Areas Related to Circles • Topic 3 of 3

Areas of segment of a circle

What is a segment of a circle? A segment is a region of a circle bounded by a straight chord and an arc. Unlike a sector, a segment does not connect back to the center of the circle. Think of cutting a small rounded piece off the side of a circular log with a single straight saw cut, or looking at the water line when a circular glass cup is tipped sideways.

Segments come in pairs:

Minor Segment: The smaller region chopped off by the chord line.
Major Segment: The massive remaining region of the circle left on the other side of the chord line.

How to Calculate the Area of a Segment We cannot find the area of a segment directly using a single basic formula. Instead, we use subtraction:

1. First, find the area of the entire sector connecting the chord ends to the circle's center point. This looks like a complete pizza slice. 2. Next, calculate the area of the triangle formed inside that slice by the two radii lines and the straight chord line. 3. Finally, subtract the area of the triangle from the area of the sector. The leftover curved piece on the edge is your segment!

Mathematical Subtraction Step Rule:

$$\text{Area of Minor Segment} = \text{Area of Sector } OAPB - \text{Area of Triangle } OAB$$

To find the area of the interior triangle with radius $r$ and central angle $\theta$, you can use the formula: $\frac{1}{2} \cdot r^2 \cdot \sin(\theta)$.

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DIAGRAM 1: THE GEOMETRIC FORMATION OF A SEGMENT

                    . - - ~ ~ - - .
                .                   .
              /                       \
             /       Major              \
            ;       Segment              ;
             \        O (Center)        /
              \      / \               /
                .   /   \            .
                  / _ _ _ \ <--- Chord Line
                 |________|
               Minor Segment

DIAGRAM 2: THE SUBTRACTION PROCESS FLOW
  
     [ Full Sector Slice ]   minus   [ Interior Triangle ]   equals   [ Edge Segment ]
              /\                             /\
             /  \                           /  \
            /    \                         /____\
           /_ _ _ \                                                   ______
          (________)                                                 (______)

Solved Examples

Worked Example

A straight chord of a circle with a radius of 10 cm forms a right angle (90 degrees) at the center of the circle. Find the exact area of the corresponding minor segment. (Use $\pi = 3.14$).

Solution

Step 1: Calculate the area of the complete minor sector.
Radius ($r$) = 10 cm, Angle ($\theta$) = 90 degrees.
Area of Sector = $(90 / 360) \cdot \pi \cdot r^2 = (1 / 4) \cdot 3.14 \cdot 10 \cdot 10$
Area of Sector = $(1 / 4) \cdot 314 = 78.5$ square cm.
Step 2: Calculate the area of the interior triangle.
Since the central angle is 90 degrees, the triangle is a right-angled triangle where the two radii act as the base and height.
Area of Triangle = $(1 / 2) \cdot \text{base} \cdot \text{height} = (1 / 2) \cdot r \cdot r$
Area of Triangle = $(1 / 2) \cdot 10 \cdot 10 = 50$ square cm.
Step 3: Subtract the triangle area from the sector area.
Area of Segment = Area of Sector - Area of Triangle
Area of Segment = $78.5 - 50 = 28.5$ square cm.
Answer: The area of the minor segment is 28.5 square cm.

Worked Example

A chord inside a circle of radius 14 cm subtends an angle of 60 degrees at the center. Find the area of the minor segment. (Take $\pi = 22/7$ and $\sqrt{3} = 1.73$).

Solution

Step 1: Compute the area of the matching circle sector.
Area of Sector = $(\theta / 360^\circ) \cdot \pi \cdot r^2 = (60 / 360) \cdot (22 / 7) \cdot 14 \cdot 14$
Area of Sector = $(1 / 6) \cdot 22 \cdot 2 \cdot 14 = (1 / 6) \cdot 616 = 102.67$ square cm.
Step 2: Compute the area of the interior triangle.
Since the radii are equal and the central angle is 60 degrees, the triangle is an equilateral triangle.
Area of Equilateral Triangle = $(\sqrt{3} / 4) \cdot \text{side}^2 = (\sqrt{3} / 4) \cdot r^2$
Area of Triangle = $(1.73 / 4) \cdot 14 \cdot 14 = (1.73 / 4) \cdot 196$
Area of Triangle = $1.73 \cdot 49 = 84.77$ square cm.
Step 3: Perform the final geometric subtraction.
Area of Segment = Area of Sector - Area of Triangle
Area of Segment = $102.67 - 84.77 = 17.9$ square cm.
Answer: The area of the minor segment is 17.9 square cm.

Worked Example

Find the area of the major segment formed by a chord in a circle of radius 7 cm, if the area of its corresponding minor segment is found to be 14 square cm. (Take $\pi = 22/7$).

Solution

Step 1: Understand the geometric relationship for major segments.
The entire circle area is split into two regions by a chord: the minor segment and the major segment.
Therefore: $\text{Area of Major Segment} = \text{Total Area of Circle} - \text{Area of Minor Segment}$.
Step 2: Calculate the total area of the complete circle.
Total Circle Area = $\pi \cdot r^2 = (22 / 7) \cdot 7 \cdot 7$
Total Circle Area = $22 \cdot 7 = 154$ square cm.
Step 3: Subtract the given minor segment value.
Area of Major Segment = $154 - 14 = 140$ square cm.
Answer: The area of the major segment is 140 square cm.
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Key Points

A segment of a circle is the space trapped between a straight chord line and an outer arc.
Segments do not naturally touch the center point of the circle on their own.
To find a minor segment's area, you must calculate Area of Sector minus Area of Triangle.
If the central angle is $90^\circ$, the interior triangle area simplifies to $\frac{1}{2}r^2$.
If the central angle is $60^\circ$, the interior triangle becomes equilateral with an area of $\frac{\sqrt{3}}{4}r^2$.
The major segment area is found by subtracting the minor segment area from the circle's total area.

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.The area of a minor segment is:

Explanation: Segment $=$ sector $-$ triangle.

Q2.The area of a major segment is:

Explanation: Whole circle minus the minor segment.

Q3.A chord divides a circle into:

Explanation: Two segments.

Q4.Segment area uses a sector and a:

Explanation: The triangle formed by the two radii and the chord.

Practice more MCQs → 📝 Assignment · 35 marks