Class 10 Mathematics · Chapter 12 NotesSurface Areas and Volumes
Revise Class 10 Mathematics Chapter 12 Surface Areas and Volumes. Learn formulas and methods for combined solids, with clear explanations and key points.
In this chapter, you learn how to find the surface areas and volumes of solids that are formed by joining two or more basic solids such as a cuboid, cone, cylinder, sphere, or hemisphere. You have already studied the surface areas and volumes of these basic solids in Class 9. Here, you extend that knowledge to everyday objects like a truck container (a cylinder with two hemispherical ends), a test tube (a cylinder with a hemispherical base), a toy (a cone on a hemisphere), a bird-bath (a cylinder with a hemispherical depression), and many more. The key idea is to break a combined solid into its basic parts, identify which surfaces are visible or hidden, and then add or subtract the appropriate areas or volumes. This chapter helps you solve real-life problems involving painting, capacity, and material required, and builds a strong foundation for further studies in geometry.
What you'll learn
1Identify the basic solids that make up a given combined solid.
2Calculate the total surface area of a combination of solids by adding only the visible curved or flat surfaces.
3Find the volume of a combined solid by adding the volumes of its constituent basic solids.
4Solve problems involving real-life objects such as tents, capsules, glasses, and toys.
5Apply the correct formulas for the surface areas and volumes of cubes, cuboids, cylinders, cones, spheres, and hemispheres.
6Determine the surface area when a part of a solid is removed, such as a hemispherical depression in a cube or cylinder.
Chapter at a glance
01Surface Area of Solids: Cubes and Cuboids
02Surface Area of Right Circular Cylinders
03Surface Area of Spheres and Hemispheres
04Surface Area of Spheres and Hemispheres
05Volume of Cubes, Cuboids, and Cylinders
06Volume of Cones and Spheres
07Real-Life Applications and Problem Solving
Detailed chapter notes
01
Surface Area of a Combination of Solids
When two basic solids are joined together, some part of their surfaces may disappear at the joint. To find the total surface area of the new solid, you add only the curved surface areas (CSA) or the exposed flat surfaces of the individual parts. For example, if a cylinder is capped with two hemispheres at its ends, the total surface area is the sum of the CSA of the two hemispheres and the CSA of the cylinder. Similarly, for a toy made of a cone and a hemisphere, the total surface area is the CSA of the hemisphere plus the CSA of the cone. You must be careful not to include the areas of the surfaces that are hidden inside the joint.
Total Surface Area (TSA) of a combined solid = sum of the curved surface areas of the visible parts.
For a solid made of a cylinder and two hemispheres: TSA = 2πrh + 2(2πr²) = 2πr(h + 2r).
For a solid made of a cone and a hemisphereTSA = πrl + 2πr².
When a hemisphere is mounted on a cube, the base area of the hemisphere is not counted because it is attached to the cube.
02
Volume of a Combination of Solids
Unlike surface area, the volume of a combined solid is simply the sum of the volumes of the individual basic solids. This is because volume is the amount of space occupied, and joining solids does not remove any space inside. For example, the volume of a solid formed by a cone on a hemisphere is the sum of the volume of the cone and the volume of the hemisphere. Similarly, if a cylinder has conical cavities hollowed out, you subtract the volume of the cavities from the volume of the original cylinder to find the volume of the remaining solid. This principle is used to solve problems about capacity, displacement, and material required.
Volume of a combined solid = sum of the volumes of the constituent solids.
Volume of a cylinder = πr²h.
Volume of a cone = (1/3)πr²h.
Volume of a sphere = (4/3)πr³.
Volume of a hemisphere = (2/3)πr³.
03
Surface Area of a Cylinder with Hemispherical Ends
A common example is a capsule or a truck container, which is a cylinder with a hemisphere at each end. The total surface area is the sum of the curved surface area of the cylinder and the curved surface areas of the two hemispheres. The flat circular faces of the cylinder are not visible because they are covered by the hemispheres. So, TSA = 2πrh + 2(2πr²) = 2πr(h + 2r). The volume of such a solid is the volume of the cylinder plus the volumes of the two hemispheres, which is πr²h + 2(2/3 πr³) = πr²h + (4/3)πr³.
