Find the length of the arc of the sector of a circle of diameter 7 cm with a central angle of 108°. [Use π = 22/7]
6.6 cm
Finding the Arc Length of a Sector
Let's find the length of the arc of the sector given the diameter of the circle and the central angle. The question provides the diameter of the circle as 7 cm and the central angle of the sector as 108°.
Understanding Arc Length
The arc length of a sector is a portion of the circle's circumference. The length of this portion depends on the central angle subtended by the arc. A full circle has a central angle of 360°.
The formula to calculate the arc length (l) of a sector with radius (r) and central angle ($\theta$) in degrees is:
\(\text{Arc Length } (l) = \frac{\theta}{360°} \times \text{Circumference}\)
Since the circumference of a circle is \(2\pi r\), the formula becomes:
\(l = \frac{\theta}{360°} \times 2\pi r\)
Step-by-Step Calculation
Given:
- Diameter = 7 cm
- Central angle (\(\theta\)) = 108°
- Use \(\pi = 22/7\)
First, we need to find the radius (r) from the diameter.
Radius \(r = \frac{\text{Diameter}}{2} = \frac{7 \text{ cm}}{2} = 3.5 \text{ cm}\)
Now, substitute the values into the arc length formula:
\(l = \frac{108°}{360°} \times 2 \times \frac{22}{7} \times 3.5 \text{ cm}\)
Simplify the fraction \(\frac{108}{360}\). Both numbers are divisible by 36.
\(\frac{108 \div 36}{360 \div 36} = \frac{3}{10}\)
Substitute the simplified fraction and \(3.5\) as \(7/2\):
\(l = \frac{3}{10} \times 2 \times \frac{22}{7} \times \frac{7}{2} \text{ cm}\)
Now, perform the multiplication. We can cancel out terms:
- The '2' in the numerator cancels with the '2' in the denominator.
- The '7' in the denominator cancels with the '7' in the numerator (from 7/2).
\(l = \frac{3}{10} \times 22 \text{ cm}\)
\(l = \frac{3 \times 22}{10} \text{ cm}\)
\(l = \frac{66}{10} \text{ cm}\)
\(l = 6.6 \text{ cm}\)
Thus, the length of the arc of the sector is 6.6 cm.
Summary of Calculation
| Parameter | Value |
|---|---|
| Diameter | 7 cm |
| Radius (r) | 3.5 cm |
| Central Angle (\(\theta\)) | 108° |
| \(\pi\) | 22/7 |
| Arc Length Formula | \(\frac{\theta}{360°} \times 2\pi r\) |
| Calculation | \(\frac{108}{360} \times 2 \times \frac{22}{7} \times 3.5 = \frac{3}{10} \times 2 \times \frac{22}{7} \times \frac{7}{2} = \frac{3}{10} \times 22 = 6.6\) |
| Arc Length | 6.6 cm |
The calculated arc length matches one of the given options.
Revision Table: Arc Length Calculation
| Concept | Formula/Definition | Notes |
|---|---|---|
| Radius (r) | Diameter / 2 | Half of the circle's diameter |
| Circumference | \(2\pi r\) or \(\pi d\) | Distance around the circle |
| Arc Length (l) | \(\frac{\theta}{360°} \times 2\pi r\) (for \(\theta\) in degrees) | Portion of circumference proportional to the central angle |
Additional Information: Sectors and Segments
A sector of a circle is a region bounded by two radii and the included arc. Its area and arc length are proportional to the central angle.
- Area of a sector: \(\frac{\theta}{360°} \times \pi r^2\) (for \(\theta\) in degrees)
- Segment of a circle: The region bounded by an arc and the chord connecting its endpoints. The area of a segment is the area of the sector minus the area of the triangle formed by the two radii and the chord.
Understanding these concepts helps in solving various problems related to parts of a circle.
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