The area of a circle is A = πr²: multiply pi (about 3.14159) by the radius squared. A circle with a 7-inch radius has an area of π × 7² = π × 49 ≈ 153.94 square inches. If you know the diameter instead, use A = πd² ÷ 4; if you know the circumference, use A = C² ÷ (4π).
The three area formulas
Which formula you use depends on what you can measure:
All three are the same formula rearranged, using d = 2r and C = 2πr. The radius is the distance from the center to the edge; the diameter goes all the way across through the center; the circumference is the distance around.
Step by step from the radius
- Measure or identify the radius.
- Square it (multiply it by itself).
- Multiply by π.
- Write the answer in square units.
Radius = 7 in → r² = 49 in² → A = π × 49 ≈ 153.94 in²
The most common mistake is doubling the radius instead of squaring it. 7 × 2 = 14 leads to 43.98, which is wrong.
Worked examples
Pizza sizes (diameter)
Pizza is sold by diameter, which makes comparisons misleading. A 12-inch pizza has a 6-inch radius:
12-inch: π × 6² ≈ 113.10 in²
Two 12-inch pizzas: 2 × 113.10 ≈ 226.19 in²
One 18-inch: π × 9² ≈ 254.47 in²
One 18-inch pizza has more food than two 12-inch pizzas, because area grows with the square of the size.
A round patio (diameter in feet)
A circular patio is 12 feet across.
A = π × 12² ÷ 4 ≈ 113.10 ft²
In square yards: 113.10 ÷ 9 ≈ 12.57 yd²
Order pavers or concrete by area plus a waste allowance. The square footage calculator combines circles with rectangles and other shapes for whole projects.
A tree trunk (circumference)
You can wrap a tape around a tree but you cannot measure through it. With a circumference of 94 inches:
A = 94² ÷ (4π) = 8,836 ÷ 12.566 ≈ 703.1 in²
Or find the radius first: r = 94 ÷ 2π ≈ 14.96 in, then π × 14.96² ≈ 703.1 in²
Quick reference table
| Radius | Area (πr²) | Circumference (2πr) |
|---|---|---|
| 1 | 3.14 | 6.28 |
| 2 | 12.57 | 12.57 |
| 3 | 28.27 | 18.85 |
| 5 | 78.54 | 31.42 |
| 10 | 314.16 | 62.83 |
| 12 | 452.39 | 75.40 |
| 20 | 1,256.64 | 125.66 |
Notice that doubling the radius from 10 to 20 multiplies the area by 4 but the circumference by only 2. At a radius of exactly 2, the area and circumference have the same number, though in different units.
The circle calculator solves for any of radius, diameter, circumference or area from any one of the others.
Working backward: radius from area
Sometimes you know how much area you want and need the size. Rearrange the formula:
A circular garden bed of 50 ft²: r = √(50 ÷ π) = √15.92 ≈ 3.99 ft
So the bed is about 8 feet across
This is handy for sizing round tables, rugs, ponds and tree rings. Because of the square root, doubling the area only increases the radius by a factor of √2, about 41%: a 100 ft² circle has a radius of about 5.64 ft.
Why the formula is πr²
Cut a circle into many thin pie slices and arrange them in a row, alternating point-up and point-down. The result looks almost like a parallelogram. Its height is the radius r, and its base is half the circumference, πr, because half the curved edges face up and half face down. The area is base × height = πr × r = πr². The more slices you use, the closer the shape gets to a true rectangle, which is the intuition behind the exact result.
Pi itself is the ratio of any circle’s circumference to its diameter. It is irrational, so its decimals never end or repeat: 3.14159265…
Parts of a circle
Semicircles and quarter circles
Take the matching fraction of the full area: a semicircle with radius 8 has area π × 64 ÷ 2 ≈ 100.53, and a quarter circle with radius 12 has area π × 144 ÷ 4 ≈ 113.10.
Sectors (pie slices)
A sector with central angle θ in degrees covers θ/360 of the circle:
A 90° sector of a circle with radius 5 has area ¼ × π × 25 ≈ 19.63. The circle sector calculator also gives the arc length. With the angle in radians, the formula is simply ½r²θ; see how to convert degrees to radians.
Rings (annulus)
For a washer or ring shape, subtract the inner circle from the outer:
A ring with outer radius 5 and inner radius 3 has area π(25 − 9) = 16π ≈ 50.27. The annulus calculator handles this, including the area of a circular path around a garden or pool.
Units and precision
- Square the units too. Radius in feet gives area in square feet. To convert, use the squared factor: 1 ft² = 144 in², and 1 yd² = 9 ft².
- Measure the diameter when possible. It is easier to find the widest point across a circle than to locate its exact center, so many practical measurements start with d.
- Pi approximations. 3.14 is about 0.05% low, and 22/7 is about 0.04% high. For a radius-10 circle, 3.14 gives 314.00 instead of 314.16.
- Ovals are not circles. For an ellipse with semi-axes a and b, the area is πab; use the ellipse calculator.
The formulas for every common flat shape, including circles, are collected on the 2D shape formulas page. To extend a circle into three dimensions, see how to calculate volume, where the cylinder and cone formulas both start from πr².
Frequently asked questions
What is the formula for the area of a circle?
A = πr², where r is the radius and π is about 3.14159. A circle with a radius of 7 inches has an area of π × 49 ≈ 153.94 square inches.
How do you find the area of a circle from the diameter?
Halve the diameter to get the radius and use A = πr², or use A = πd² ÷ 4 directly. A 12-foot-diameter patio has an area of π × 144 ÷ 4 ≈ 113.10 square feet.
How do you find the area from the circumference?
Use A = C² ÷ (4π). A tree trunk with a circumference of 94 inches has a cross-sectional area of 94² ÷ (4π) ≈ 703.1 square inches. Alternatively, find the radius first with r = C ÷ 2π.
What happens to the area if you double the radius?
The area quadruples, because the radius is squared. A 20-inch circle has four times the area of a 10-inch circle, not twice. Tripling the radius multiplies the area by nine.
Is it OK to use 3.14 for pi?
For rough work, yes. Using 3.14 instead of π makes the area about 0.05% too small, which is 0.16 square units for a circle of radius 10. For precise work, use the π key on a calculator.