NCERT Class 10 Maths Solutions: Areas Related to Circles

Question:

The area of a segment of a circle is the area of the corresponding sector minus the area of the triangle formed by the chord and the two radii to the endpoints of the chord. The area of a sector is given by ($\theta$/360°) * $\pi r^2$, where $\theta$ is the angle in degrees and r is the radius. The area of a triangle can be calculated using (1/2)ab sin C, where a and b are two sides and C is the angle between them.


Given:
Radius of the circle, r = 12 cm
Angle subtended by the chord at the centre, $\theta$ = 120°
$\pi$ = 3.14
$\sqrt{3}$ = 1.73

Step 1: Calculate the area of the sector.
The area of the sector with angle 120° and radius 12 cm is:
Area of sector = ($\theta$/360°) * $\pi r^2$
Area of sector = (120°/360°) * 3.14 * (12 cm)$^2$
Area of sector = (1/3) * 3.14 * 144 cm$^2$
Area of sector = 3.14 * 48 cm$^2$
Area of sector = 150.72 cm$^2$

Step 2: Calculate the area of the triangle formed by the chord and the radii.
The triangle formed by the chord and the two radii to its endpoints is an isosceles triangle with two sides equal to the radius (12 cm) and the angle between them is 120°.
Area of triangle = (1/2) * r * r * sin $\theta$
Area of triangle = (1/2) * 12 cm * 12 cm * sin 120°
We know that sin 120° = sin (180° – 60°) = sin 60° = $\sqrt{3}$/2
Area of triangle = (1/2) * 144 cm$^2$ * ($\sqrt{3}$/2)
Area of triangle = 72 cm$^2$ * ($\sqrt{3}$/2)
Area of triangle = 36 * $\sqrt{3}$ cm$^2$
Substitute the value of $\sqrt{3}$:
Area of triangle = 36 * 1.73 cm$^2$
Area of triangle = 62.28 cm$^2$

Step 3: Calculate the area of the corresponding segment.
Area of segment = Area of sector – Area of triangle
Area of segment = 150.72 cm$^2$ – 62.28 cm$^2$
Area of segment = 88.44 cm$^2$

The area of the corresponding segment of the circle is 88.44 cm$^2$.

Question:

To find the area of a minor segment, you need to subtract the area of the triangle formed by the chord and the two radii from the area of the sector formed by the same radii. The area of a sector is a fraction of the circle’s area, determined by the angle at the center. The area of a triangle can be calculated if you know its base and height, or using trigonometric formulas.


The problem asks for the area of the minor segment of a circle.
We are given:
Radius of the circle (r) = 10 cm
The angle subtended by the chord at the center (θ) = 90 degrees (a right angle)
We need to use π = 3.14.

First, let’s find the area of the sector formed by the radii and the arc.
The area of a sector is given by the formula: (θ/360°) * πr²
Area of sector = (90°/360°) * 3.14 * (10 cm)²
Area of sector = (1/4) * 3.14 * 100 cm²
Area of sector = 0.25 * 314 cm²
Area of sector = 78.5 cm²

Next, let’s find the area of the triangle formed by the chord and the two radii.
Since the angle at the center is 90 degrees, the two radii are perpendicular to each other. We can consider these radii as the base and height of the triangle.
Area of triangle = (1/2) * base * height
Area of triangle = (1/2) * 10 cm * 10 cm
Area of triangle = (1/2) * 100 cm²
Area of triangle = 50 cm²

Finally, the area of the minor segment is the difference between the area of the sector and the area of the triangle.
Area of minor segment = Area of sector – Area of triangle
Area of minor segment = 78.5 cm² – 50 cm²
Area of minor segment = 28.5 cm²

Therefore, the area of the minor segment is 28.5 cm².

Question:

The length of an arc of a circle is a fraction of the circle’s circumference. This fraction is determined by the angle subtended by the arc at the center, relative to the total angle of a circle (360 degrees). The formula for arc length is: Arc Length = (θ/360°) * 2πr, where θ is the angle in degrees and r is the radius.