TSA = 2πrh + 4πr² = 2πr(h + 2r).
Volume = πr²h + (4/3)πr³.
04
Surface Area of a Cone on a Hemisphere
A toy or a top is often shaped like a cone mounted on a hemisphere. The total surface area is the sum of the curved surface area of the cone and the curved surface area of the hemisphere. The base of the cone and the flat face of the hemisphere are joined together, so they are not part of the outer surface. Thus, TSA = πrl + 2πr², where l is the slant height of the cone. The volume is the sum of the volume of the cone and the volume of the hemisphere: V = (1/3)πr²h + (2/3)πr³.
TSA = πrl + 2πr².
Volume = (1/3)πr²h + (2/3)πr³.
05
Surface Area of a Cylinder with a Hemispherical Depression
When a hemisphere is scooped out from one end of a cylinder, as in a bird-bath, the total surface area includes the curved surface area of the cylinder, the curved surface area of the hemisphere, and the area of the remaining flat circular base of the cylinder. The flat circular area of the hemisphere is not exposed because it is inside the cylinder. So, TSA = 2πrh + 2πr² + πr² = 2πrh + 3πr². The volume of such a solid is the volume of the cylinder minus the volume of the hemispherical depression: V = πr²h - (2/3)πr³.
TSA = 2πrh + 3πr².
Volume = πr²h - (2/3)πr³.
06
Real-Life Applications and Problem Solving
The concepts of surface area and volume of combined solids are used in many real-life situations. For example, calculating the canvas required for a tent (cylinder with a conical top), finding the capacity of a glass with a hemispherical raised bottom, determining the amount of wood in a pen stand with conical depressions, or finding the volume of water left in a cylinder after placing a solid in it. To solve such problems, first identify the basic solids involved, then decide whether to add or subtract their surface areas or volumes. Always use the correct formulas and take care to use consistent units. In many problems, you may need to find the slant height of a cone using the Pythagorean theorem: l = √(r² + h²).
Identify the basic solids and how they are joined.
For surface area, include only the exposed surfaces.
For volume, add or subtract the volumes of the basic solids.
Use l = √(r² + h²) for the slant height of a cone.
Ensure all measurements are in the same unit before calculating.
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Two cubes each of volume 64 cm³ are joined end to end. Find the surface area of the resulting cuboid.
A128 cm²
B160 cm²
C192 cm²
D256 cm²
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Answer: (B) 160 cm²
Each cube has side 4 cm (since volume = 64 cm³). When joined, the cuboid has dimensions 8 cm × 4 cm × 4 cm. Surface area = 2(lb + bh + hl) = 2(8×4 + 4×4 + 4×8) = 2(32+16+32) = 160 cm².
Question 02
What is the formula for the curved surface area (CSA) of a right circular cylinder?
Aπr²h
B2πrh
C2πr(h + r)
Dπr² + 2πrh
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Answer: (B) 2πrh
The curved surface area of a cylinder is given by 2πrh, where r is the radius and h is the height.
Question 03
A cuboid has dimensions 10 cm × 5 cm × 2 cm. What is its volume?
A100 cm³
B17 cm³
C1000 cm³
D70 cm³
Show answer
Answer: (A) 100 cm³
Volume of a cuboid = length × breadth × height = 10 × 5 × 2 = 100 cm³.
Question 04
A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. What is the volume of the toy? (Take π = 3.14)
A25.12 cm³
B33.49 cm³
C16.75 cm³
D41.87 cm³
Show answer
Answer: (A) 25.12 cm³
Volume of toy = volume of hemisphere + volume of cone = (2/3)πr³ + (1/3)πr²h = (2/3)π(2)³ + (1/3)π(2)²(2) = 25.12 cm³.
