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Surface Areas and Volumes

Action Plan Notes Questions Mock-Test

Study Action Plan (Sorted by Difficulty)

Hollow and Cut-out Solids

Accuracy: 0%

Total Attempted1

Total Correct0

Total Incorrect1

Difficulty Level:

High

Boundary Conditions in Composite Solids

Accuracy: 33%

Total Attempted3

Total Correct1

Total Incorrect2

Difficulty Level:

High

Mathematical Constraints in Volume Calculations

Accuracy: 44%

Learning Objectives

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Learning Objectives

Calculate the total surface area of composite solids formed by combining basic shapes like cylinders, cones, and hemispheres, focusing on curved surface areas and boundary conditions.
Determine the volume of composite solids by summing the individual volumes of basic shapes, ensuring no overlap in calculations.
Derive and apply formulas for the curved surface areas of cylinders, cones, and hemispheres, using $\pi$ and specific dimensions.
Understand the impact of boundary conditions on surface area calculations, particularly where surfaces are internal or not visible in composite solids.
Apply mathematical constraints and assumptions in volume calculations to accurately represent the structure of combined solid shapes.

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Revision Notes & Summary

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Chapter Notes

Surface Area of Composite Solids

Total Surface Area (TSA): For a composite solid made by combining basic solids, the TSA is the sum of the curved surface areas (CSA) of the individual components that are visible.
Example: A cylinder with two hemispheres at its ends:
- TSA = CSA of one hemisphere + CSA of cylinder + CSA of other hemisphere.

Volume of Composite Solids

Total Volume: The volume of a composite solid is the sum of the volumes of the individual solids.
Example: A cuboid surmounted by a half-cylinder:
- Volume = Volume of cuboid + Volume of half-cylinder.

Curved Surface Area Calculations

Cylinder: CSA = $2\pi rh$ , where $r$ is the radius and $h$ is the height.
Cone: CSA = $\pi rl$ , where $l$ is the slant height.
Hemisphere: CSA = $2\pi r^2$ , where $r$ is the radius.

Boundary Conditions in Composite Solids

When solids are joined, internal surfaces are not included in the TSA.
Example: A hemisphere attached to a cube does not include the base area of the hemisphere in the TSA.

Mathematical Constraints in Volume Calculations

Assumptions: Ensure no overlap in the volume of combined solids.
Example: A cylinder with a conical cavity:
- Volume = Volume of cylinder - Volume of cone.

Examples

Example 1: A toy shaped like a cone on a hemisphere:
- TSA = CSA of hemisphere + CSA of cone.
Example 2: A decorative block with a cube and hemisphere:
- TSA = TSA of cube - base area of hemisphere + CSA of hemisphere.
Example 3: A rocket with a cone on a cylinder:
- Orange area = CSA of cone + base area of cone - base area of cylinder.
- Yellow area = CSA of cylinder + area of one base of the cylinder.
Example 4: A bird-bath shaped like a cylinder with a hemispherical depression:
- TSA = CSA of cylinder + CSA of hemisphere.

Exercises

Exercise 1: Two cubes joined end to end, find the surface area of the resulting cuboid.
Exercise 2: A vessel with a hemispherical base and cylindrical top, find the inner surface area.
Exercise 3: A toy with a cone on a hemisphere, find the TSA.
Exercise 4: A cube with a hemisphere on top, find the maximum diameter of the hemisphere and TSA.
Exercise 5: A cube with a hemispherical depression, find the remaining surface area.
Exercise 6: A capsule with two hemispheres and a cylinder, find the surface area.

Summary

Surface Area: Combine the CSA of visible parts.
Volume: Sum the volumes of individual solids without overlap.

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Practice Test – MCQs, True/False

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Multiple Choice Questions

CSA of hemisphere + CSA of cone

CSA of hemisphere + TSA of cone

TSA of hemisphere + CSA of cone

TSA of hemisphere + TSA of cone

Correct Answer: A

Solution:

The total surface area is the sum of the curved surface areas of the hemisphere and the cone.

