Calculating the Perimeter of a Circular Sector

This problem asks us to find the perimeter of a sector formed by a circular arc. We are given the radius of the circle and the angle subtended by the arc at the centre.

Here are the given details:

• Radius of the circle (\(r\)) = 4 cm
• Angle subtended at the centre (\(\theta\)) = 45°
• Value of \(\pi\) to use = \(\frac{22}{7}\)

The perimeter of a sector is the total length of its boundary. The boundary consists of the circular arc and the two radii connecting the endpoints of the arc to the centre. Therefore, the formula for the perimeter of a sector is:

\(\text{Perimeter of Sector} = \text{Length of Arc} + 2 \times \text{Radius}\)

Step 1: Find the Length of the Arc

The length of a circular arc is a fraction of the circle's circumference. The fraction is determined by the ratio of the central angle to the total angle in a circle (360°).

The formula for the length of an arc is:

\(\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r\)

Let's plug in the given values:

\(\text{Arc Length} = \frac{45^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 4\)

Simplify the fraction \(\frac{45^\circ}{360^\circ}\):

\(\frac{45}{360} = \frac{45}{45 \times 8} = \frac{1}{8}\)

Now substitute this back into the arc length formula:

\(\text{Arc Length} = \frac{1}{8} \times 2 \times \frac{22}{7} \times 4\)

Combine the terms:

\(\text{Arc Length} = \frac{1}{8} \times 8 \times \frac{22}{7}\)

\(\text{Arc Length} = \frac{22}{7}\) cm

Step 2: Calculate the Perimeter of the Sector

Now that we have the arc length, we can find the perimeter of the sector using the formula:

\(\text{Perimeter of Sector} = \text{Arc Length} + 2 \times \text{Radius}\)

Substitute the calculated arc length and the given radius:

\(\text{Perimeter of Sector} = \frac{22}{7} + 2 \times 4\)

\(\text{Perimeter of Sector} = \frac{22}{7} + 8\)

To add these values, we need a common denominator. Convert 8 into a fraction with denominator 7:

\(8 = 8 \times \frac{7}{7} = \frac{56}{7}\)

Now add the fractions:

\(\text{Perimeter of Sector} = \frac{22}{7} + \frac{56}{7}\)

\(\text{Perimeter of Sector} = \frac{22 + 56}{7}\)

\(\text{Perimeter of Sector} = \frac{78}{7}\) cm

The perimeter of the sector formed is \(\frac{78}{7}\) cm.

Revision Table: Circular Sector Formulas

Additional Information on Circular Sectors

A circular sector is a portion of a disk (a circle plus its interior) enclosed by two radii and a circular arc. It looks like a slice of pizza or pie.

• The angle \(\theta\) at the centre is called the central angle of the sector.
• When the central angle is 360°, the sector is the entire circle.
• When the central angle is 180°, the sector is a semicircle.
• The units for perimeter and arc length are units of length (like cm, m, inches).
• The units for area of a sector are square units (like cm², m², square inches).
• Understanding the relationship between the central angle and the full circle is key to calculating arc length and sector area.