Area of Arc

An arc is a portion of the circumference of a circle. It can be thought of as the 'curved edge' of a sector of the circle. The area of the arc refers to the area of the sector of the circle that the arc is part of.

The area of the arc is the total space it encloses.

To find the area of an arc, we use the formula: (θ/360) × π × r², where θ is the central angle in degrees, and r is the radius of the circle. Now let’s see how the formula is derived. Derivation of the formula: The circle's total area is π × r².

The arc is part of the circle, proportional to the angle θ compared to the full circle of 360 degrees. Thus, the area of the arc (sector) is (θ/360) × π × r². Therefore, the area of the arc = (θ/360) × π × r²

We can find the area of the arc using this formula, which involves knowing the circle's radius and the central angle. Here’s how:

Method Using the Central Angle and Radius If the central angle θ and radius r are given, we find the area of the arc using the formula: Area = (θ/360) × π × r² For example, if θ is 60 degrees and r is 10 cm, what will be the area of the arc? Area = (60/360) × π × 10² = (1/6) × π × 100 ≈ 52.36 cm² The area of the arc is approximately 52.36 cm²

We measure the area of an arc in square units. The measurement depends on the system used:

In the metric system, the area is measured in square meters (m²), square centimeters (cm²), and square millimeters (mm²).

In the imperial system, the area is measured in square inches (in²), square feet (ft²), and square yards (yd²).

Since an arc is part of a circle, its area calculation involves the central angle and radius. Here are some special cases:

Case 1: Using the Central Angle and Radius If the central angle and radius are given, use the formula: Area = (θ/360) × π × r².

Case 2: Using a Full Circle If the arc is a full circle (θ = 360°), the area becomes the area of the circle itself: Area = π × r².

To make sure that you get correct results while calculating the area of an arc, there are some tips and tricks you should know about.

Here are some: Ensure the angle θ is in degrees when using the formula. The larger the central angle, the larger the area of the arc. If the angle is given in radians, convert it to degrees by multiplying by (180/π).

• Area: The total space occupied by a shape. It tells us how much surface a shape covers.

• Arc: A portion of the circumference of a circle.

• Central Angle: The angle subtended by the arc at the center of the circle.

• Radius: The distance from the center of the circle to any point on its circumference.

• Sector: The region enclosed by two radii of a circle and the arc between them.