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Mensuration - Arc Length and Sector Area

Grade 12A Level

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The arc length is the distance along the curved part of a sector. When the central angle θ\theta is in degrees, it is calculated as a fraction of the total circumference: s=θ360×2πrs = \frac{\theta}{360} \times 2\pi r. When θ\theta is in radians, it is simply s=rθs = r\theta.

A circular sector showing radius r, central angle theta, and arc length s.
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The area of a sector is the space enclosed by two radii and the arc. In degrees, it is A=θ360×πr2A = \frac{\theta}{360} \times \pi r^2. In radians, the formula simplifies to A=12r2θA = \frac{1}{2}r^2\theta.

A circle with a shaded sector area highlighted.
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A segment is the region between a chord and the arc. Its area is found by subtracting the area of the triangle formed by the radii and the chord from the total sector area: Areasegment=12r2(θ−sin⁡θ)Area_{segment} = \frac{1}{2}r^2(\theta - \sin\theta) (where θ\theta is in radians).

A sector with a chord showing the segment area at the top.
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The perimeter of a sector is the sum of the arc length and the two bounding radii: P=s+2rP = s + 2r.

📐Formulae

s=rθs = r\theta (Arc length where θ\theta is in radians)

A=12r2θA = \frac{1}{2}r^2\theta (Sector area where θ\theta is in radians)

s=θ360×2πrs = \frac{\theta}{360} \times 2\pi r (Arc length where θ\theta is in degrees)

A=θ360×πr2A = \frac{\theta}{360} \times \pi r^2 (Sector area where θ\theta is in degrees)

Areasegment=12r2(θ−sin⁡θ)Area_{segment} = \frac{1}{2}r^2(\theta - \sin\theta) (Segment area where θ\theta is in radians)

💡Examples

Problem 1:

A sector of a circle has a radius of 66 cm and an angle of 1.51.5 radians. Calculate the arc length and the area of the sector.

Solution:

s=6×1.5=9s = 6 \times 1.5 = 9 cm; A=12×62×1.5=27A = \frac{1}{2} \times 6^2 \times 1.5 = 27 cm²

Explanation:

To find the arc length, use s=rθs = r\theta. To find the area, use A=12r2θA = \frac{1}{2}r^2\theta. Substitute r=6r=6 and θ=1.5\theta=1.5 directly into the radian-based formulas.

Problem 2:

A sector has a perimeter of 2424 cm and a radius of 55 cm. Find the area of the sector.

Solution:

s=24−(2×5)=14s = 24 - (2 \times 5) = 14 cm; θ=145=2.8\theta = \frac{14}{5} = 2.8 rad; A=12×52×2.8=35A = \frac{1}{2} \times 5^2 \times 2.8 = 35 cm²

Explanation:

First, find the arc length ss by subtracting the two radii from the total perimeter (P=s+2rP = s + 2r). Then, find the angle θ\theta using s=rθs = r\theta. Finally, use the area formula A=12r2θA = \frac{1}{2}r^2\theta.

Problem 3:

Find the area of the segment cut off by a chord in a circle of radius 1010 cm, where the chord subtends an angle of 22 radians at the center.

Solution:

Areasector=12(102)(2)=100Area_{sector} = \frac{1}{2}(10^2)(2) = 100 cm²; Areatriangle=12(102)sin⁡(2)≈45.47Area_{triangle} = \frac{1}{2}(10^2)\sin(2) \approx 45.47 cm²; Areasegment=100−45.47=54.53Area_{segment} = 100 - 45.47 = 54.53 cm²

Explanation:

The segment area is the difference between the sector area (0.5r2θ0.5r^2\theta) and the area of the triangle formed by the two radii and the chord (0.5r2sin⁡θ0.5r^2\sin\theta). Ensure the calculator is in Radian mode when calculating sin⁡(2)\sin(2).

Problem 4:

A circular garden plot is designed as a sector with a radius of 1212 m and a central angle of 150∘150^\circ. Calculate the length of the fence required to enclose the curved boundary only, and the total area of the garden.

A sector with a 150 degree angle and a 12m radius.

Solution:

  1. Calculate arc length (curved fence): s=θ360×2πrs = \frac{\theta}{360} \times 2\pi r s=150360×2×π×12s = \frac{150}{360} \times 2 \times \pi \times 12 s=512×24π=10π≈31.42 ms = \frac{5}{12} \times 24\pi = 10\pi \approx 31.42 \text{ m}

  2. Calculate total area: A=θ360×πr2A = \frac{\theta}{360} \times \pi r^2 A=150360×π×122A = \frac{150}{360} \times \pi \times 12^2 A=512×144π=60π≈188.50 m2A = \frac{5}{12} \times 144\pi = 60\pi \approx 188.50 \text{ m}^2

Explanation:

Since the angle is given in degrees, we use the degree-based formulae for arc length and sector area. The fence for only the curved boundary refers to the arc length.

Problem 5:

A sector of a circle has an area of 50 cm250 \text{ cm}^2 and a central angle of 44 radians. Determine the radius of the circle and the perimeter of the sector.

A large sector representing an angle of 4 radians.

Solution:

  1. Use sector area formula to find radius: A=12r2θA = \frac{1}{2}r^2\theta 50=12×r2×450 = \frac{1}{2} \times r^2 \times 4 50=2r250 = 2r^2 r2=25  ⟹  r=5 cmr^2 = 25 \implies r = 5 \text{ cm}

  2. Find arc length: s=rθ=5×4=20 cms = r\theta = 5 \times 4 = 20 \text{ cm}

  3. Calculate perimeter: P=s+2rP = s + 2r P=20+2(5)=30 cmP = 20 + 2(5) = 30 \text{ cm}

Explanation:

We first use the area formula in radians to isolate rr. Once the radius is known, we calculate the arc length and then add the two radii to get the total perimeter.