1.3 Area of Sector of a Circle
A sector of a circle is the region bounded by an arc and two radii. When dealing with circular objects, such as cutting slices of a pizza, measuring regions on a decorative table cloth, or determining the cross-sectional area of water flowing in a pipe, understanding how to calculate the area of sectors and segments is essential. In circular measure, using angles measured in radians simplifies the relationship between the radius, angle subtended at the centre, and the resulting area.
Learning Objectives
- Determine the area of a sector, radius, and the angle subtended at the centre of a circle.
- Determine the area of a segment of a circle using sector and triangle formulae.
- Solve real-world problems involving areas of sectors and segments of circles.
Components of a Circular Sector and Segment
A segment is obtained by subtracting the area of a triangle from the area of a sector.
Short Explanation
Use radians to calculate sector and segment areas accurately.
1. Area of a Sector in Radians
When the central angle \(\theta\) is measured in radians, the area of a sector is \(A = \dfrac{1}{2}r^2\theta\).
2. Area of a Sector Using Arc Length
Since \(s = r\theta\), the area can also be written as \(A = \dfrac{1}{2}rs\).
3. Area of a Sector in Degrees
If the angle is given in degrees, use \(A = \dfrac{\theta}{360^\circ} \times \pi r^2\).
4. Area of a Segment
Area of segment \(=\) Area of sector \(-\) Area of triangle.
5. Area of Triangle Methods
Use \(\dfrac{1}{2}ab\sin C\), \(\dfrac{1}{2} \times \text{base} \times \text{height}\), or Heron's formula \(\sqrt{s(s-a)(s-b)(s-c)}\).
6. Finding Radius or Angle
From \(A = \dfrac{1}{2}r^2\theta\): \(r = \sqrt{\dfrac{2A}{\theta}}\) and \(\theta = \dfrac{2A}{r^2}\text{ rad}\).
| Quantity | Formula | Condition |
|---|---|---|
| Area of sector | \(A = \dfrac{1}{2}r^2\theta\) | \(\theta\) in radians |
| Area of sector | \(A = \dfrac{1}{2}rs\) | Arc length \(s\) known |
| Area of sector | \(A = \dfrac{\theta}{360^\circ} \times \pi r^2\) | \(\theta\) in degrees |
| Area of segment | \(A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}}\) | Chord and arc bound the segment |
Steps for Solving Problems Involving Areas of Sectors
Identify known dimensions: radii, chord lengths, ratios, and sector angles.
Plan the strategy: identify composite regions and convert angles to radians if needed.
Substitute values accurately into \(A = \dfrac{1}{2}r^2\theta\) or appropriate trigonometric formulae.
Check and reflect using alternative methods such as degree ratios.
Area of a sector in radians
\(A = \dfrac{1}{2}r^2\theta\)\(\theta\) must be in radians.
Area of a segment
\(A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}}\)Subtract the triangle from the sector.
Try to Answer First
Answer in your mind, then press "Check Answer".
Find the area of a sector of a circle with a radius of \(12\text{ cm}\) and a subtended central angle of \(1.7\text{ rad}\).
A sector \(POQ\) with centre \(O\) has an area of \(35\text{ cm}^2\) and a central angle \(\theta = 0.7\text{ rad}\). Find the radius \(r\) of the sector.
A circle with centre \(O\) has a radius of \(6\text{ cm}\). A chord \(PQ\) subtends an angle of \(2.2\text{ rad}\) at the centre. Given \(\pi = 3.142\), find the area of the segment bounded by chord \(PQ\) and minor arc \(PQ\).
Common Mistakes
- Substituting an angle given in degrees directly into \(A = \dfrac{1}{2}r^2\theta\) without converting it to radians first.
- Forgetting to set the scientific calculator to Radian mode when evaluating \(\sin\theta\) directly from an angle given in radians.
- Confusing the area of a sector (bounded by two radii and an arc) with the area of a segment (bounded by a chord and an arc).
- Forgetting to square the radius in the sector formula, writing \(\dfrac{1}{2}r\theta\) instead of \(\dfrac{1}{2}r^2\theta\).
Concept Misunderstandings
The formula \(A = \dfrac{1}{2}r^2\theta\) can be used with any angle measurement, whether in degrees or radians.
\(A = \dfrac{1}{2}r^2\theta\) is derived specifically from the full turn of \(2\pi\) radians. If the angle is in degrees, use \(A = \dfrac{\theta}{360^\circ} \times \pi r^2\) or convert the angle to radians first.
The formula \(A = \dfrac{1}{2}rs\) is an approximation.
It is an exact formula derived directly by substituting \(s = r\theta\) into \(A = \dfrac{1}{2}r^2\theta = \dfrac{1}{2}r(r\theta) = \dfrac{1}{2}rs\).
Summary
- Area of sector in radians: \(A = \dfrac{1}{2}r^2\theta\).
- Area of sector using arc length: \(A = \dfrac{1}{2}rs\).
- Area of sector in degrees: \(A = \dfrac{\theta}{360^\circ} \times \pi r^2\).
- Area of segment \(=\) Area of sector \(-\) Area of triangle.
- Area of triangle: \(\dfrac{1}{2}ab\sin C\) or \(\sqrt{s(s-a)(s-b)(s-c)}\) where \(s = \dfrac{a+b+c}{2}\).
Short Activity
Objective: Identify sector and segment area formulae and solve related problems.
A. Objective Quiz
1 What is the area of a sector of a circle with a radius of \(8\text{ cm}\) and a subtended angle of \(2.2\text{ rad}\) at the centre?
2 Given that a sector of a circle has an arc length \(s = 14\text{ m}\) and a radius \(r = 16\text{ m}\), what is the area of the sector?
B. Fill in the Blanks
3 If a sector has a radius of \(r\) and subtends an angle of \(\theta\) radians at the centre, the formula for its area is \(A = \dfrac{1}{2}r^2\) ________.
4 The area of a segment of a circle is calculated by subtracting the area of the triangle from the area of the corresponding ________.
C. Matching / Drag and Drop
Drag each concept or formula name to the correct mathematical expression or definition. On a phone, tap a card first, then tap the matching box.
