Area of a Circle: Formula & Examples (With Steps)

The area of a circle is A = πr², where r is the radius and π is about 3.14159. Quick check: a 12-inch pizza has a radius of 6 inches, so its area is 3.14159 × 6² = about 113 square inches. If you only know the diameter, use A = (π/4)d². If you only know the distance around the edge (the circumference), use A = C² ÷ (4π). This guide covers all three, with worked examples you can verify with objects in your own home.

The Area of a Circle Formula

Area measures the flat space inside the circle. It is always given in square units, such as cm², m², in² or ft².

The main formula is:

A = π × r²

What r and π mean

• r (radius) is the distance from the center of the circle to its edge.
• π (pi) is a constant that links a circle’s size to its area and edge. It is roughly 3.14159, and it never ends.

The radius is the part people most often get wrong. The radius is half the diameter. If a plate is 10 inches across, its radius is 5 inches, not 10.

Three ways to write the formula

You will not always be handed the radius. Use the version that matches what you already know.

You know the…	Use this formula	Why
Radius (r)	A = πr²	The standard form
Diameter (d)	A = (π ÷ 4) × d²	Because r = d ÷ 2
Circumference (C)	A = C² ÷ (4π)	Because r = C ÷ (2π)

All three give the same answer for the same circle. They are just written to save you a step.

How to Calculate the Area Step by Step

1. Find the radius. If you were given the diameter, divide it by 2.
2. Square the radius. Multiply r by itself.
3. Multiply by π. Use 3.14159, or the π button on your calculator.
4. Attach the correct square units.
5. Do a quick sanity check. The area should look reasonable for the object’s size.

Worked Examples

Each example below uses either a plain radius or a real object with a published size, so you can repeat the math yourself.

Example 1: Metric (radius given)

A circular garden bed has a radius of 5 cm on a scale drawing.

A = π × r²
A = 3.14159 × 5²
A = 3.14159 × 25
A = 78.54 cm²  (rounded to 2 decimal places)

The bed covers about 78.54 square centimeters.

Example 2: Imperial (a 12-inch pizza)

A “12-inch” pizza is 12 inches across, so the diameter is 12 in and the radius is 6 in.

A = π × r²
A = 3.14159 × 6²
A = 3.14159 × 36
A = 113.10 in²  (rounded to 2 decimal places)

That is why a 12-inch pizza gives you about 113 square inches of food. Bump the size up to a 16-inch pizza and the area jumps to about 201 in², nearly double, because area grows with the square of the radius.

Example 3: Real object, from the diameter (a US quarter)

A US quarter has a diameter of 24.26 mm (per U.S. Mint coin specifications). Grab one and follow along.

First find the radius:

r = d ÷ 2 = 24.26 ÷ 2 = 12.13 mm

Now the area:

A = π × r²
A = 3.14159 × 12.13²
A = 3.14159 × 147.1369
A = 462.24 mm²  (rounded to 2 decimal places)

You can reach the same answer in one line with the diameter formula:

A = (π ÷ 4) × d²
A = 0.785398 × 24.26²
A = 0.785398 × 588.5476
A = 462.24 mm²

Both routes land on 462.24 mm², which is about 4.62 cm². The match is a good sign the math is right.

Example 4: From the circumference (a round table)

Sometimes you can only measure around the edge. Wrap a string or tape around a round dining table and suppose it reads 440 cm.

A = C² ÷ (4π)
A = 440² ÷ (4 × 3.14159)
A = 193,600 ÷ 12.56636
A = 15,406.6 cm²

The tabletop is about 15,407 cm², or roughly 1.54 m². Sanity check: that radius works out to about 70 cm, and π × 70² is about 15,394 cm², which agrees once you allow for rounding.

Which Formula Should You Use?

Most guides show only πr² and stop. In real problems the trouble is not the formula, it is knowing which one fits your data. Use this quick guide.

