Mensuration
154. Area of a Circle · πr²: The magic formula
The rectangle's area always equals the circle's area.
Area of a circle = π × r². The classic visualization: cut the circle into thin pizza-slices and rearrange them into an approximate rectangle of width π·r (half the circumference) and height r. Area = πr² regardless of orientation.
What this lesson covers
Try to break it
Drag R outward to grow the radius or inward to shrink it. The rectangle's width stays at πr and height at r, so its area πr² always equals the circle's area. Try to break the equality by dragging R — you can't.
How you build it
Draw a circle and an unrolled rectangle.
- Mark point O — the centre of your circle.
- Draw a circle centred at O — drag outward to set the radius r. The area will equal π r².
The proof, step by step
Prove that the area of a circle is π × radius².
- Cut the circle into many equal sectors.
- Arrange the sectors alternately (up-down) to form a shape.
- The shape approximates a rectangle with length = half circumference = πr.
- The width of this rectangle equals the radius r.
- Area of rectangle = length × width = πr × r = πr².
Worked example
Find the area of a circle whose radius is 7 cm. (Take π = 22/7)
Area = πr² = (22/7) × 7 × 7 = 154 cm².
- 44 cm²
- 154 cm² — correct
- 22 cm²
- 110 cm²