Pengi Editor's Note: This article was originally published by Think Academy. We're sharing it here for educational value. Think Academy is a leading K-12 math education provider.
Area of a Circle: Definition, Formula and Example
Definition of a Circle
A circle is the set of all points in a plane that are the same distance from a fixed point.
- Center: The fixed point, usually labeled O.
- Radius (r): The distance from the center to any point on the circle.
- Diameter (d): The distance across the circle through the center, and d=2r.
📌 A circle is a perfectly symmetrical shape — the radius is the same in all directions.
Circle Area Formula: Introduction & Derivation
Formula
To calculate the area of a circle, the formula is:
Area = π × r²
Where:
- r = radius
- π ≈ 3.1416 (an irrational number that represents the ratio of a circle’s circumference to its diameter)
Derivation Method: Rearranging Sectors
- Divide the circle into many small sectors (like pizza slices).
- Rearrange these sectors alternately to form a shape that approximates a parallelogram or rectangle.
- The base of this shape is approximately half the circumference: base = ½ × (2πr) = πr
- The height = radius = r
- So, area = base × height = πr × r = πr²
Example: Applying the Area Formula
Problem
A circle has a radius of 7 cm. Find its area (in terms of π and also as a decimal)?
Solution
Using the formula – Area = π × r², substitute r=7:
Area = π × r² = π × 7² = π × 49 = 49π cm²
Answer
The area of the circle is 49π cm².
If you need an approximate value:
Area = 49π≈ 49 × 3.14 =153.86 cm²
Download Free WorkSheets
Middle School Geometry Resources Ready for You
We’ve created a full library of printable resources — from foundational math to logic-based challenges — organized by grade to support every stage of your child’s journey.
Think Academy Math Program
If you’re looking to build on math strengths or close math gaps, we offer comprehensive K–12 math programs designed to:
- Catch up: Fill in gaps and strengthen foundational skills
- Stay ahead: Maintain a competitive edge and accelerate math progress
- Go further: Take on challenge math competitions, Math kangaroo, AMC competitions
Next Step: Take a personalized evaluation to understand your child’s current math progress and get a tailored roadmap for courses, pacing, and goals.
Try Pengi AI — Smarter Math Practice for Students
Pengi AI supports K–12 learners with personalized math practice, guided explanations, and feedback designed to help them build confidence and improve steadily.