Chapter Concept:

Surface Area of Composite Solids

220 mm²

235 mm²

250 mm²

265 mm²

Correct Answer: B

Solution:

The capsule consists of a cylindrical part and two hemispherical ends. The radius

r = 2.5

mm and the height of the cylinder

h = 14 - 2 \times 2.5 = 9

mm. Surface area of the cylinder =

2 \pi r h = 2 \times 3.14 \times 2.5 \times 9 = 141.3

mm². Surface area of two hemispheres =

2 \times 2 \pi r^2 = 2 \times 2 \times 3.14 \times 2.5^2 = 78.5

mm². Total surface area =

141.3 + 78.5 = 219.8

mm², approximately 220 mm².

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Surface Areas and Volumes

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True or False

Correct Answer: False

Solution:

When calculating the surface area of a composite solid, overlapping areas are not counted twice. Therefore, the total surface area is not simply the sum of the total surface areas of the individual components.

Chapter Concept :

Mathematical Constraints in Volume Calculations

Correct Answer: True

Solution:

When calculating the volume of a composite solid, the volume is the sum of the volumes of its individual components, as no volume is lost in the process of combining them.

Chapter Concept :

Mathematical Constraints in Volume Calculations

Correct Answer: True

Solution:

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Surface Areas and Volumes

Study Action Plan (Sorted by Difficulty)

Hollow and Cut-out Solids

High

Boundary Conditions in Composite Solids

High

Mathematical Constraints in Volume Calculations

Learning Objectives

Learning Objectives

Practice, Analyze & Improve 🚀

Revision Notes & Summary

Chapter Notes

Surface Area of Composite Solids

Volume of Composite Solids

Curved Surface Area Calculations

Boundary Conditions in Composite Solids

Mathematical Constraints in Volume Calculations

Examples

Exercises

Summary

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Practice Test – MCQs, True/False

Multiple Choice Questions

Q1.A toy is made by combining a hemisphere and a cone. If the radius of the hemisphere is equal to the base radius of the cone, how do you calculate the total surface area of the toy?

Correct Answer: A

Q2.A medicine capsule is in the shape of a cylinder with two hemispheres attached at each end. The total length of the capsule is 14 mm and the diameter is 5 mm. What is the surface area of the capsule? (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: B

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True or False

Q1.The total surface area of a composite solid is always the sum of the total surface areas of its individual components.

Correct Answer: False

Q2.The volume of a solid formed by combining two basic solids is the sum of the volumes of the individual solids.

Correct Answer: True

Q3.The total surface area of a solid formed by a cylinder with two hemispheres at its ends is the sum of the curved surface areas of the cylinder and the two hemispheres.

Correct Answer: True

High

Curved Surface Area Calculations

Medium

Surface Area of Composite Solids

Medium

Capacity and Displacement Problems

Low

Volume of Composite Solids

Low

Q3.A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of the same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy.

Correct Answer: A

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Q4.A decorative block is made of a cube with edge length 5 cm and a hemisphere with a diameter of 4.2 cm placed on top. Calculate the total surface area of the block, excluding the base of the hemisphere.

Correct Answer: A

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Q5.A solid is formed by joining a cylinder with a height of 10 cm and radius 3 cm, and a hemisphere with the same radius on top of the cylinder. What is the total volume of the solid? (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: B

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Q6.A toy is made by mounting a cone on a hemisphere. If the radius of both the cone and hemisphere is 2 cm and the height of the cone is 2 cm, what is the total surface area of the toy? (Take π=3.14\pi = 3.14π=3.14)

Correct Answer: A

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Q7.A toy is made by combining a hemisphere and a cone with equal radii. If the radius of both the hemisphere and the cone is 3 cm, and the height of the cone is 4 cm, what is the total surface area of the toy?

Correct Answer: C

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Q8.A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm14 \text{ cm}14 cm and the total height of the vessel is 13 cm13 \text{ cm}13 cm. Find the inner surface area of the vessel.

Correct Answer: B

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Q9.A cylindrical tank with a hemispherical top is filled with water. The height of the cylindrical part is 1.45 m and the radius is 30 cm. Calculate the total surface area of the tank excluding the base. (Take π=227\pi = \frac{22}{7}π=722​)

Correct Answer: A

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Q10.What is the formula for the curved surface area (CSA) of a cylinder with radius rrr and height hhh?

Correct Answer: B

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