Practice Questions

A pendulum of length 21 cm swings through an angle of $60^\circ$. Its tip describes an arc. Formulate the problem to find the area of the sector formed by this swing and evaluate it. (Use $\pi = \frac{22}{7}$)

Explain what a 'quadrant' of a circle is and state the measure of its angle.

A student calculates the area of a major sector by subtracting the minor sector's area from the circle's circumference formula, stating Area(Major) = $2\pi r - \text{Area(Minor)}$. Evaluate this method and propose the correct formula.

State the formula for the length of an arc of a sector with angle $\theta$ (in degrees) and radius $r$.

Calculate the length of the arc of a circle with a radius of 14 cm that subtends an angle of 45° at the center. (Use $\pi = \frac{22}{7}$)

A sector is cut from a circle of diameter 42 cm. If the angle of the sector is 150°, calculate its area. (Use $\pi = \frac{22}{7}$)

What is the name given to the portion of a circular region enclosed between a chord and its corresponding arc?

Define a 'sector' of a circle.

State the formula for the area of a sector of a circle with radius $r$ and angle $\theta$ in degrees.

Recall the formula for the area of a sector. What does each variable ($\theta, r, \pi$) in the formula represent?

Justify, using the principle of proportionality, why the formula for the length of an arc of a sector with angle $\theta$ (in degrees) is given by $L = \frac{\theta}{360} \times 2\pi r$.

A circular park has a radius of 20 m. A path of width 5 m is laid inside the park along the boundary. Calculate the cost of paving the path at a rate of ₹25 per m². (Use $\pi = 3.14$)

If the circumference of a circle is $44$ cm, describe the steps to find the radius of the circle. (Use $\pi = \frac{22}{7}$)

A circle has a radius of $10$ cm. Find the area of a sector with an angle of $90^{\circ}$. Explain your steps. (Use $\pi = 3.14$)

Describe how the area of a segment of a circle is calculated.

A chord of a circle of radius 10 cm subtends an angle of 60° at the center. Analyze the triangle formed by the chord and the radii, and then calculate the area of the corresponding minor segment. (Use $\pi = 3.14$ and $\sqrt{3} = 1.73$)

If the angle of a minor sector of a circle is $x^{\circ}$, what is the angle of the corresponding major sector?

Explain the difference between a minor sector and a major sector of a circle.

Using a diagram, identify and describe the four key regions related to a chord in a circle: minor arc, major arc, minor segment, and major segment.

Explain how the area of a major sector and the area of a major segment are related to the total area of the circle. Provide the formulas used to find them, assuming the areas of the minor sector and minor segment are known.

The minute hand of a wall clock is 10 cm long. Calculate the area swept by it in 12 minutes. (Use $\pi = 3.14$)

A circle has an area of 154 cm². Calculate the area of a sector of this circle that subtends an angle of 72° at the center. (Use $\pi = \frac{22}{7}$)

If the length of an arc of a sector of a circle with radius 21 cm is 22 cm, calculate the area of the sector.

A car has two windshield wipers that do not overlap. Each wiper has a blade of length 35 cm and sweeps through an angle of 120°. Calculate the total area cleaned in one sweep by both blades. (Use $\pi = \frac{22}{7}$)

A cow is tethered to a peg at a corner of a rectangular field of dimensions 50 m by 40 m with a rope of length 14 m. Calculate the area of the field the cow can graze. Also, calculate the increase in the grazing area if the length of the rope is increased to 21 m. (Use $\pi = \frac{22}{7}$)

A silver wire, when bent in the form of a square, encloses an area of 484 cm². The same wire is now bent into the form of a circle. Calculate the area of the circle. Compare the two areas and determine which shape encloses more area for the same perimeter. (Use $\pi = \frac{22}{7}$)

Critique the following statement: 'The area of a segment is always less than the area of its corresponding sector.' Is this statement universally true? Justify your answer.

Justify whether the area of a segment can ever be equal to zero. If so, under what conditions?

Design a circular logo with a diameter of 28 cm, divided into three distinct sectors. The angles of these sectors are in the ratio 2:3:4. Formulate the steps to find the area of the largest sector and then evaluate it. (Use $\pi = \frac{22}{7}$)

Derive a simplified formula for the area of a segment of a circle where the corresponding chord forms an equilateral triangle with the center and two radii. Express the area in terms of the radius $r$.

Two concentric circles have radii of $r_1 = 7$ cm and $r_2 = 14$ cm. A sector with a central angle of $45^\circ$ is marked in both circles. Evaluate the area of the ring-shaped region between the arcs of these two sectors. Propose a general formula for this area.

An athletic track is designed with two parallel straight sections and two semi-circular ends. The total perimeter of the inner edge must be exactly 400 m. The straight sections are each 90 m long. Create a mathematical model to find the radius of the inner semi-circular ends. Further, if the track is 14 m wide everywhere, justify the calculation for the total area of the track. (Use $\pi = \frac{22}{7}$)

Describe the complete procedure for finding the area of a minor segment of a circle if you are given the radius $r$ and the central angle $\theta$. List all the formulas you would need.

Formulate a single expression for the perimeter of a minor segment of a circle with radius $r$ and a central angle $\theta$ (in degrees).

Summarize the unitary method used to derive the formula for the area of a sector. Start with the area of a full circle and explain the logical steps to arrive at the formula for a sector with angle $\theta$.

In a square of side 14 cm, four quadrants of circles of radius 7 cm are drawn from the four corners. Calculate the area of the region left in the center, not covered by the quadrants. (Use $\pi = \frac{22}{7}$)

Propose a method to find the area of the region common to two intersecting circles of equal radius $r$, where the center of each circle lies on the circumference of the other. Justify your method with formulas.

An athletic track 14 m wide consists of two straight sections 120 m long and two semicircular ends. The inner boundary has a radius of 35 m. Calculate the area of the track. (Use $\pi = \frac{22}{7}$)

Derive a general formula for the area of the triangle formed by two radii and a chord in a circle of radius $r$ with central angle $\theta$ using trigonometry. Using this, propose a single, unified formula for the area of the minor segment. Justify that your proposed formula is valid.

A decorative window is designed in the shape of a rectangle surmounted by a semicircle. The total perimeter of the window is 10 m. Formulate an expression for the total area of the window in terms of its radius, $r$. Then, create a plan to find the dimensions (radius and height of the rectangular part) that will maximize the light admitted (i.e., maximize the area). You do not need to solve for the maximum, just formulate the problem.

In the given figure, a circle is inscribed in an equilateral triangle ABC of side 12 cm. Calculate the area of the region inside the triangle but outside the circle. (Use $\pi = 3.14$ and $\sqrt{3} = 1.73$)

A decorative round tablecloth has a radius of 32 cm. A design is formed by leaving an equilateral triangle in the middle with its vertices on the circumference of the circle. Calculate the area of the design (the region inside the circle but outside the triangle). (Use $\pi = 3.14$ and $\sqrt{3} = 1.73$)

The perimeter of a sector of a circle of radius 5.2 cm is 16.4 cm. Calculate the area of the sector.

A square with side length 'a' is inscribed within a circle. Justify, with calculations, the exact ratio of the area of the circle to the area of the square.