The problem asks us to find the length of an arc in a circle.
We are given:
Radius of the circle (r) = 21 cm
Angle subtended by the arc at the center (θ) = 60°
We are also given to use π = 22/7.

The formula for the length of an arc is:
Arc Length = (θ/360°) × 2πr

Now, substitute the given values into the formula:
Arc Length = (60°/360°) × 2 × (22/7) × 21

First, simplify the fraction 60°/360°:
60/360 = 1/6

Now, substitute this simplified fraction back into the formula:
Arc Length = (1/6) × 2 × (22/7) × 21

We can perform the multiplication. It’s often helpful to cancel out common factors before multiplying to simplify calculations.
Arc Length = (1/6) × 2 × 22 × (21/7)

Calculate 21/7:
21/7 = 3

Now the expression becomes:
Arc Length = (1/6) × 2 × 22 × 3

Multiply the numbers:
Arc Length = (1/6) × 44 × 3
Arc Length = (1/6) × 132

Now, divide 132 by 6:
132 / 6 = 22

So, the length of the arc is 22 cm.

Final Answer Check:
The angle is 60°, which is 1/6 of the total circle (360°).
The circumference of the circle is 2πr = 2 * (22/7) * 21 = 2 * 22 * 3 = 132 cm.
The arc length should be 1/6 of the circumference: (1/6) * 132 cm = 22 cm.
The calculation matches the expected result.

The final answer is $\boxed{22 cm}$.

Question:

Area of a sector of a circle: The area of a sector is a fraction of the total area of the circle, determined by the angle subtended by the arc at the center. The formula is (θ/360°) * πr², where θ is the angle in degrees and r is the radius.


We are given the radius of the circle (r = 21 cm) and the angle subtended by the arc at the center (θ = 60°).
The formula for the area of a sector is:
Area of sector = (θ/360°) * πr²

Substitute the given values into the formula:
Area of sector = (60°/360°) * (22/7) * (21 cm)²

Simplify the fraction 60°/360°:
60°/360° = 1/6

Now, calculate the square of the radius:
(21 cm)² = 21 cm * 21 cm = 441 cm²

Substitute these simplified values back into the formula:
Area of sector = (1/6) * (22/7) * 441 cm²

Now, perform the multiplication. We can cancel out common factors. Notice that 441 is divisible by 7 (441 = 7 * 63).
Area of sector = (1/6) * 22 * (441/7) cm²
Area of sector = (1/6) * 22 * 63 cm²

Now, we can also simplify by dividing 63 by 6. Both are divisible by 3.
63 / 3 = 21
6 / 3 = 2
Area of sector = (1/2) * 22 * (63/3) cm²
Area of sector = (1/2) * 22 * 21 cm²

Now, multiply 22 by 21:
22 * 21 = 462

So, the area of the sector is:
Area of sector = (1/2) * 462 cm²

Finally, divide by 2:
Area of sector = 231 cm²

Therefore, the area of the sector formed by the arc is 231 cm².

Question:

The area of a sector of a circle is a fraction of the total area of the circle, determined by the angle of the sector. The formula for the area of a sector is (θ/360°) * πr², where θ is the central angle of the sector and r is the radius of the circle.


The problem asks us to find the area of a sector of a circle.
We are given the radius of the circle, r = 6 cm.
We are also given the angle of the sector, θ = 60°.
We are instructed to use the value of π = 22/7.

The formula for the area of a sector is:
Area of Sector = (θ / 360°) * πr²

Substitute the given values into the formula:
Area of Sector = (60° / 360°) * (22/7) * (6 cm)²

Simplify the fraction (60° / 360°):
60/360 = 1/6

Calculate the square of the radius:
(6 cm)² = 36 cm²

Now, substitute these simplified values back into the formula:
Area of Sector = (1/6) * (22/7) * 36 cm²

Multiply the numbers:
Area of Sector = (1/6) * 36 * (22/7) cm²
Area of Sector = 6 * (22/7) cm²
Area of Sector = (6 * 22) / 7 cm²
Area of Sector = 132 / 7 cm²

To express the answer as a mixed number or decimal:
132 ÷ 7 = 18 with a remainder of 6.
So, Area of Sector = 18 6/7 cm²

Alternatively, as a decimal:
132 / 7 ≈ 18.857 cm² (rounded to three decimal places)

The area of the sector is 132/7 cm² or approximately 18.86 cm².

