Math of Space: Surface Area and Volume: CBSE Class 9 Mathematics practice questions

Practise 50 questions on Math of Space: Surface Area and Volume, free, with an explanation for each answer.

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Sample questions

The volume of a cube is $64\ \text{cm}^3$. What is its total surface area?

1. $24\ \text{cm}^2$
2. $96\ \text{cm}^2$
3. $384\ \text{cm}^2$
4. $16\ \text{cm}^2$

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B. $96\ \text{cm}^2$

Since $a^3 = 64$, the side is $a = 4\ \text{cm}$. The cube has six square faces, so $\text{TSA} = 6a^2 = 6 \times 16 = 96\ \text{cm}^2$. The trap is stopping early: $16\ \text{cm}^2$ is only one face, $24$ comes from $6a$ (forgetting to square) and $384$ from $6 \times 64$ (using the volume instead of $a^2$).

A cube of side $4\ \text{cm}$ is cut into cubes of side $1\ \text{cm}$. What is the ratio of the surface area of the original cube to the total surface area of all the small cubes?

1. $1 : 4$
2. $4 : 1$
3. $1 : 64$
4. $1 : 16$

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A. $1 : 4$

Original: $6 \times 4^2 = 96\ \text{cm}^2$. Number of small cubes $= 4^3 = 64$, each with area $6 \times 1^2 = 6$, so total $= 384\ \text{cm}^2$. Ratio $96 : 384 = 1 : 4$. The volume has not changed but the surface area has grown four times; reversing the ratio gives $4:1$, and $1:64$ would compare only counts of cubes.

The radius of a cylinder is increased by $10\%$ and the height is decreased by $x\%$. The volume stays the same. What is $x$?

1. about 17.4%
2. 10%
3. 21%
4. about 9.1%

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A. about 17.4%

$(1.1)^2 \times (1 - \tfrac{x}{100}) = 1$, so $1 - \tfrac{x}{100} = \tfrac{1}{1.21} \approx 0.826$ and $x \approx 17.4$. The radius enters squared, so $10\%$ does not cancel $10\%$. $21\%$ is the rise in $r^2$, and $9.1\%$ comes from using $1/1.1$ instead of $1/1.21$.

Topics covered

• Surface area of a cube
• Packing cuboids in a cuboid
• Cube versus cuboid
• Surface area when a solid is cut up
• Volume from face areas
• Painted cube cut into small cubes
• Painted cube counts
• Choosing a swimming pool
• Parts of a cylinder
• Curved surface of a cylinder
• Curved surface area of a cylinder
• Cylinder closed at one end
• Changing a cylinder
• Volume kept constant
• Melting and recasting
• Steps of a recasting problem
• Slant height of a cone
• Height of a cone
• Cone triples
• Total surface area of a cone
• Radius from volume of a cone
• Cloth for a conical tent
• Steps for a conical tent
• Solid formed by rotating a triangle
• Water in a conical cup
• Cone and cylinder volumes
• Cone and its net
• Parts of a pyramid
• Volume of a pyramid
• Volume formulas
• Surface area formulas
• Archimedes and the sphere
• Parts of a hemisphere
• Total surface area of a hemisphere
• Spheres packed in a box
• Painting a hemispherical dome
• Scaling a sphere
• Comparing volumes of solids
• Ratios of solids
• Guesstimate: golgappas
• Packing tennis balls
• How to make a guesstimate
• Litres of milk
• Marbles raise the water level
• Cone and sphere together
• Comparing planets
• String round a sphere
• Which quantity?

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More CBSE Class 9 Mathematics chapters

• Orienting Yourself: The Use of Coordinates
• Introduction to Linear Polynomials
• The World of Numbers
• Exploring Algebraic Identities
• I’m Up and Down, and Round and Round
• Measuring Space: Perimeter and Area
• The Mathematics of Maybe: Introduction to Probability
• Predicting What Comes Next: Exploring Sequences and Progressions
• Propositions and Their Converses
• How Quantities Combine: Understanding Data
• The World of Algorithms
• Quadrilaterals
• Two Variables, One Line