Question

Area of the sector of a circle with radius \(7 \text{ cm}\) and the angle at the centre is \(270^\circ\) is _____ \(\text{cm}^2\).

• A. \(115.5\)
• B. \(7\)
• C. \(551.1\)
• D. \(27\)

Hint:

A full circle has: \[ 360^\circ \] So: \[ 270^\circ = \frac{3}{4} \] of the entire circle. Therefore, this sector is simply: \[ \frac{3}{4}\times \pi r^2 \]

Answer 1

The Correct Option is A

Concept: A sector is a portion of a circle enclosed by:
• two radii
• and the corresponding arc The area of a sector depends on the central angle. Formula: \[ \text{Area of Sector} = \frac{\theta}{360^\circ}\times \pi r^2 \] where:
• \(r\) = radius
• \(\theta\) = central angle

Step 1: Write the given values. Radius: \[ r = 7 \text{ cm} \] Central angle: \[ \theta = 270^\circ \] Using: \[ \pi = \frac{22}{7} \]

Step 2: Substitute values into the formula. \[ \text{Area} = \frac{270}{360}\times \frac{22}{7}\times 7^2 \] Since: \[ 7^2 = 49 \] therefore: \[ \text{Area} = \frac{270}{360}\times \frac{22}{7}\times 49 \]

Step 3: Simplify the fraction \(\frac{270}{360}\). Divide numerator and denominator by 90: \[ \frac{270}{360} = \frac{3}{4} \] So: \[ \text{Area} = \frac{3}{4}\times \frac{22}{7}\times 49 \]

Step 4: Simplify the numerical calculation. Cancel 49 with 7: \[ 49 \div 7 = 7 \] Thus: \[ \text{Area} = \frac{3}{4}\times 22\times 7 \] Multiply: \[ 22\times 7 = 154 \] So: \[ \text{Area} = \frac{3}{4}\times 154 \] \[ = \frac{462}{4} \] \[ = 115.5 \]

Step 5: Write the final unit. \[ \boxed{115.5 \text{ cm}^2} \]