Surface Area & Volume: Combination Solids MCQs Quiz | Class 10
This quiz covers important Multiple Choice Questions (MCQs) for Class X Mathematics (Code 041), focusing on Unit VI: Mensuration. Specifically, it tests your understanding of Surface Area and Volume calculations for Combination Solids, including cubes, cuboids, spheres, hemispheres, cylinders, and cones. Attempt all 10 questions and then submit to view your score. You can also download a detailed answer PDF for revision.
Understanding Surface Areas and Volumes of Combination Solids
In Class X Mathematics, a significant part of Mensuration involves calculating the surface areas and volumes of solids that are formed by combining two or more basic three-dimensional shapes. These ‘combination solids’ are common in real-world objects, from everyday toys to complex engineering structures.
Key Concepts for Combination Solids:
- Surface Area: When calculating the surface area of a combined solid, you must be careful not to simply add the surface areas of the individual solids. Instead, consider only the ‘visible’ surfaces that make up the exterior of the new combined solid. Any surfaces that are “hidden” inside the combination are not included. For example, if a cone is placed on a cylinder, the base of the cone and the top circular face of the cylinder are no longer exposed and thus are not part of the total surface area.
- Volume: Unlike surface area, the volume of a combined solid is always the sum or difference of the volumes of its constituent parts. If solids are joined together, their volumes add up. If a part is hollowed out (like a conical cavity from a cylinder), the volume of the hollowed part is subtracted from the volume of the original solid.
Common Combinations and Their Approaches:
- Cylinder + Hemisphere / Cone: Often seen in toys, decorative pieces, or vessels.
- Surface Area: Curved surface area of cylinder + curved surface area of hemisphere/cone. (Base area of cylinder or hemisphere will be exposed if it is at the bottom/top respectively, depending on combination.)
- Volume: Volume of cylinder + Volume of hemisphere/cone.
- Cuboid + Hemisphere / Cone / Cylinder: Examples include pen stands, storage blocks.
- Surface Area: Surface area of cuboid – Area of contact surface + Curved surface area of added solid.
- Volume: Volume of cuboid +/- Volume of the added/removed solid.
- Sphere / Hemisphere within or outside a Cylinder / Cone: Water displacement problems, melting and recasting.
- Volume: Volume of the outer solid – Volume of the inner solid (for water displacement) or equating volumes for melting/recasting.
- Surface Area: Depends heavily on what part is exposed.
Important Formulas for Basic Solids:
| Solid | Curved Surface Area (CSA) | Total Surface Area (TSA) | Volume |
|---|---|---|---|
| Cube (side ‘a’) | 4a^2 | 6a^2 | a^3 |
| Cuboid (l, b, h) | 2h(l+b) | 2(lb+bh+hl) | lbh |
| Cylinder (r, h) | 2 pi r h | 2 pi r (h+r) | pi r^2 h |
| Cone (r, h, l) | pi r l | pi r (l+r) | (1/3) pi r^2 h |
| Sphere (r) | – | 4 pi r^2 | (4/3) pi r^3 |
| Hemisphere (r) | 2 pi r^2 | 3 pi r^2 | (2/3) pi r^3 |
Quick Revision Tips:
- Always draw a clear diagram of the combined solid.
- Identify which surfaces are exposed for surface area calculation.
- For volume, identify the individual solids and whether their volumes should be added or subtracted.
- Pay close attention to units (cm, m, cubic cm, sq cm, etc.).
- Use the value of pi as specified in the question (22/7 or 3.14).
Practice Questions (No Options):
- A storage tank is made of a cylindrical portion and a hemispherical portion. If the diameter is 14 m and the total height is 18 m, find the total surface area of the tank.
- A wooden toy rocket is in the shape of a cone mounted on a cylinder. The height of the conical part is 6 cm and its base diameter is 5 cm, while the cylindrical part has a height of 12 cm and base diameter of 3 cm. Find the volume of the wooden rocket. (Use pi = 3.14)
- A building is in the form of a cylinder surmounted by a hemispherical dome. The diameter of the dome is equal to the height of the cylinder. If the total height of the building is 10 m, find the volume of the building. (Use pi = 22/7)
- A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of pi.
- From a solid right circular cylinder with height 10 cm and radius of base 6 cm, a right circular cone of the same height and base radius is removed. Find the volume of the remaining solid. (Use pi = 3.14)
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