Arc length = rθ. Sector area = (1/2)r²θ. Convert degrees to radians.

Length of Arc and Area of Sector — Circle Formulas

Question

A circle has radius 7 cm. A sector is cut from it with a central angle of 60°. Find:

The length of the arc
The area of the sector

(Use π = 22/7)

Solution — Step by Step

Before we use any formula, we need the angle in radians — not degrees. The conversion is:

\theta = \frac{\pi}{180} \times 60° = \frac{\pi}{3} \text{ radians}

This matters because the arc length and sector area formulas only work directly with radians.

Arc length uses the formula $L = r\theta$ , where $r$ is the radius and $\theta$ is in radians.

L = 7 \times \frac{\pi}{3} = \frac{7\pi}{3} = \frac{7 \times 22/7}{3} = \frac{22}{3} \approx 7.33 \text{ cm}

Arc length = 22/3 cm ≈ 7.33 cm

Sector area uses $A = \frac{1}{2}r^2\theta$ . Plug in $r = 7$ and $\theta = \pi/3$ :

A = \frac{1}{2} \times 7^2 \times \frac{\pi}{3} = \frac{1}{2} \times 49 \times \frac{22/7}{3} = \frac{1}{2} \times \frac{49 \times 22}{21} = \frac{1}{2} \times \frac{1078}{21}

A = \frac{1}{2} \times \frac{154}{3} = \frac{77}{3} \approx 25.67 \text{ cm}^2

Sector area = 77/3 cm² ≈ 25.67 cm²

Why This Works

Think of a full circle as a sector with $\theta = 2\pi$ radians. The full circumference is $2\pi r$ , and arc length $L = r\theta$ gives exactly that when $\theta = 2\pi$ . So the formula is just “what fraction of the full circle is this sector?” scaled to the circumference.

Same logic for area: full circle area is $\pi r^2$ . Our formula $\frac{1}{2}r^2\theta$ at $\theta = 2\pi$ gives $\frac{1}{2}r^2 \cdot 2\pi = \pi r^2$ . Everything checks out.

The radian conversion isn’t magic — it’s just the natural unit for measuring rotation. When we write $60° = \pi/3$ , we’re saying “this angle is $1/6$ of a half-turn,” which is exactly the fraction of the circle we’re cutting.

Alternative Method

CBSE Class 10 NCERT also teaches the degree-based versions of these formulas directly:

\text{Arc Length} = \frac{\theta}{360°} \times 2\pi r

\text{Sector Area} = \frac{\theta}{360°} \times \pi r^2

For our problem:

L = \frac{60}{360} \times 2 \times \frac{22}{7} \times 7 = \frac{1}{6} \times 44 = \frac{22}{3} \text{ cm}

A = \frac{60}{360} \times \frac{22}{7} \times 49 = \frac{1}{6} \times 154 = \frac{77}{3} \text{ cm}^2

Same answers, different route. For board exams, this degree-based form is perfectly acceptable and avoids the radian conversion step.

The degree-based formulas are safer for CBSE Class 10 since the NCERT introduces them first. Save the radian forms for Class 11 and JEE prep — they’re the same thing, just written differently.

Common Mistake

Forgetting to convert degrees to radians before using $L = r\theta$ .

Students plug in $\theta = 60$ directly and get $L = 7 \times 60 = 420$ cm — which is absurd (longer than the radius by 60 times!). The formula $L = r\theta$ demands $\theta$ in radians. If your angle is in degrees, either convert first or use the degree-based formula $\frac{\theta}{360} \times 2\pi r$ . Both routes give the same answer; picking one and sticking with it prevents this error entirely.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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