Area of a Circle Formula: A = πr² Explained Simply
The area of a circle formula is A = πr²: multiply pi (about 3.14) by the radius, then by the radius again. In this guide you will see what each letter means, why the formula works, three worked examples, and the mistakes that cost students marks.
Circle
Every point on a circle is the same distance r from the centre. π (pi) ≈ 3.14159 links the radius to the circumference and area.
- Area
- A = πr2
- Circumference
- C = 2πr
- Diameter
- d = 2r
- r radius
- d diameter
- π ≈ 3.14159
What the formula means
- A is the area: the amount of flat space inside the circle. It is always in square units, such as cm² or m².
- π (pi) is a fixed number, about 3.14159. It is the same for every circle, big or small.
- r is the radius: the distance from the centre of the circle to its edge.
- r² means r × r (radius times radius), not 2 × r.
If you are given the diameter (d) instead, halve it first: r = d ÷ 2. You can also use the diameter version of the formula: A = πd² ⁄ 4.
Why A = πr² works
Cut a circle into many thin slices, like a pizza, and lay them side by side, pointing up and down in turn. The slices form a shape that looks more and more like a rectangle as the slices get thinner. Its height is the radius r, and its length is half the circumference: ½ × 2πr = πr. Area of a rectangle = length × height = πr × r = πr².
Worked example 1: given the radius
A circle has a radius of 7 cm. Find its area.
- Write the formula: A = πr²
- Substitute: A = π × 7²
- Square the radius: 7² = 49
- Multiply: A = π × 49 = 153.94 cm² (to 2 decimal places)
Worked example 2: given the diameter
A round table has a diameter of 10 cm on a scale drawing. Find the area.
- Find the radius: r = 10 ÷ 2 = 5 cm
- Substitute: A = π × 5²
- Square: 5² = 25
- Multiply: A = π × 25 = 78.54 cm²
Worked example 3: working backwards from the area
A circle has an area of 50 cm². What is its radius?
- Start with the formula: 50 = πr²
- Divide both sides by π: r² = 50 ÷ π = 15.92
- Take the square root: r = √15.92 = 3.99 cm
Check your answer with the circle calculator
Area of a circle in real life: the pizza test
Is one 12-inch pizza more food than two 8-inch pizzas? Pizza sizes are diameters, so halve them first.
- One 12-inch pizza: r = 6, A = π × 6² = 113.10 in²
- Two 8-inch pizzas: r = 4, A = 2 × π × 4² = 100.53 in²
The single 12-inch pizza is bigger, because area grows with the square of the radius. Doubling the radius makes the area four times larger.
5 common mistakes with the area of a circle
- Using the diameter as the radius. Always halve the diameter first.
- Calculating 2r instead of r². r² means r × r.
- Mixing up area and circumference. Area is πr² (square units); circumference is 2πr (plain units).
- Forgetting square units. Write cm², m² or in², not cm.
- Rounding π too early. Using 3.14 for r = 7 gives 153.86 instead of 153.94. Use the π button and round only at the end.
Try it yourself
A circular garden pond has a radius of 3 m. What is its area?
Show the answer
A = π × 3² = π × 9 = 28.27 m²
Area of a circle: FAQ
What is the formula for the area of a circle?
A = πr², where r is the radius. With the diameter, use A = πd² ⁄ 4.
How do you find the area of a circle with the diameter?
Divide the diameter by 2 to get the radius, then use A = πr². For a diameter of 10 cm: r = 5 cm and A = 78.54 cm².
Should I use 3.14 or the π button?
Use the π button on your calculator unless the question tells you to use 3.14 or 22⁄7. It gives a more accurate answer.
What units is the area of a circle measured in?
Square units, because area measures a flat surface: mm², cm², m², in² or ft².
More circle formulas (circumference, diameter, arc length), a calculator and a printable sheet are on the circle formulas page. For every other shape, see the full list of geometry formulas.