Final Answer is 132/7 cm².

Question:

The problem requires finding the area of a sector of a circle. The formula for the area of a sector is (θ/360°) * πr², where θ is the angle of the sector in degrees and r is the radius of the circle.


The lighthouse warns ships over a sector of a circle.
The angle of the sector is given as θ = 80°.
The distance to which the light spreads is the radius of the sector, r = 16.5 km.
We need to find the area of this sector.
Using the formula for the area of a sector:
Area = (θ/360°) * πr²
Substitute the given values:
Area = (80°/360°) * 3.14 * (16.5 km)²
Simplify the fraction 80°/360°:
80/360 = 8/36 = 2/9
Calculate the square of the radius:
(16.5 km)² = 16.5 * 16.5 km²
16.5 * 16.5 = 272.25
Now, substitute these values back into the area formula:
Area = (2/9) * 3.14 * 272.25 km²
Calculate the product of 3.14 and 272.25:
3.14 * 272.25 = 854.865
Now, multiply by 2/9:
Area = (2 * 854.865) / 9 km²
Area = 1709.73 / 9 km²
Perform the division:
1709.73 / 9 = 189.97 km²
Therefore, the area of the sea over which the ships are warned is approximately 189.97 km².

Question:

The area swept by the minute hand of a clock in a certain time is a sector of a circle. The area of a sector is given by the formula: Area = (θ/360°) * πr², where θ is the angle subtended by the sector at the center and r is the radius of the circle. We need to determine the angle swept by the minute hand in 5 minutes.


The length of the minute hand is given as 14 cm, which is the radius (r) of the circle.
The minute hand completes a full circle (360°) in 60 minutes.
Therefore, in 1 minute, the minute hand sweeps an angle of 360°/60 = 6°.
In 5 minutes, the minute hand sweeps an angle of 5 minutes * 6°/minute = 30°.
So, the angle (θ) of the sector is 30°.
We are given to use π = 22/7.
Now, we can calculate the area of the sector using the formula:
Area = (θ/360°) * πr²
Area = (30°/360°) * (22/7) * (14 cm)²
Area = (1/12) * (22/7) * (196 cm²)
Area = (1/12) * 22 * (196/7) cm²
Area = (1/12) * 22 * 28 cm²
Area = (1/6) * 11 * 28 cm²
Area = 11 * (28/6) cm²
Area = 11 * (14/3) cm²
Area = 154/3 cm²
Area ≈ 51.33 cm²

Question:

The area of a sector of a circle is given by the formula (θ/360°) * πr², where θ is the angle subtended by the sector at the centre and r is the radius of the circle. A major sector corresponds to the larger part of the circle when a chord divides it. The angle of the major sector is 360° minus the angle of the minor sector.


The radius of the circle is given as r = 10 cm.
The chord subtends a right angle (90°) at the centre. This refers to the minor sector.
The angle of the minor sector is θ_minor = 90°.
The angle of the corresponding major sector is θ_major = 360° – θ_minor.
θ_major = 360° – 90° = 270°.
The area of the major sector is given by the formula:
Area_major_sector = (θ_major / 360°) * πr²
Substituting the values:
Area_major_sector = (270° / 360°) * 3.14 * (10 cm)²
Area_major_sector = (3/4) * 3.14 * 100 cm²
Area_major_sector = 0.75 * 3.14 * 100 cm²
Area_major_sector = 0.75 * 314 cm²
Area_major_sector = 235.5 cm²

The area of the corresponding major sector is 235.5 cm².