The area of a circle measures how much flat space is inside its boundary. It is one of the most important formulas in geometry because circles appear in wheels, pipes, plates, planets, and many designs. The formula A = πr^2 lets you find the area when you know the radius.
Understanding where the formula comes from makes it easier to remember and use correctly.
A circle can be divided into many thin wedge-shaped sectors and rearranged into a shape that looks like a parallelogram. The height of this new shape is about the radius r, and the base is about half the circumference, or πr. Since the area of a parallelogram is base times height, the circle area is A = πr × r = πr^2.
As the slices become thinner, the rearranged shape becomes closer to a true parallelogram or rectangle, making the formula more exact.
Key Facts
- Area of a circle: A = πr^2
- Circumference of a circle: C = 2πr
- Diameter and radius: d = 2r and r = d/2
- In the slicing model, the rearranged base is half the circumference: base = C/2 = πr
- Parallelogram area: A = base × height
- If the radius is doubled, the area becomes 4 times as large because A depends on r^2
Vocabulary
- Radius
- The radius is the distance from the center of a circle to any point on the circle.
- Diameter
- The diameter is the distance across a circle through its center, equal to twice the radius.
- Area
- Area is the amount of two-dimensional space inside a shape.
- Circumference
- Circumference is the distance around the outside of a circle.
- Sector
- A sector is a wedge-shaped part of a circle bounded by two radii and an arc.
Common Mistakes to Avoid
- Using the diameter instead of the radius in A = πr^2. This is wrong because the formula requires r, so a diameter of 10 gives r = 5, not r = 10.
- Forgetting to square the radius. This is wrong because circle area grows with r^2, so A = πr is not an area formula.
- Confusing area and circumference. Area uses square units and A = πr^2, while circumference uses linear units and C = 2πr.
- Writing the final answer without square units. This is wrong because area measures two-dimensional space, so units should be cm^2, m^2, in^2, or another squared unit.
Practice Questions
- 1 A circle has radius 6 cm. Find its area in terms of π, then approximate using π = 3.14.
- 2 A circular garden has diameter 14 m. Find its area using π = 22/7.
- 3 Explain why rearranging many thin circle sectors into a parallelogram helps show that the area of a circle is A = πr^2.