Area of a Segment of a Circle: Practice Problems
The area of a segment of a circle is the region bounded by a chord and the corresponding arc. In real terms, this concept appears frequently in geometry, engineering, and design, where calculating the space between a straight line and a curved edge is essential. Whether you are solving textbook problems or tackling real‑world applications, mastering the formulas and step‑by‑step approach will help you find the exact area quickly and accurately Not complicated — just consistent. Worth knowing..
Introduction
A circular segment is formed when a chord cuts a circle, creating two distinct parts: a minor segment (the smaller region) and a major segment (the larger region). The key variables are the radius (r) of the circle and the central angle (θ) in degrees or radians. To compute the area of a segment, you must first determine the area of the corresponding sector and then subtract the area of the isosceles triangle formed by the two radii and the chord. By following a systematic method, you can solve any practice problem involving the area of a segment, whether the angle is given in degrees or radians, and whether you need the area of the minor or major segment The details matter here..
Understanding the Concept
The area of a sector is proportional to its central angle. If the central angle is θ (in degrees), the sector’s area is:
Area of sector = (θ / 360) × πr²
If θ is expressed in radians, the formula simplifies to:
Area of sector = (1/2) θ r²
The triangle inside the sector is isosceles, with two sides equal to the radius and the included angle equal to the central angle. Its area can be found using:
Area of triangle = (1/2) r² sin θ (when θ is in radians)
or
Area of triangle = (1/2) r² sin(θ°) (when θ is in degrees)
The area of the segment is then:
Area of segment = Area of sector – Area of triangle
For the major segment, simply subtract the minor segment’s area from the total circle area (πr²).
Step‑by‑Step Method to Find the Area
- Identify the given values – radius (r) and central angle (θ). Note whether θ is in degrees or radians.
- Calculate the sector area using the appropriate formula.
- Compute the triangle area with the sine formula.
- Subtract the triangle area from the sector area to obtain the minor segment area.
- If needed, find the major segment by subtracting the minor segment area from the total circle area (πr²).
Example: Find the area of a segment with radius 6 cm and central angle 60° (minor segment) And that's really what it comes down to..
- Sector area = (60/360) × π × 6² = (1/6) × π × 36 = 6π cm².
- Triangle area = (1/2) × 6² × sin 60° = 18 × (√3/2) = 9√3 cm².
- Segment area = 6π – 9√3 ≈ 18.85 – 15.59 = 3.26 cm².
Practice Problems
Below are ten practice problems covering various scenarios. Solve each step‑by‑step, then check the provided solutions to verify your understanding.
Problem 1 – Radius = 8 cm, central angle = 90° (minor segment).
Problem 2 – Radius = 5 m, central angle = π/3 rad (minor segment).
Problem 3 – Radius = 12 in, central angle = 120° (major segment).
Problem 4 – Radius = 7 mm, central angle = 45° (minor segment).
Problem 5 – Radius = 10 cm, central angle = 2π/5 rad (minor segment).
Problem 6 – Radius = 9 ft, central angle = 300° (major segment).
Problem 7 – Radius = 4 km, central angle = 30° (minor segment).
Problem 8 – Radius = 15 cm, central angle = π rad (minor segment).
Problem 9 – Radius = 6 m, central angle = 180° (major segment).
Problem 10 – Radius = 11 in, central angle = 270° (major segment).
Solutions
- Sector = (90/360)π·8² = 16π in²; Triangle = (1/2)·8²·sin 90° = 32 in²; Segment = 16π – 32 ≈ 1.27 in².
- Sector = (1/2)·(π/3)·5² = (25π/6) m²; Triangle = (1/2)·5²·sin(π/3) = (25√3/4) m²; Segment = (25π/6) – (25√3/4) ≈ 2.18 m².
- Minor segment first: Sector = (120/360)π·12² = 48π in²; Triangle = (1/2)·12²·sin 120° = 72·(√3/2) = 36√3 in²; Minor = 48π – 36√3 ≈ 150.80 – 62.35 = 88.45 in². Major = π·12² – 88.45 = 452.39 – 88.45 = 363.94 in².
- Sector = (45/360)π·7² = (π·49)/8 ≈ 19.24 mm²; Triangle = (1/2)·7²·sin 45° = 24.5·(√2/2) ≈ 17.32 mm²; Segment ≈ 1.92 mm².
- Sector = (1/2)·(2π/5)·10² = (100π/5) = 20π cm²; Triangle = (1/2)·10²·sin(2π/5) = 50·sin 72° ≈ 50·0.9511 = 47.56 cm²; Segment ≈ 20π – 47.56 ≈ 19.82 cm².
- Central angle = 300° → minor segment = (60/360)π·9² = 13.5π
