How to Calculate the Area of a Circle
Which gives you more pizza: one 18-inch pizza, or two 12-inch pizzas?
Two 12-inch pizzas feel like more. They're not. One 18-inch pizza has about 254 square inches. Two 12-inch pizzas together have about 226 square inches. The single large pizza wins by more than 12%.
That surprise comes straight from the area formula for a circle. Here's how to use it, whether you know the radius, the diameter or the circumference.
Area Grows With the Square of the Size
When a circle's diameter grows by 50%, from 12 inches to 18 inches, its area doesn't grow by 50%. It grows by 125%, because area depends on the radius squared:
(18 / 12)² = 1.5² = 2.25
So one 18-inch pizza has 2.25 times the area of one 12-inch pizza. Two 12-inch pizzas only give you 2 times. Menus price pizzas by diameter, but you eat area.
The Formula
A = π × r²
- A is the area
- r is the radius, the distance from the center to the edge
- π ≈ 3.14159
The result is in square units: square inches, square meters, and so on. See the geometry formulas.
Example 1: From the Radius
A circular garden has a radius of 5 meters.
A = π × 5² = π × 25 ≈ 78.54 m²
Example 2: From the Diameter
The diameter is twice the radius, so first divide by 2, or use the diameter version of the formula:
A = π × (d/2)² = π × d² / 4
An 18-inch pizza:
r = 18 / 2 = 9 inches
A = π × 9² = π × 81 ≈ 254.47 in²
A 12-inch pizza:
r = 6 inches
A = π × 36 ≈ 113.10 in²
Two of them ≈ 226.19 in²
Check the squares with the square numbers list.
Example 3: From the Circumference
If you can only measure around the outside, use:
A = C² / (4π)
A tree trunk with a circumference of 31.4 inches:
A = 31.4² / (4π) = 985.96 / 12.566 ≈ 78.46 in²
This works because C = 2πr, so r = C / (2π), and substituting into πr² gives C² / (4π).
Working Backward: From Area to Radius
Rearrange the formula:
r = √(A / π)
A circular rug covering 50 ft² has a radius of:
r = √(50 / π) = √15.92 ≈ 3.99 ft
So its diameter is about 8 feet. Use the square roots list or the scientific calculator.
Partial Circles
Semicircle: half the area, πr² / 2.
Sector (a pizza slice) with a central angle θ in degrees:
A = (θ / 360) × π r²
A 45° slice of an 18-inch pizza: (45/360) × 254.47 ≈ 31.8 in².
Ring (annulus) between radii R and r:
A = π(R² − r²)
Where You'll Use It
- Home projects: how much seed, mulch or paint a round garden bed, patio or tabletop needs
- Cooking: swapping a 9-inch round cake pan (about 63.6 in²) for an 8-inch one (about 50.3 in²) means roughly 21% less batter
- Engineering: the cross-sectional area of a pipe determines how much water it can carry; doubling its diameter quadruples its capacity
- Irrigation: a sprinkler that throws water 10 meters covers about 314 m²
An Insider Reference: The Egyptian Shortcut
The Rhind Mathematical Papyrus, copied by a scribe named Ahmes around 1550 BC from an older text, includes a method for finding the area of a circular field: take the diameter, subtract one-ninth of it, and square the result.
A ≈ (8d/9)²
For a diameter of 9, that gives 8² = 64. The true area is π × 4.5² ≈ 63.62. The Egyptian rule is off by just 0.6%. It's equivalent to using π ≈ 256/81 ≈ 3.1605, remarkably accurate for over 3,500 years ago.
About 1,300 years later, Archimedes proved that a circle's area equals a triangle with the radius as its height and the circumference as its base, which is exactly πr². See Why Does the Area of a Circle Equal πr²?
Common Mistakes
1. Using the diameter as the radius. This makes the answer four times too big.
2. Squaring π instead of r. It's π × r², not (π × r)².
3. Mixing up area and circumference. Circumference is 2πr, a length. Area is πr², square units.
4. Forgetting square units. 78.54 m², not 78.54 m.
Two Concepts Worth Knowing
Pi
π is the ratio of any circle's circumference to its diameter, about 3.14159. It never ends or repeats. See Pi.
Scaling
When every length of a shape is multiplied by k, its area is multiplied by k². Doubling a circle's radius quadruples its area.
Quick Answer: How Do You Calculate the Area of a Circle?
Use A = πr², where r is the radius. For a radius of 5 m, A = π × 25 ≈ 78.54 m². If you know the diameter, halve it first or use A = πd²/4. If you know the circumference C, use A = C²/(4π).
Try Them Yourself
- Geometry Formulas: circle, sector and ring formulas
- Square Numbers List: quick r² values
- Square Roots List: work backward from area
- Pi: the constant in every circle formula
- Area Unit Converter: convert square inches, feet and meters
- How to Calculate the Volume of a Sphere: the 3D follow-up
Next time you order pizza, compare the price per square inch of each size. The largest pizza is almost always the best deal.