Area of Parts of Circle : Sector and Segment

Overview

**Arc Length and Area of a Sector: ** The arc-length and area are proportion to the angle subtended by the sector.

Arc length of a sector of $\theta}^{\circ$ angle $=2\pi r\times \frac{\theta}{360}$

Area of a sector of $\theta}^{\circ$ angle $=\pi {r}^{2}\times \frac{\theta}{360}$
**Area of a Segment**: is the difference between area of sector and the triangle.

Area of a Segment of $\theta}^{\circ$ angle = area of the corresponding sector $-$ the triangle

recap circle

The perimeter and area of a circle of radius $r$ is "$2\pi r$ and $\pi {r}^{2}$"

The angle subtended by a circle on the center is "$360}^{\circ$"

sector

Consider the sector of a circle shown in the figure. The parameters required to define the sector are "the radius and the angle subtended by the sector"

We learned that

Perimeter of a circle is $2\pi r$

Area of a circle is $\pi {r}^{2}$

Angle subtended by the entire circle is $360}^{\circ$

By the symmetry in the circle, it is found that the sector of angle $\theta$ has

Arc length of a sector of $\theta$ is $2\pi r\times \frac{\theta}{360}$. That is, the ratio of angle of the sector to full-angle is the ratio of perimeter of the sector to that of the full-circle.

Area of a sector of $\theta$ is $\pi {r}^{2}\frac{\theta}{360}$. That is, the ratio of angle of the sector to full-angle is the ratio of area of the sector to that of the full-circle.

measures of sector

Consider the sector given in figure. The arc length of minor arc $AB$ is "$2\pi r\times \frac{\theta}{360}$"

The perimeter of the shaded sector $OAB$ is curve plus two radius length = $2\pi r\times \frac{\theta}{360}+2r$

The area of the shaded sector $OAB$ is "$\pi {r}^{2}\times \frac{\theta}{360}$".

The area of the shaded segment is "Area of triangle $OAB$ is subtracted from area of sector $OAB$"

What is the area of a sector of $180}^{\circ$ and segment of $180}^{\circ$ in a circle?

The answer is "area of sector equals the area of segment $=\pi r$"

summary

**Arc Length and Area of a Sector: **
The arc-length and area are proportion to the angle subtended by the sector.

Arc length of a sector of $\theta}^{\circ$ angle $=2\pi r\times \frac{\theta}{360}$

Area of a sector of $\theta}^{\circ$ angle $=\pi {r}^{2}\times \frac{\theta}{360}$

**Area of a Segment**: is the difference between area of sector and the triangle.
Area of a Segment of $\theta}^{\circ$ angle = area of the corresponding sector $-$ the triangle

Outline

The outline of material to learn *Mensuration : Length, Area, and Volume* is as follows.

Note 1: * click here for the detailed overview of Mensuration High *

Note 2: * click here for basics of mensuration, which is essential to understand this. *

• ** Basics of measurement**

→ __Summary of Measurement Basics__

→ __Measurement by superimposition__

→ __Measurement by calculation__

→ __Measurement by equivalence__

→ __Measurement by infinitesimal pieces__

→ __Cavalieri's Principle (2D)__

→ __Cavalieri's Principle (3D)__

• **Perimeter & Area of 2D shapes**

→ __Circumference of Circles__

→ __Area of Circles__

• **Surface area & Volume of 3D shapes**

→ __Prisms : Surface Area & Volume__

→ __Pyramids : Surface Area & Volume__

→ __Cone : Surface Area & Volume__

→ __Sphere : Surface Area & Volume__

• **Part Shapes **

→ __Understanding part Shapes__

→ __Circle : Sector and Segment__

→ __Frustum of a Cone__