Mathematics Notes for Class 10 Areas Related-To-Circles

Areas Related to Circles

In this chapter, you will learn how to find the area and perimeter (circumference) of circles, as well as the areas of sectors and segments. By the end of this chapter, you will be able to solve problems involving circles and their parts using formulas.

Key Concepts

Circle: A round shape with all points at the same distance from the center.
Radius (r): The distance from the center to any point on the circle.
Diameter (d): A line passing through the center, touching two points on the circle. d = 2r
Circumference: The distance around the circle. C = 2πr
Area of Circle: The space inside the circle. A = πr²
Sector: A part of a circle made by two radii and the arc between them.
Segment: A region between a chord and the corresponding arc.

Formulas

Circumference of a Circle: C = 2πr or πd
Area of a Circle: A = πr²
Area of a Sector (with angle θ in degrees): (θ/360) × πr²
Length of an Arc (with angle θ in degrees): (θ/360) × 2πr
Area of a Segment: Area of sector − Area of triangle (formed by the two radii and the chord)

Examples

Example 1: Find the area of a circle with radius 7 cm.
Area = π × 7² = 22/7 × 49 = 154 cm²
Example 2: Find the circumference of a circle with diameter 10 cm.
Circumference = π × 10 = 31.4 cm (use π ≈ 3.14)
Example 3: Find the area of a sector of a circle with radius 6 cm and angle 60°.
Area = (60/360) × π × 6² = (1/6) × π × 36 = 6π ≈ 18.84 cm²

Applications

Finding the area of circular gardens, playgrounds, and fields.
Calculating the length of fences around circular objects.
Solving real-life problems involving wheels, clocks, and circular plates.

Practice Questions

Find the area of a circle with radius 14 cm.
Calculate the circumference of a circle with radius 5 cm.
Find the area of a sector of a circle with radius 10 cm and angle 90°.
A circular park has a diameter of 28 m. What is its area?
Find the length of an arc of a circle with radius 8 cm and angle 45°.

Challenge Yourself

Find the area of a segment of a circle with radius 12 cm and angle 60°.
A wheel makes 100 revolutions. If its radius is 35 cm, how much distance does it cover?

Did You Know?

π (pi) is an irrational number and is approximately 3.14159.
The area of a circle increases four times if the radius is doubled!

Glossary

Radius: Distance from the center to the edge of the circle.
Diameter: Distance across the circle through the center.
Sector: A part of a circle made by two radii and the arc.
Segment: A region between a chord and the arc.
π (pi): A special number used in circle formulas, about 3.14 or 22/7.

Answers to Practice Questions

Area = π × 14² = 22/7 × 196 = 616 cm²
Circumference = 2 × π × 5 = 31.4 cm
Area = (90/360) × π × 10² = (1/4) × π × 100 = 25π ≈ 78.5 cm²
Radius = 14 m, Area = π × 14² = 616 m²
Arc length = (45/360) × 2π × 8 = (1/8) × 16π = 2π ≈ 6.28 cm

Practice solving problems on circles to master geometry and real-life applications!

Mathematics Class 10 - Areas Related-To-Circles Notes