Area of a Trapezium and a Polygon
196. The Circle's Area · how radius unlocks the formula
The area formula πr² holds for any radius size.
Area of a circle = πr². Imagine slicing the circle into thin wedges and rearranging them into a near-rectangle of width πr (half circumference) and height r. The constant π ≈ 3.14159 is the ratio of any circle's circumference to its diameter.
What this lesson covers
Try to break it
Drag R to grow or shrink the radius. The area readout always equals πr². Try to find a radius where the displayed area doesn't match the formula — you can't. πr² is locked in for every circle.
How you build it
Draw a circle with a radius OR.
- Mark point O at the centre.
- Use the compass to draw a circle with centre O.
- Draw segment RO to mark the radius.
The proof, step by step
Prove that the area of a circle is π × radius².
- Cut the circle into many equal sectors (like pizza slices).
- Rearrange the sectors alternately upside down to form a shape resembling a rectangle.
- The shape's height is r and width is πr (half the circumference). Area = r × πr = πr².
Worked example
Find the area of a circle whose radius is 14 cm. (Take π = 22/7)
Using the formula Area = πr², substitute r = 14 and π = 22/7. Area = (22/7) × 14 × 14 = 616 cm².
- 440 cm²
- 616 cm² — correct
- 308 cm²
- 154 cm²