CBSE Class 10 Mathematics (2026–27)
Chapter 12: Surface Areas and Volumes
20 Important Questions and Answers
Q1. What is the difference between Curved Surface Area (CSA) and Total Surface Area (TSA) of a cylinder?
Answer:
The Curved Surface Area (CSA) of a cylinder refers to the area covered by its curved outer surface only. It does not include the areas of the two circular bases. The formula for CSA is (2\pi rh), where (r) is the radius and (h) is the height. The Total Surface Area (TSA) includes both the curved surface and the two circular bases. Its formula is (2\pi r(h+r)). CSA is used when only the side surface needs covering, such as painting a pipe, while TSA is used when the entire cylinder, including the top and bottom, must be covered or painted.
Q2. Explain how to find the volume of a cylinder.
Answer:
The volume of a cylinder represents the amount of space enclosed within it. To calculate the volume, first find the area of the circular base using the formula (\pi r^2), where (r) is the radius. Then multiply this area by the height (h) of the cylinder. Thus, the volume formula becomes (V=\pi r^2h). The volume is measured in cubic units such as cm³ or m³. This formula is useful in finding the capacity of tanks, pipes, and containers. A larger radius or height increases the volume of the cylinder significantly.
Q3. State the formulae for the surface area and volume of a cone.
Answer:
A cone is a three-dimensional figure with a circular base and a pointed vertex. Its Curved Surface Area (CSA) is given by (\pi rl), where (r) is the radius and (l) is the slant height. The Total Surface Area (TSA) is (\pi r(l+r)), which includes the base area. The volume of a cone is (\frac{1}{3}\pi r^2h), where (h) is the vertical height. The factor (\frac{1}{3}) indicates that a cone occupies one-third the volume of a cylinder having the same base and height. These formulas are widely used in practical measurements.
Q4. Why is the slant height important in finding the surface area of a cone?
Answer:
The slant height of a cone is the distance from the vertex to any point on the edge of the circular base measured along the sloping surface. It is important because the curved surface area of a cone depends on the slant height rather than the vertical height. The formula for curved surface area is (\pi rl). If only the radius and vertical height are known, the slant height can be calculated using the Pythagoras theorem: (l=\sqrt{r^2+h^2}). Without knowing the slant height, the surface area of the cone cannot be accurately determined.
Q5. How is the volume of a cone related to the volume of a cylinder?
Answer:
A cone and a cylinder having the same base radius and height are closely related in volume. The volume of a cylinder is (\pi r^2h), while the volume of a cone is (\frac{1}{3}\pi r^2h). Therefore, the volume of a cone is exactly one-third the volume of the corresponding cylinder. This relationship can be verified experimentally by filling a cone with sand or water and pouring it into a cylinder of the same dimensions. Three conefuls are required to fill the cylinder completely. This concept is important for solving comparison-based questions.
Q6. Define a sphere and write its surface area and volume formulas.
Answer:
A sphere is a perfectly round three-dimensional solid in which every point on the surface is at the same distance from the center. This constant distance is called the radius. The surface area of a sphere is given by (4\pi r^2), and the volume is given by (\frac{4}{3}\pi r^3). These formulas show that both surface area and volume depend on the radius. Spheres are commonly found in everyday life, such as balls, globes, and bubbles. The formulas help calculate the material required to cover or fill spherical objects.
Q7. What is a hemisphere? How does it differ from a sphere?
Answer:
A hemisphere is exactly half of a sphere obtained by cutting the sphere through its center. Unlike a sphere, a hemisphere has one curved surface and one flat circular base. The curved surface area of a hemisphere is (2\pi r^2), while its total surface area is (3\pi r^2), including the base. The volume of a hemisphere is (\frac{2}{3}\pi r^3). Since it is half of a sphere, its volume is half the volume of a sphere. Hemispherical shapes are commonly seen in bowls, domes, and satellite dishes.
Q8. Explain the concept of conversion of solids.
Answer:
Conversion of solids means changing one solid shape into another without any loss or gain of material. In such problems, the volume remains constant before and after conversion. For example, a cylindrical metal rod may be melted and recast into spherical balls. To solve such questions, we equate the volume of the original solid with the volume of the new solid. This concept is important in manufacturing industries where metals are melted and reshaped. Understanding volume conservation helps solve practical problems involving casting, molding, and reshaping of materials.
Q9. Why are units important in surface area and volume calculations?
Answer:
Units play a vital role in mathematics because they indicate the measurement scale being used. Surface area is measured in square units such as cm², m², or km² because it represents two-dimensional coverage. Volume is measured in cubic units such as cm³ or m³ because it represents three-dimensional space. Using incorrect units can lead to wrong answers even if calculations are correct. Before solving a problem, all measurements should be converted into the same unit system. Proper unit usage ensures accuracy and helps avoid confusion in practical applications.
Q10. What is the difference between capacity and volume?
Answer:
Volume refers to the amount of space occupied by a solid object, whereas capacity refers to the amount of liquid a container can hold. Volume is usually measured in cubic units such as cm³ or m³. Capacity is measured in liters or milliliters. The relationship between them is important: 1000 cm³ equals 1 liter. For example, the volume of a water tank indicates its size, while its capacity indicates how much water it can store. Understanding this difference is useful in daily life and in solving measurement-based mathematical problems.
