To find the area of a sector of a circle with a radius of 6 cm and an angle of 60°, use the formula: Area = (θ/360) × π × r², where θ is the angle in degrees and r is the radius. Plugging in the values, we get: Area = (60/360) × π × 6². Simplifying this, Area = (1/6) × π × 36. Therefore, the area of the sector is 6π cm². This formula calculates the proportional area of the circle that the sector occupies, based on the fraction of the full 360° circle that the 60° sector represents.

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## Introduction to Sector Area Calculation

Understanding the area of a sector of a circle is a fundamental concept in geometry, often applied in various fields including engineering, architecture, and design. A sector is a portion of a circle enclosed by two radii and an arc. The area of a sector is proportional to the angle it subtends at the center of the circle. To calculate this area, one must understand the relationship between the angle of the sector and the total area of the circle. The formula for the area of a circle is πr², where r is the radius. However, for a sector, this formula is adjusted to account for the fraction of the circle the sector represents.

### Understanding the Formula

The formula for calculating the area of a sector is derived from the area of a circle. Since a circle is 360 degrees, the area of a sector depends on what fraction of 360 degrees its central angle represents. The formula is given as Area = (θ/360) × π × r², where θ is the central angle in degrees and r is the radius of the circle. This formula essentially calculates the area of the entire circle and then multiplies it by the fraction of the circle that the sector represents.

#### Calculating Sector Area with a 60° Angle

When calculating the area of a sector with a specific angle, such as 60°, it’s important to substitute the values accurately into the formula. For a circle with a radius of 6 cm and a sector angle of 60°, the calculation becomes straightforward. The angle of 60° is one-sixth of 360°, indicating that the sector occupies one-sixth of the total area of the circle. This understanding is crucial in simplifying the calculation process.

Step-by-Step Calculation
To calculate the area of the sector, first determine the area of the full circle using the formula πr². For a radius of 6 cm, the area of the circle is π × 6² = 36π cm². Since the sector’s angle is 60°, which is one-sixth of a full circle, the sector’s area is one-sixth of the circle’s total area. Therefore, the area of the sector is (1/6) × 36π = 6π cm². This step-by-step approach helps in understanding how the sector’s area relates to the circle’s total area.

##### Practical Implications

The calculation of a sector’s area has practical implications in various real-world scenarios. For instance, in architectural design, understanding the area of a sector can help in designing rounded elements or segments of a structure. In fields like engineering, it can assist in calculating material requirements for circular components. The concept is also used in everyday situations, such as determining the size of a slice of pizza or a pie, where each slice can be considered a sector of a circle.

###### Conclusion of Solutions

In conclusion, calculating the area of a sector of a circle requires understanding the relationship between the sector’s angle and the total area of the circle. By applying the formula (θ/360) × π × r², one can easily determine the area of a sector for any given radius and angle. This concept not only has academic significance in geometry but also practical applications in various professional and everyday contexts. Understanding and applying this formula is a key skill in geometry, providing a foundation for more complex geometrical calculations and applications.

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Questions of 10th Maths Exercise 11.1 in Detail

 Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60°. Find the area of a quadrant of a circle whose circumference is 22 cm. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes. A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding (i) minor segment (ii) major sector. In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find: (i) the length of the arc (ii) area of the sector formed by the arc (iii) area of the segment formed by the corresponding chord. A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the corresponding minor and major segments of the circle. A chord of a circle of radius 12 cm subtends an angle of 120° at the centre. Find the area of the corresponding segment of the circle. A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope. Find (i) the area of that part of the field in which the horse can graze. (ii) the increase in the grazing area if the rope were 10 m long instead of 5 m. A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors. Find (i) the total length of the silver wire required. (ii) the area of each sector of the brooch. An umbrella has 8 ribs which are equally spaced. Assuming umbrella to be a flat circle of radius 45 cm, find the area between the two consecutive ribs of the umbrella. A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of 115°. Find the total area cleaned at each sweep of the blades. To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle 80° to a distance of 16.5 km. Find the area of the sea over which the ships are warned. A round table cover has six equal designs. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of ₹0.35 per cm².

Last Edited: June 11, 2024