This chapter explores how to calculate the areas of two specific regions associated with circles: sectors and segments.
Formulas
- Area of a Circle: A = πr² (where π is a mathematical constant approximately equal to 3.14159 and r is the circle’s radius)
- Area of a Sector:
- A_sector = (θ / 360°) * πr² (or A_sector = θ * πr² / 360)
- θ (theta) represents the central angle of the sector in degrees.
- Area of a Segment:
- A_segment = A_sector – A_triangle
- A_triangle represents the area of the triangle formed by the chord and the two radii drawn from the center of the circle to the chord’s endpoints.
Understanding Sectors
- A sector is a portion of a circle enclosed by two radii and the arc they intercept.
- The central angle (θ) of the sector is the angle formed at the center of the circle by these two radii.
- The ratio between the sector’s central angle (θ) and 360 degrees (or 2π radians) represents the fractional area of the circle that the sector occupies.
Examples
1. Finding the Area of a Sector:
A circle has a radius of 6 cm. A sector of this circle has a central angle of 90°. Find the area of the sector.
Solution:
- Use the formula for the area of a sector: A_sector = (θ / 360°) * πr²
- Substitute the known values: A_sector = (90° / 360°) * π * (6 cm)²
- Calculate the area: A_sector ≈ 8.49 cm² (using a calculator for π)
Therefore, the area of the sector is approximately 8.49 cm².
2. Determining the Area of a Minor Segment:
In the same circle from example 1 (radius 6 cm), a chord is drawn that creates a minor segment (shorter arc between the chord’s endpoints). The central angle associated with the minor segment is also 90°. How can we find the area of this segment?
Solution:
- Follow a two-step approach:
- First, find the area of the sector using the formula from example 1 (A_sector ≈ 8.49 cm²).
- Since this is a minor segment, the corresponding triangle formed by the chord is a right triangle (with a right angle at the center of the circle). We would need the length of the base or height of this triangle to calculate its area (A_triangle).
- Without the triangle’s side lengths, we cannot directly calculate the segment’s area (A_segment) using the formula A_segment = A_sector – A_triangle.
Note: In later chapters, you might learn methods to find properties of triangles formed within circles, which would allow you to solve for the missing side lengths and consequently the area of the triangle and the segment.
3. Calculating the Area of a Major Segment:
Consider the same circle from example 1 (radius 6 cm) again. If we draw a chord that creates a major segment (longer arc between the chord’s endpoints), the central angle associated with this major segment would be more than 180° (but less than 360°).
Solution:
- Similar to the previous example, first find the area of the whole circle using A = πr² (A_circle ≈ 113.09 cm²).
- Then, calculate the area of the minor segment using the formula from example 1 (assuming we know the minor segment’s central angle and can find its area).
- Finally, the area of the major segment (A_segment) can be found using the following relationship:
- A_segment = A_circle – A_minor_segment
1. Using Inscribed Angles
- An inscribed angle is an angle formed inside a circle by two chords that intersect within the circle.
- The measure of an inscribed angle is half the measure of the central angle that intercepts the same arc. (This can be proven using geometric theorems)
Example:
In the circle from previous examples (radius 6 cm), a chord is drawn, and we want to find the area of the segment it creates. We know the central angle of the sector (θ) is 120°, but we lack information about the triangle’s sides.
Solution:
- Identify the Inscribed Angle: The central angle associated with the sector is 120°, and the inscribed angle formed by the chord at the center of the circle will be half of that (120° / 2 = 60°).
- Triangle Properties: Since we know one angle of the triangle (60°) and the triangle is formed within a semicircle (meaning the third angle must be 90°), we can classify it as a 30-60-90 triangle.
- Triangle Side Ratios: In a 30-60-90 triangle, the ratio between the sides is specific: short leg : long leg : hypotenuse = 1 : √3 : 2.
- Relating to Segment: The hypotenuse of the triangle is the chord, and once we know its length, we can find the area of the triangle using the formula for right triangles (A_triangle = 1/2 * base * height).
- Segment Area: Finally, with the triangle’s area (A_triangle), we can apply the formula: A_segment = A_sector – A_triangle to find the area of the segment.
2. Using Pythagorean Theorem
- Applicable when the distance from the center of the circle to the midpoint of the chord (also the perpendicular bisector of the chord) is known.
Example:
Consider the same circle (radius 6 cm) with a chord. We are given that the distance from the center to the midpoint of the chord is 4 cm.
Solution:
- Apply Pythagorean Theorem:
- The distance from the center to the midpoint (4 cm) forms one leg of a right triangle.
- The other leg is half the length of the chord (which we want to find).
- Use the Pythagorean theorem: (radius)² = (distance to midpoint)² + (half chord length)².
- Substitute the known radius value (6 cm) and solve for the half chord length.
- Complete Chord Length: Once you have the half chord length, multiply it by 2 to find the entire chord length.
- Triangle Area: With the chord length, you can potentially find other side lengths of the triangle (using inscribed angles or other geometric relationships) and calculate the triangle’s area (A_triangle) using the right triangle formula.
- Segment Area: Finally, use the formula A_segment = A_sector – A_triangle to find the area of the segment.
This chapter focused on calculating the areas of two specific shapes associated with circles: sectors and segments.
Key Formulas
- Area of a Circle: A = πr²
- Area of a Sector: A_sector = (θ / 360°) * πr² (θ is the central angle)
- Area of a Segment: A_segment = A_sector – A_triangle (A_triangle is the area of the triangle formed by the chord)
Short Notes
- Sectors are portions of a circle enclosed by two radii and their intercepted arc.
- The sector’s area is a fraction of the whole circle’s area based on the ratio of its central angle (θ) to 360°.
- Segments are the regions bounded by a chord and the arc it cuts off. Calculating their area requires the sector’s area and the area of the triangle formed by the chord.
Examples
- We can find the area of a sector directly using the formula if the central angle is known.
- Calculating segment area might require additional information about the triangle formed by the chord (e.g., side lengths or inscribed angles).
Advanced Techniques
- Using Inscribed Angles: Relates the inscribed angle’s measure (half of the central angle) to triangle properties (like 30-60-90 triangles) to find side lengths and the triangle’s area.
- Using Pythagorean Theorem: Applicable when the distance from the centre to the chord’s midpoint is known. This helps solve for the chord length and potentially other triangle side lengths for area calculation.
Practice Questions (with Explanations)
- Find the area of a sector with a central angle of 60° and a radius of 8 cm.
- Explanation: Area of Sector = (60/360) * π * 8² = 8π cm².
- A circle has a radius of 7 cm. If a chord divides the circle into two segments with a central angle of 120°, find the area of the larger segment.
- Explanation: First, calculate the area of the whole circle (πr²). Then, find the area of the smaller sector using (θ/360) * πr². Finally, subtract the smaller sector’s area from the whole circle’s area to get the larger segment’s area.