Q11. What happens to the volume of a sphere if its radius is doubled?
Answer:
The volume of a sphere depends on the cube of its radius according to the formula (\frac{4}{3}\pi r^3). If the radius is doubled, the new radius becomes (2r). Substituting into the formula gives (\frac{4}{3}\pi(2r)^3 = \frac{4}{3}\pi(8r^3)). Thus, the volume becomes eight times the original volume. This shows that volume increases much faster than the radius. Understanding this relationship helps solve questions involving changes in dimensions and their effects on volume in real-life situations.
Q12. What is the importance of surface area in daily life?
Answer:
Surface area is important because it helps determine the amount of material needed to cover an object. For example, when painting a wall, covering a tank with metal sheets, or wrapping a gift box, surface area calculations are required. Engineers and architects use surface area to estimate construction materials and costs. Manufacturers also calculate surface area when designing containers and packages. Understanding surface area helps reduce wastage and improve efficiency. Therefore, it has practical applications in construction, packaging, painting, and many other fields.
Q13. How can you find the slant height of a cone?
Answer:
The slant height of a cone can be found using the Pythagoras theorem when the radius and vertical height are known. A right triangle is formed by the radius, height, and slant height. Therefore, the formula is (l=\sqrt{r^2+h^2}), where (l) is the slant height, (r) is the radius, and (h) is the height. The slant height is necessary for calculating the curved surface area and total surface area of a cone. It represents the actual distance along the sloping side of the cone from the base edge to the vertex.
Q14. Explain the formula for the volume of a sphere.
Answer:
The volume of a sphere is given by the formula (\frac{4}{3}\pi r^3), where (r) is the radius. This formula shows that the volume depends on the cube of the radius. Therefore, even a small increase in radius causes a significant increase in volume. The formula is derived using advanced geometric principles but is widely used in school mathematics. It helps calculate the amount of space inside spherical objects such as balls, tanks, and globes. The result is always expressed in cubic units.
Q15. What is the difference between CSA and TSA of a hemisphere?
Answer:
The Curved Surface Area (CSA) of a hemisphere includes only the curved portion of the surface and is given by (2\pi r^2). The Total Surface Area (TSA) includes both the curved surface and the circular base. Therefore, TSA is (3\pi r^2). The difference between TSA and CSA is the area of the circular base, which equals (\pi r^2). Understanding this distinction is important because some questions require only the curved surface while others require the entire surface. Careful reading of the problem helps choose the correct formula.
Q16. Why is volume measured in cubic units?
Answer:
Volume measures the space occupied by a three-dimensional object. Since length, breadth, and height are all involved, the units are multiplied three times. For example, if measurements are in centimeters, the volume is expressed in cubic centimeters (cm³). Cubic units represent three-dimensional space accurately. Using square units for volume would be incorrect because square units measure area, not space. Cubic measurements are commonly used in construction, packaging, storage, and engineering. They provide a standard way to compare the capacities and sizes of different objects.
Q17. What do you understand by recasting of solids?
Answer:
Recasting of solids refers to melting a solid object and reshaping it into another form without changing the amount of material. Since no material is lost, the volume remains constant. For example, a metal cube may be melted and recast into a cylinder or sphere. In mathematical problems, the volume of the original solid is equated to the volume of the new solid. This principle helps determine unknown dimensions of the newly formed object. Recasting is widely used in industries for manufacturing tools, machine parts, and decorative items.
Q18. How does increasing the radius affect the surface area of a sphere?
Answer:
The surface area of a sphere is given by (4\pi r^2). Since the radius is squared, any increase in radius causes the surface area to increase rapidly. For example, if the radius is doubled, the surface area becomes four times the original surface area because ((2r)^2 = 4r^2). This relationship is important in understanding how dimensions affect measurements. Engineers and designers use this concept when creating spherical objects such as storage tanks, balls, and domes. It helps estimate material requirements accurately.
Q19. Why do we use π in surface area and volume formulas?
Answer:
The symbol π (pi) is a mathematical constant approximately equal to 3.14 or 22/7. It appears in formulas involving circles, cylinders, cones, and spheres because these shapes contain circular parts. Pi represents the ratio of a circle’s circumference to its diameter. Since surface area and volume formulas of many solids are derived from circles, π naturally becomes part of the calculations. It helps provide accurate measurements of curved shapes. Without π, calculations involving circular and spherical objects would not be mathematically correct.
Q20. Mention some real-life applications of surface areas and volumes.
Answer:
Surface areas and volumes are used extensively in daily life and professional fields. Surface area helps determine the amount of paint needed for walls, metal sheets for tanks, or wrapping paper for gifts. Volume helps calculate the capacity of water tanks, fuel containers, and storage units. Engineers use these concepts in designing buildings, roads, and machines. Manufacturers calculate surface area and volume while producing containers and packaging materials. These applications make the chapter highly practical and relevant, helping students understand how mathematics is applied in real-world situations.