ 78.5398.
3. Diameter mode (d): enter the full width across the circle through the center. The calculator halves it to get the radius, then returns circumference and area. A 12 inch pizza has d = 12, r = 6, and A ≈ 113.10 in².
4. Circumference mode (C): enter the perimeter, the length all the way around the edge. The calculator solves r = C ÷ (2π), then fills in diameter and area. Wrap a string around a tree trunk, measure 47 inches, enter 47, and read diameter ≈ 14.96 in.
5. Area mode (A): enter the flat surface area of the circle. The calculator solves r = √(A ÷ π), then returns the other three properties. Enter 100 and you get r ≈ 5.6419, d ≈ 11.2838, C ≈ 35.4491.

The result panel shows all four values to six decimal places. Units stay consistent with whatever you entered: centimeters in gives centimeters out, with area in square centimeters. Because the circle calculator is unit agnostic you can use inches, meters, feet, or any other length unit without changing anything.

Given radius r:
  Diameter      d = 2r
  Circumference C = 2πr  (π ≈ 3.14159265)
  Area          A = πr²

Solving for radius from other measurements:
  From diameter:       r = d / 2
  From circumference:  r = C / (2π)
  From area:           r = √(A / π)

Example with radius = 5:
  Diameter      = 2 × 5        = 10
  Circumference = 2 × π × 5  ≈ 31.4159
  Area          = π × 5²     ≈ 78.5398

Core Circle Formulas at a Glance

Three formulas cover almost every circle calculation you will run into. If you know the radius r, everything else drops out:

Diameter       d = 2r
Circumference  C = 2πr   = πd
Area           A = πr²   = πd² / 4

These are the same formulas the area of a circle calculator and the circumference calculator use internally. \u03c0 (pi) is the constant ratio between a circle's circumference and its diameter, roughly 3.14159265.

Sector Area and Arc Length

A sector is a pie-slice shape cut from a circle by two radii. The central angle between those radii is \u03b8 (theta). When \u03b8 is measured in radians, the formulas are compact:

Sector area   A_sector = ½ r² θ
Arc length    L_arc    = r θ

Conversion:   angle_in_radians = angle_in_degrees × π / 180

Example: a circle with r = 10 and a 60\u00b0 central angle. 60\u00b0 in radians is \u03c0 \u00f7 3 \u2248 1.0472. Arc length = 10 \u00d7 1.0472 \u2248 10.472. Sector area = 0.5 \u00d7 100 \u00d7 1.0472 \u2248 52.36. In degrees the same formulas become A_sector = (\u03b8 \u00f7 360) \u00d7 \u03c0r\u00b2 and L_arc = (\u03b8 \u00f7 360) \u00d7 2\u03c0r, which some students find easier to remember.

How Radius, Diameter, Circumference, and Area Relate

Any one of the four measurements determines the other three. This table shows how to derive each property from each input, which is exactly what the radius calculator above does behind the scenes:

The diameter to circumference calculator route uses the middle row: multiply diameter by \u03c0. A 24 inch wheel has a circumference of about 75.40 inches, which is how far it travels per revolution on the ground.

Quick Reference: Common Radii

For back-of-the-envelope work, these pre-computed values cover most everyday radii. Values are rounded to two decimals:

Notice the area jumps fast. Doubling the radius from 5 to 10 quadruples the area, from about 78.54 to 314.16. That scaling matters every time you compare two circles of different size, from pizzas to pipes.

Plugging numbers into a circle calculator is easy. Knowing when area matters vs circumference, how tight an approximation of \u03c0 you need, and why a 14 inch pizza is a much better deal than a 10 inch pizza is where the useful thinking happens. The sections below cover the practical angles a pure formula sheet leaves out.

Where Circle Math Actually Shows Up

Circumference governs anything that wraps, rolls, or loops. Area governs anything that fills, covers, or flows. A few concrete cases:

• Pipe flow: water flow rate through a round pipe scales with the cross-sectional area, \u03c0r\u00b2. A 2 inch diameter pipe carries 4 times the flow of a 1 inch pipe at the same pressure, not twice as much.
• Wheels and tires: one revolution covers a distance equal to the circumference. A 26 inch bike wheel (d = 26) rolls forward 26\u03c0 \u2248 81.68 inches, or about 6.81 feet, per turn.
• Pizza and pie sizing: area tells you how much food is on the pan. A 16 inch pizza has an area of \u03c0 \u00d7 8\u00b2 \u2248 201 in\u00b2, vs 113 in\u00b2 for a 12 inch pizza. That is 78% more pizza, not 33% more.
• Landscaping and patios: circular flower beds, firepits, and garden edging all use A = \u03c0r\u00b2 for materials (mulch, stone, sod) and C = 2\u03c0r for edging length.
• Tank volume: a cylindrical tank holds \u03c0r\u00b2h, so any water tank, storage drum, or silo volume starts with a circle area calculation on the base.

Pi (\u03c0): What It Is, and When Precision Matters

Pi is the ratio of a circle's circumference to its diameter. It is the same number for every circle, 3.14159265358979... and so on forever. It is irrational, meaning its decimal expansion never ends and never repeats. For most everyday work you do not need more than a few digits:

The circle calculator above uses JavaScript's Math.PI, which is accurate to about 15 significant digits. That is far more precision than any physical measurement you will ever make, so the result is always limited by how accurately you measured the input, not by the value of \u03c0.

Inscribed vs Circumscribed: Circles and Squares Together

Two classic geometry setups come up constantly in design and packaging problems. A circle inscribed in a square just touches all four sides; the circle's diameter equals the square's side. A circle circumscribed around a square passes through all four corners; the square's diagonal equals the circle's diameter.

Square side s, circle inscribed inside the square:
  Circle diameter d = s
  Circle fills π/4 ≈ 78.54% of the square

Square side s, circle drawn around (circumscribing) the square:
  Circle diameter d = s√2
  Square fills 2/π ≈ 63.66% of the circle

Practical use: if you are cutting round disks from a square sheet, you lose about 21.5% of the material no matter how tightly you pack them in a single circle. If you are fitting a square box inside a cylindrical container, the box can only be about 64% of the cylinder's cross-section.

The Pizza Size Trap: Why Bigger Is Way Bigger

Because area scales with r\u00b2, a small jump in diameter means a big jump in food. This is the single most common consumer pricing trap with circles:

A 14 inch pizza gives you almost twice as much pizza as a 10 inch, yet it rarely costs twice as much. The same trap applies to cake pans, skillets, round tables, and any other circle priced by nominal diameter rather than area. Run the numbers through the circle area calculator above before paying extra for two smaller units when one larger unit would do.