Question 05
A toy is in the form of a cone mounted on a hemisphere. The total height of the toy is 15.5 cm and the radius is 3.5 cm. What is the total surface area of the toy? (Take π = 22/7)
A214.5 cm²
B137.5 cm²
C39.6 cm²
D163.86 cm²
Show answer
Answer: (A) 214.5 cm²
The total surface area of the toy is the sum of the curved surface area of the cone and the curved surface area of the hemisphere. Using the given dimensions and π = 22/7, the calculated area is 214.5 cm².
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Q1. Two cubes each of volume 64 cm³ are joined end to end. Find the surface area of the resulting cuboid.
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Model answer
Edge of each cube = cube root of 64 = 4 cm. When joined, length = 8 cm, breadth = 4 cm, height = 4 cm. Surface area = 2(lb + bh + hl) = 2(8×4 + 4×4 + 4×8) = 2(32 + 16 + 32) = 2×80 = 160 cm².
Sample question3 marks
Q2. A right circular cylinder has a radius of 7 cm and a height of 10 cm. Find its curved surface area. (Take π = 22/7)
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Model answer
Curved surface area of a cylinder = 2πrh = 2 × (22/7) × 7 × 10 = 440 cm².
Sample question3 marks
Q3. Find the surface area of a sphere of radius 7 cm.
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Model answer
The surface area of a sphere is given by 4πr². Substituting r = 7 cm and π = 22/7, we get 4 × (22/7) × 7 × 7 = 4 × 22 × 7 = 616 cm².
Sample question3 marks
Q4. A cuboid has dimensions 10 cm × 8 cm × 5 cm. Find its volume.
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Model answer
Volume of a cuboid = length × breadth × height. Here, length = 10 cm, breadth = 8 cm, height = 5 cm. So, volume = 10 × 8 × 5 = 400 cm³.
Sample question3 marks
Q5. A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. Find the volume of the toy. (Take π = 3.14)
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Model answer
Radius r = 2 cm. Volume of hemisphere = (2/3)πr³ = (2/3)×3.14×8 = 16.75 cm³. Volume of cone = (1/3)πr²h = (1/3)×3.14×4×2 = 8.37 cm³. Total volume = 16.75 + 8.37 = 25.12 cm³.
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What is the formula for the surface area of a combination of solids?
The total surface area of a combination of solids is the sum of the curved surface areas of the individual visible parts. For example, for a cylinder with two hemispherical ends, TSA = 2πrh + 4πr² = 2πr(h + 2r). You do not include the areas of the surfaces that are hidden at the joints.
How do you find the volume of a solid made by joining two basic solids?
The volume of a combined solid is simply the sum of the volumes of the individual basic solids. For instance, if a cone is mounted on a hemisphere, the total volume is (1/3)πr²h + (2/3)πr³. If a part is removed, subtract its volume from the original solid's volume.
What is the difference between surface area and volume of a combined solid?
Surface area measures the total area of the outer surfaces that are exposed, so you add only the visible curved or flat surfaces. Volume measures the space occupied by the solid, so you add the volumes of all constituent parts, even those hidden inside.
How do you find the surface area of a bird-bath shaped like a cylinder with a hemispherical depression?
The total surface area includes the curved surface of the cylinder, the curved surface of the hemisphere, and the flat circular base of the cylinder. So, TSA = 2πrh + 2πr² + πr² = 2πrh + 3πr². The flat circular area of the hemisphere is not exposed.
What is the formula for the slant height of a cone?
The slant height l of a cone is given by l = √(r² + h²), where r is the base radius and h is the height of the cone. This comes from the Pythagorean theorem applied to the right triangle formed by the radius, height, and slant height.
How do you solve problems involving a cylinder with conical cavities?
First, find the volume of the original cylinder using πr²h. Then, find the volume of one conical cavity using (1/3)πr²h, and multiply by the number of cavities. Subtract the total cavity volume from the cylinder's volume to get the volume of the remaining solid.