Mathematics worksheet on surface area and volumes for Class 1X & X students.

Here are the step-by-step solutions for each question on the worksheet.

1) If the radius of cylinder is halved and height is doubled, then what will be the curved Surface area?
* Step 1: The formula for the Curved Surface Area (CSA) of a cylinder is $2\pi rh$.
* Step 2: Let the new radius be $r/2$ and the new height be $2h$.
* Step 3: Calculate the new CSA: $2 \times \pi \times (r/2) \times (2h)$.
* Step 4: The $2$ in the denominator and the $2$ in the numerator cancel out. You are left with $2\pi rh$.
* Conclusion: The new area is exactly the same as the old area.
* Answer: b) same

2) If a right circular cone has radius 4cm and slant height 5cm then what is its volume?
* Step 1: We need the height ($h$) to find the volume. We have radius ($r=4$) and slant height ($l=5$).
* Step 2: Use the Pythagorean theorem: $l^2 = r^2 + h^2$. So, $5^2 = 4^2 + h^2$.
* Step 3: $25 = 16 + h^2$, which means $h^2 = 9$, so $h = 3$ cm.
* Step 4: Volume formula is $\frac{1}{3}\pi r^2 h$.
* Step 5: Calculate: $\frac{1}{3} \times \pi \times 4^2 \times 3$. The $3$s cancel out.
* Step 6: Result is $16\pi$ cm³.
* Answer: a) $16\pi$ cm³

3) The radius of a hemisphere is r. what is its volume?
* Step 1: The volume of a full sphere is $\frac{4}{3}\pi r^3$.
* Step 2: A hemisphere is half of a sphere.
* Step 3: Divide the sphere's volume by 2: $\frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$.
* Answer: b) $\frac{2}{3}\pi r^3$

4) What is the total surface area of a hemisphere of radius?
* Step 1: Total Surface Area includes the curved part plus the flat circular base.
* Step 2: Curved part is half of a sphere's surface area ($4\pi r^2$), which is $2\pi r^2$.
* Step 3: The flat base is a circle with area $\pi r^2$.
* Step 4: Add them together: $2\pi r^2 + \pi r^2 = 3\pi r^2$.
* Answer: d) $3\pi r^2$

5) If the radius of a sphere is doubled, then what is the ration of their surface area?
* Step 1: Surface Area formula is $4\pi r^2$. Area is proportional to the square of the radius ($r^2$).
* Step 2: If you double the radius ($2r$), you square that change: $2^2 = 4$.
* Step 3: The new area is 4 times the original area.
* Step 4: The ratio of Original : New is $1 : 4$.
* Answer: c) 1:4

6) Two right circular cones of equal of curved surface areas have slant heights in the ratio of 3:4, then the ratio of their radii____
* Step 1: Formula for Curved Surface Area is $\pi rl$.
* Step 2: Since the areas are equal: $\pi r_1 l_1 = \pi r_2 l_2$.
* Step 3: This simplifies to $r_1 l_1 = r_2 l_2$, or $\frac{r_1}{r_2} = \frac{l_2}{l_1}$.
* Step 4: The ratio of radii is the *inverse* of the ratio of slant heights.
* Step 5: Inverse of $3:4$ is $4:3$.
* Note: Looking at the options provided in the image (4:1, 3:5, 5:3, 4:5), none match the mathematically correct answer of 4:3. However, based on standard test patterns, if there is a typo in the question or options, the intended logic is the inverse ratio.
* Answer: *Note: The correct mathematical answer is 4:3, which is not listed in the options.*

7) In the cylindrical container, the base radius is 8 cm. If the height of the water level Is 20 cm, then the volume of the water in the container____?
* Step 1: Volume formula is $\pi r^2 h$.
* Step 2: Calculation: $\frac{22}{7} \times 8^2 \times 20$.
* Step 3: $\frac{22}{7} \times 64 \times 20 = \frac{28160}{7} \approx 4022.8$ cm³.
* Step 4: Convert to Liters (divide by 1000): $4.0228$ L.
* Step 5: This matches closest to option (b).
* Answer: b) 4.0218 L

8) The total surface area of a cube is 96 cm² ,then the volume of a cube____
* Step 1: Total Surface Area of a cube is $6a^2$ (where $a$ is the side length).
* Step 2: $6a^2 = 96$. Divide by 6: $a^2 = 16$.
* Step 3: Square root of 16 is 4. So, side $a = 4$ cm.
* Step 4: Volume is $a^3$. $4^3 = 4 \times 4 \times 4 = 64$ cm³.
* Answer: b) 64 cm³

9) A cylinder and cone have bases of equal radii and are of equal heights then the Volumes are in the ratio ______
* Step 1: Volume of Cylinder = $\pi r^2 h$.
* Step 2: Volume of Cone = $\frac{1}{3} \pi r^2 h$.
* Step 3: Ratio Cylinder : Cone is $1 : \frac{1}{3}$.
* Step 4: Multiply both sides by 3 to remove the fraction: $3 : 1$.
* Answer: c) 3:1

10) The volume of right circular cone with radius 6 cm and height 7 cm____
* Step 1: Volume formula is $\frac{1}{3}\pi r^2 h$.
* Step 2: Calculation: $\frac{1}{3} \times \frac{22}{7} \times 6^2 \times 7$.
* Step 3: The $7$ in the denominator and the $7$ in the height cancel out.
* Step 4: You are left with $\frac{1}{3} \times 22 \times 36$.
* Step 5: $36$ divided by $3$ is $12$. So, $22 \times 12$.
* Step 6: $22 \times 12 = 264$ c.c.
* Answer: b) 264 c.c.

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Final Answer:
1) b
2) a
3) b
4) d
5) c
6) [Correct answer is 4:3, not listed in options]
7) b
8) b
9) c
10) b