Key takeaway: The area of a circle is the number of square units inside it, found by multiplying pi by the square of the radius.
Grade 77.G.B.4Geometry and Measurement
Definition of Area of a Circle
The area of a circle measures the flat space enclosed by its edge: the amount of grass inside a circular lawn or the cheese on a round pizza. It is counted in square units, such as square meters.
The formula comes from a neat fact. A square built on the radius has area \(r^2\), and exactly π of those squares, about 3.14, fit inside the circle. So the area equals π times the radius squared. Because the radius is squared, doubling it makes the area four times as big.
Formula and notation
With radius \(r\):
\[A = \pi r^2\]
When only the diameter is given, halve it first: \(A = \pi (\frac{d}{2})^2\). The answer is in square units, such as \(\text{m}^2\).
Key facts
- In the area of a circle, only the radius is squared; the π factor is not squared.
- Tripling the radius multiplies the area by 9, since \(3^2 = 9\).
- Using the diameter by mistake makes the answer four times too large.
- The area of a circle is always in square units, because it covers a surface.
Area of a Circle example
A circular herb garden has a diameter of 12 meters. Find its area using 3.14 for pi.
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Halve the diameter to get the radius: \(r = 12 : 2 = 6\) meters.
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Square the radius: \(6^2 = 36\).
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Multiply: \(A \approx 3.14 \times 36 = 113.04\) square meters.
The garden covers about 113.04 square meters, or exactly \(36\pi\) square meters.
Common mistake
Watch out
Using the diameter in place of the radius, Ahmed writes \(A = 3.14 \times 12^2 = 452.16\) square meters.
Why it is wrong
The formula calls for the radius, which is 6. Using 12 makes the result four times too large: \(452.16 : 4 = 113.04\). Problems often hand you the diameter, because it is what you can measure across a garden or a table, so halve it before squaring. A sanity check also helps: the garden fits inside a 12-by-12 square of area 144, so an answer of 452 is impossible.
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Geometry and Measurement