Derivation of Formula for Arc Length of a Sector
Consider a circle with centre O and radius r. Let OAPB be a sector of the circle, and θ° be the angle subtended by the arc at the centre O.
We know that the circumference of the whole circle is given by 2πr. If the subtended angle is 360°, the arc length of the sector is equal to the circumference of the whole circle, which is 2πr.
To find the arc length for any angle θ, we can set up a proportion using the unitary method:
If the subtended angle is 360°, the arc length of the sector is 2πr.
If the subtended angle is θ°, the arc length of the sector is x.
Using proportions we get
θ°/360° = x/2πr
⇒ x = θ°/360° × 2πr
x = θ°/360° × πd
Where d = 2r is the diameter of the circle.
This derives the formula for the arc length of a sector of a circle.
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Sector of a Circle
Sector of a Circle is one of the components of a circle like a segment which students learn in their academic years as it is one of the important geometric shapes. The sector of a circle is a section of a circle formed by the arc and its two radii and it is produced when a section of the circle’s circumference and two radii meet at both extremities of the arc. From a slice of pizza to a region between two fan blades, we can see sectors of the circle in our daily lives everywhere.
In this article, we will explore the geometric shape of the sector which is derived from the circle in detail including its areas, perimeter, and all the formulas related to the sector of a circle.
Table of Content
- What is Sector of a Circle?
- Sector of a Circle Definition
- Sector Angle
- Sector of a Circle Examples
- Sector of a Circle Area
- Formula for Area of a Sector
- Derivation of Formula for Area of a Sector
- Area of Minor Sector
- Area of Major Sector
- Arc Length of Sector of a Circle
- Formula for Arc Length of a Sector
- Derivation of Formula for Arc Length of a Sector
- Sector of a Circle Perimeter
- Perimeter of a Sector Formula
- Sample Problems Sector of a Circle
- Summarizing Important Formulas of Sector of a Circle