Area of Arc
2026-02-28 09:04 Diff
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Last updated on September 17, 2025

The area is the space inside the boundaries of a two-dimensional shape or surface. There are different formulas for finding the area of various shapes/figures. These are widely used in architecture and design. In this section, we will find the area of an arc.

What is the 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.

Area of the Arc Formula

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

How to Find the Area of Arc?

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

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Unit of Area of Arc

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).

Special Cases or Variations for the Area of Arc

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.

Tips and Tricks for Area of Arc

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).

Common Mistakes and How to Avoid Them in Area of Arc

It is common for people to make mistakes while finding the area of an arc. Let’s take a look at some mistakes made.

Problem 1

The central angle θ of an arc is 90 degrees, and the radius r is 7 m. What will be the area?

Okay, lets begin

We will find the area as 38.48 m²

Explanation

Here, the central angle θ is 90 degrees, and the radius r is 7 m.

The area of the arc = (90/360) × π × 7² = (1/4) × π × 49 ≈ 38.48 m²

Well explained 👍

Problem 2

What will be the area of the arc if the central angle is 120 degrees and the radius is 5 cm?

Okay, lets begin

We will find the area as 26.18 cm²

Explanation

If the central angle is 120 degrees and the radius is 5 cm, we use the formula:

Area = (120/360) × π × 5² = (1/3) × π × 25 ≈ 26.18 cm²

Well explained 👍

Problem 3

The area of an arc is 78.54 m² and the radius is 10 m. Find the central angle.

Okay, lets begin

We find the central angle θ as 90 degrees

Explanation

To find the central angle, use the formula:

Area = (θ/360) × π × 10².

Here, the area is 78.54 m², and the radius is 10 m: 78.54 = (θ/360) × π × 100 θ = (78.54 × 360)/(π × 100) θ ≈ 90 degrees

Well explained 👍

FAQs on Area of Arc

1.Is it possible for the area of the arc to be negative?

No, the area of the arc can never be negative. The area of any shape will always be positive.

2.How to find the area of an arc if the central angle and radius are given?

If the central angle and radius are given, find the area using the formula: Area = (θ/360) × π × r².

3.What if the central angle is given in radians?

If the central angle is in radians, convert it to degrees by multiplying by (180/π) before using the formula.

4.What is meant by the area of the arc?

The area of the arc is the total space occupied by the sector of the circle that the arc is part of.

Seyed Ali Fathima S

About the Author

Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.

Fun Fact

: She has songs for each table which helps her to remember the tables