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Mensuration - Arc Length and Sector Area

Grade 10IGCSE

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

🔑Concepts

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Understanding the relationship between a circle's circumference/area and its fractional parts (arcs and sectors).

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Identifying the difference between a Minor Arc/Sector (angle < 180°) and a Major Arc/Sector (angle > 180°).

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Recognizing that the angle at the center (theta) is proportional to the total 360 degrees of a circle.

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Calculating the perimeter of a sector, which must include the arc length plus the two radii.

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Using the value of Pi (π) as approximately 3.142 or using the calculator button for higher precision.

📐Formulae

Circumference of a Circle=2πr\text{Circumference of a Circle} = 2\pi r

Area of a Circle=πr2\text{Area of a Circle} = \pi r^2

Arc Length=θ360×2πr\text{Arc Length} = \frac{\theta}{360} \times 2\pi r

Sector Area=θ360×πr2\text{Sector Area} = \frac{\theta}{360} \times \pi r^2

Perimeter of a Sector=Arc Length+2r\text{Perimeter of a Sector} = \text{Arc Length} + 2r

Angle of Sector θ=Arc Length×3602πr\text{Angle of Sector } \theta = \frac{\text{Arc Length} \times 360}{2\pi r}

💡Examples

Problem 1:

A sector has a radius of 12 cm and a central angle of 60°. Calculate the length of the arc. Give your answer in terms of π\pi.

Solution:

ArcLength=60360×2×π×12=16×24π=4π cmArc Length = \frac{60}{360} \times 2 \times \pi \times 12 = \frac{1}{6} \times 24\pi = 4\pi \text{ cm}

Explanation:

To find the arc length, we take the fraction of the circle represented by the angle (60/360) and multiply it by the full circumference formula (2πr).

Problem 2:

Calculate the area of a sector with a radius of 10 cm and a central angle of 144°. Round your answer to 3 significant figures.

Solution:

Area=144360×π×102=0.4×100π=40π≈126 cm2Area = \frac{144}{360} \times \pi \times 10^2 = 0.4 \times 100\pi = 40\pi \approx 126 \text{ cm}^2

Explanation:

The area is found by multiplying the fraction (144/360) by the total area of the circle (πr²). 144/360 simplifies to 0.4.

Problem 3:

A sector has an arc length of 11 cm and a radius of 7 cm. Find the perimeter of the sector.

Solution:

Perimeter=ArcLength+2×radius=11+2(7)=11+14=25 cmPerimeter = Arc Length + 2 \times radius = 11 + 2(7) = 11 + 14 = 25 \text{ cm}

Explanation:

The perimeter of a sector consists of the curved arc plus the two straight edges (radii) that meet at the center.

Problem 4:

The area of a sector is 20π cm220\pi \text{ cm}^2 and its radius is 6 cm. Find the central angle θ\theta.

Solution:

20π=θ360×π×62⇒20π=θ360×36π⇒20=θ10⇒θ=200∘20\pi = \frac{\theta}{360} \times \pi \times 6^2 \Rightarrow 20\pi = \frac{\theta}{360} \times 36\pi \Rightarrow 20 = \frac{\theta}{10} \Rightarrow \theta = 200^{\circ}

Explanation:

Substitute the known values into the sector area formula and solve for the unknown angle θ by rearranging the equation.