1. Areas of Sector and Segment of a Circle
A slice of pizza looks like a sector. When one end of a sector is joined with the other end, a cone is formed. Let's look at the formula to compute the area of the sector and its corresponding segment.
Sector of a circle
A sector of a circle is the portion (or part) of the circular region enclosed by two radii and the corresponding arc.
- OAPB is called the minor sector
- OAQB is called the major sector
- ∠ AOB is called the angle of the minor sector
- Angle of the major sector is 360° – ∠ AOB
Area of the sector of angle span id="MathJax-Element-1-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="">θθ = span id="MathJax-Element-2-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="()360">(θ)360(θ)360x span id="MathJax-Element-3-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="r2">πr2πr2 (span id="MathJax-Element-4-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="">θθ is in degrees).
Length of the arc of a sector of angle span id="MathJax-Element-5-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="">θθ = span id="MathJax-Element-6-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="()360">(θ)360(θ)360x 2span id="MathJax-Element-7-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="r">πrπr
Segment of a circle
The portion (or part) of the circular region enclosed between a chord and the corresponding arc.
Areas of sectors and segments
- APB is called the minor segment
- AQB is called the major segment
Area of segment of a circle = Area of the corresponding sector – Area of the corresponding triangle
2. Areas of Combinations of Plane Figures
Suppose a square is drawn and a semicircle is drawn on each side of the square with the diameter as the side of the square. What would be the area of the figure drawn?
We should add the area of the square and the area of the 4 semi-circles.
In reality, there will be a lot of situations where you will see a combination of figures. For example, the walls of a room are rectangles, doors and windows are also rectangles. If we are trying to find the area to be painted, we would subtract the area of the windows and doors from the area of the wall.
Area of combinations of plane figures
Some standard plane figures with shaded area marked:
To find the area of the shaded region in
1. Shape A: find the difference between the area of the larger circle and the area of the smaller circle
2. Shape B: find the area of the square and then subtract the area of the circle
3. Shape C: Subtract the area of the rectangle from the area of the circle
4. Shape D: Difference between the area of the rectangle and the triangle gives the area of the shaded region