CBSE-Class-10-Maths-Assertion-and-Reason-Chapter-12-Areas-Related-to-circles

Class 10^th Maths Chapter Assertion and Reason Questions

Question 1

Assertion (A): The area of a circle is directly proportional to the square of its radius.
Reason (R): The formula for the area of a circle is $A = \pi r^2$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The area formula shows that area increases with the square of the radius.

Question 2

Assertion (A): The circumference of a circle is independent of the radius.
Reason (R): The formula for the circumference of a circle is $C = 2\pi r$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: D) A is false, but R is true.
Explanation: The circumference is directly proportional to the radius, contrary to the assertion.

Question 3

Assertion (A): A semicircle has half the area of a full circle with the same radius.
Reason (R): The area of a semicircle is given by $A = \frac{1}{2}\pi r^2$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula confirms that the area of a semicircle is indeed half of that of a full circle.

Question 4

Assertion (A): The area of a sector of a circle is proportional to the angle of the sector.
Reason (R): The area of a sector is given by $A = \frac{\theta}{360^\circ} \times \pi r^2$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula shows that the area of a sector increases with the angle.

Question 5

Assertion (A): The radius of a circle can be found if the area is known.
Reason (R): The radius can be calculated using the formula $r = \sqrt{\frac{A}{\pi}}$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: This formula allows the radius to be derived from the area.

Question 6

Assertion (A): The area of a circle can be expressed in terms of its diameter.
Reason (R): The formula for area can be rewritten as $A = \frac{\pi d^2}{4}$ , where $d$ is the diameter.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The area can indeed be calculated using the diameter.

Question 7

Assertion (A): The total area of a circle increases as the radius increases.
Reason (R): The area is calculated using the formula $A = \pi r^2$ , which is an increasing function of $r$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: As the radius increases, the area also increases due to the square relationship.

Question 8

Assertion (A): The area of a circle can be expressed as a function of its circumference.
Reason (R): The formula for circumference is $C = 2\pi r$ and can be used to derive the area.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The circumference can be used to find the radius, which in turn gives the area.

Question 9

Assertion (A): If the radius of a circle is doubled, its area becomes four times the original area.
Reason (R): The area of a circle is proportional to the square of its radius.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Doubling the radius quadruples the area, as shown in the area formula.

Question 10

Assertion (A): The area of a circle can be calculated if the arc length and the angle of the sector are known.
Reason (R): The area of a sector can be calculated using the arc length and angle.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: B) Both A and R are true, but R is not the correct explanation of A.
Explanation: While the area can be derived from the sector, the assertion applies only to sectors, not circles in general.

Question 11

Assertion (A): The area of a circle is always greater than the area of a square whose side is equal to the radius of the circle.
Reason (R): The area of the circle is $A = \pi r^2$ and the area of the square is $A = r^2$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Since $\pi > 1$ , the area of the circle is indeed greater than that of the square.

Question 12

Assertion (A): The area of a circle can never be negative.
Reason (R): Area is defined as a positive quantity in geometry.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: By definition, area cannot be negative, which holds true for the area of a circle.

Question 13

Assertion (A): The area of a circle can be approximated using $A \approx 3.14r^2$ .
Reason (R): $\pi$ is approximately equal to $3.14$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The approximation of $\pi$ allows for estimation of area using this formula.

Question 14

Assertion (A): A circle with a radius of 0 has an area of 0.
Reason (R): The area of a circle is calculated using the formula $A = \pi r^2$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: A radius of 0 results in an area of 0, as per the area formula.

Question 15

Assertion (A): The radius of a circle can be found if the circumference is known.
Reason (R): The radius can be calculated using the formula $r = \frac{C}{2\pi}$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The circumference formula directly leads to the calculation of the radius.

Question 16

Assertion (A): The area of a circle can be compared to the area of a triangle.
Reason (R): A triangle can be inscribed within a circle.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: B) Both A and R are true, but R is not the correct explanation of A.
Explanation: While triangles can be inscribed in circles, the areas cannot be directly compared due to differing formulas.

Question 17

Assertion (A): The area of a circle increases exponentially as the radius increases.
Reason (R): The area of a circle is a quadratic function of the radius.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: C) A is true, but R is false.
Explanation: The area increases quadratically, not exponentially, even though it does increase with larger radii.

Question 18

Assertion (A): The area of a circle can be represented as $A = \pi r^2$ for any positive $r$ .
Reason (R): $\pi$ is a constant approximately equal to 3.14.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The area formula holds true for all positive radii, and $\pi$ being constant validates this.

Question 19

Assertion (A): The area of a sector can be calculated if the radius and angle are known.
Reason (R): The area of a sector is given by the formula $A = \frac{r^2\theta}{2}$ for angle $\theta$ in radians.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula confirms that both radius and angle are necessary for sector area calculation.

Question 20

Assertion (A): The area of a circle with a diameter of $d$ can be expressed as $A = \frac{\pi d^2}{4}$ .
Reason (R): The radius $r$ is half of the diameter $d$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The area formula in terms of diameter is derived correctly using the relationship between radius and diameter.

Question 21

Assertion (A): The area of a circle remains the same regardless of the unit of measurement used.
Reason (R): Area is a derived unit and is independent of the measurement system.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: C) A is true, but R is false.
Explanation: While the area is indeed calculated the same way, the numerical value changes with the unit (e.g., cm² vs. m²).

Question 22

Assertion (A): The area of a circle can be calculated without knowing the radius if the circumference is given.
Reason (R): The formula $A = \frac{C^2}{4\pi}$ allows the area to be derived from circumference.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The area can indeed be calculated from the circumference using this derived formula.

Question 23

Assertion (A): The area of a circle cannot be expressed in terms of the chord length.
Reason (R): The chord length does not provide enough information to determine the radius directly.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: A chord length alone does not yield sufficient data to calculate the area without knowing additional parameters.

Question 24

Assertion (A): A circle can be divided into sectors of equal area.
Reason (R): The area of each sector depends solely on the total area of the circle divided by the number of sectors.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The total area can indeed be evenly split among sectors.

Question 25

Assertion (A): The area of a circle is larger than the area of an inscribed polygon with the same radius.
Reason (R): A circle represents the maximum area for a given perimeter.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The circle's property of maximizing area relative to perimeter confirms that it will always have a greater area than any inscribed polygon.

Chapter 12 Areas Related to circles

Class 10th Maths Chapter Assertion and Reason Questions