Circles – Area & Circumference Quiz - ICSE Class 9 Mathematics
Free practice quiz: Circles – Area & Circumference (ICSE Class 9 Mathematics). 41 multiple-choice questions with answers and explanations. Play online, no sign-up needed.
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A circular park has a radius of 21 meters. What is its circumference?
- 132 meters
- 66 meters
- 264 meters
- 42 meters
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Answer: A. 132 meters
The circumference of a circle is given by the formula $C = 2 \pi r$. Given $r = 21$ meters and $\pi \approx \frac{22}{7}$, the circumference is $C = 2 \times \frac{22}{7} \times 21 = 2 \times 22 imes 3 = 132$ meters.
The ratio of the radii of two circles is 2:3. What is the ratio of their areas?
- 2:3
- 4:9
- 8:27
- 2:5
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Answer: B. 4:9
Let the radii of the two circles be $r_1$ and $r_2$. We are given $\frac{r_1}{r_2} = \frac{2}{3}$. The area of the first circle is $A_1 = \pi r_1^2$ and the second is $A_2 = \pi r_2^2$. The ratio of their areas is $\frac{A_1}{A_2} = \frac{\pi r_1^2}{\pi r_2^2} = \left(\frac{r_1}{r_2}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9}$.
If the circumference of a circle is doubled, by what factor does its area increase?
- 2
- 4
- 8
- 16
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Answer: B. 4
Let the initial circumference be $C_1 = 2 \pi r_1$ and the initial area be $A_1 = \pi r_1^2$. If the circumference is doubled, the new circumference $C_2 = 2 C_1 = 4 \pi r_1$. Let the new radius be $r_2$. Then $C_2 = 2 \pi r_2$. So, $4 \pi r_1 = 2 \pi r_2 \implies r_2 = 2 r_1$. The new area is $A_2 = \pi r_2^2 = \pi (2 r_1)^2 = \pi (4 r_1^2) = 4 (\pi r_1^2) = 4 A_1$. Therefore, the area increases by a factor of 4.
A goat is tied to a stake at the center of a circular field of radius 14 meters. If the goat can graze an area of 616 sq meters, what is the radius of the area it can graze?
- 7 meters
- 14 meters
- 21 meters
- 28 meters
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Answer: B. 14 meters
The area the goat can graze is a circle. We are given the area grazed $A = 616$ sq meters. The formula for the area of a circle is $A = \pi r^2$. Using $\pi = \frac{22}{7}$, we have $616 = \frac{22}{7} r^2$. Solving for $r^2$: $r^2 = 616 imes \frac{7}{22} = 28 imes 7 = 196$. So, $r = \sqrt{196} = 14$ meters. The radius of the field (14 meters) matches the radius of the area the goat can graze, meaning the goat can graze the entire field if its rope is long enough. The question asks for the radius of the area it can graze, which is calculated to be 14 meters. The radius of the field being 14 meters is a condition, not necessarily the limit of grazing. The rope length determines the grazing area. If the rope is shorter than 14m, it can't graze the entire field. The question implies the rope length is such that it can graze 616 sq m. So we calculate the radius from the area. The radius of the field is given as 14 meters. This means the goat's rope length is at least 14 meters. If the rope length were shorter than 14m, then the grazing area radius would be the rope length. However, the calculation shows the radius of the grazing area is exactly 14 meters. So the rope length is exactly 14 meters.
The diameter of a circle is 7 cm. What is its circumference?
- 7 cm
- 11 cm
- 22 cm
- 154 cm
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Answer: C. 22 cm
The diameter of the circle is $d = 7$ cm. The circumference of a circle is given by the formula $C = \pi d$. Using $\pi \approx \frac{22}{7}$, the circumference is $C = \frac{22}{7} imes 7 = 22$ cm.