Situation	Given quantity	Best formula	First step
Drawing shows a radius line	radius	A = πr²	Square the radius
Object measured across the middle	diameter	A = (π ÷ 4) × d²	Or halve d, then use πr²
You wrapped a tape around the edge	circumference	A = C² ÷ (4π)	No need to find r first
Coordinate geometry, two endpoints	endpoints of a diameter	A = (π ÷ 4) × d²	Find d with the distance formula

Pick the row that matches your measurement and you avoid the most common error, which is squaring the wrong number.

From Circle Area to Cylinder Volume

Here is the part general geometry pages skip. The area of a circle is the starting point for the volume of a cylinder.

A cylinder is simply a circle pushed straight up through a height. Find the area of the circular base, then multiply by the height:

V = πr² × h

So πr² is the base, and h stretches that flat disc into a solid. Get the circle area right and the volume is one short step away. If you would rather skip the arithmetic, the free cylinder volume calculator does both steps in one go. You can also read the full cylinder volume formula, or a plain explainer on what a cylinder is if you want to see how the parts of a cylinder fit together.

Common Mistakes to Avoid

• Using the diameter as the radius. Plug a diameter into πr² and your answer will be four times too big. Halve it first.
• Forgetting to square. The r is squared, not just multiplied once. πr and πr² are very different numbers.
• Mixing up area and circumference. Area is the space inside, in square units. Circumference is the distance around, in plain units like cm or inches.
• Rounding π too soon. Keep a few decimals of π until the final step, then round the answer.

Area of a Circle in Different Units

For a circle with a 10 cm radius, the area is π × 100, about 314.16 cm². Here is the same area shown in common units.

Unit	Area of a 10 cm radius circle
cm²	314.16
mm²	31,415.9
m²	0.0314
in²	48.69
ft²	0.338

Conversion keys: 1 cm² = 100 mm², 1 m² = 10,000 cm², and 1 cm² = 0.155 in².

Frequently Asked Questions

What is the formula for the area of a circle?
The area of a circle is A = πr², where r is the radius and π is about 3.14159.

What is πr²?
It means π multiplied by the radius squared. Square the radius first, then multiply by π to get the area.

How do you find the area of a circle with the diameter?
Use A = (π ÷ 4) × d², or halve the diameter to get the radius and use πr². For a 10 cm diameter: r = 5 cm, so A = 3.14159 × 25 = 78.54 cm².

Can you find the area of a circle from the circumference?
Yes. Use A = C² ÷ (4π). For a circumference of 31.4 cm, A = 31.4² ÷ (4 × 3.14159) = 78.5 cm². This is handy when you can measure around an object but not across it.

Why is the area of a circle πr²?
Slice the circle into many thin sectors, like tiny pizza slices, then lay them side by side pointing up and down. They line up into a shape close to a parallelogram. Its base is half the circumference, which is πr, and its height is the radius r. Multiply base by height and you get πr × r, which is πr². Archimedes used a version of this idea over 2,000 years ago.

What units is the area of a circle measured in?
Always square units, such as cm², m², mm², in² or ft². If your answer is not in square units, something went wrong.

What is the area of a circle with radius 5?
A = 3.14159 × 5² = 78.54. If the radius is 5 cm, the area is 78.54 cm².

Is the area the same as the surface area of a circle?
For a flat 2D circle, yes. “Surface area of a circle” usually just means the flat area inside it, which is πr². A sphere is a different shape and uses 4πr².

Should I use 3.14, 3.14159, or 22/7 for pi?
For everyday schoolwork, 3.14 or 22/7 is fine. Note that 22/7 is about 3.142857, a tiny bit high. For accurate results, use 3.14159 or your calculator’s π button.

How does the area of a circle relate to cylinder volume?
The circle’s area is the base of a cylinder. Multiply that base by the height and you get the volume: V = πr² × h. So finding the circle area is the first half of finding a cylinder’s volume.

Sources & Verification

The π value used is 3.14159, rounded where noted. Coin dimensions follow U.S. Mint coin specifications (quarter diameter 24.26 mm). Pizza sizing uses the standard industry format, where the stated size is the diameter. All calculations in this article were worked step by step and cross-checked using more than one formula where possible.

For further reading, see Math is Fun: Area of a Circle, Wolfram MathWorld: Circle, and Khan Academy: Geometry.