Practice Questions

Identify the solid shape that has a curved surface and two identical, parallel circular faces.

Explain the difference between 'area' and 'perimeter' for a closed plane figure.

Define the term 'mensuration'.

Calculate the area of a rhombus whose diagonals are 16 cm and 10 cm long.

Calculate the volume of a cube with a side length of 6 cm.

Calculate the area of a trapezium whose parallel sides are 12 cm and 8 cm, and the height is 5 cm.

State the formula for the area of a trapezium.

List the number of faces, vertices, and edges for a cube.

Justify why the formula for the area of a trapezium, $A = \frac{1}{2}(a+b)h$, can be interpreted as multiplying the average of the parallel sides by the height.

The volume of a cuboid is 360 m³. If its length is 12 m and breadth is 5 m, solve for its height.

A student claims that if you double the radius of a cylinder while keeping its height constant, its volume also doubles. Critique this statement.

You need to create a closed cylindrical can that holds exactly $1.54$ litres of juice. Design the can by determining its radius if the height is fixed at $10 \text{ cm}$. Then, formulate the total area of the metal sheet required to make it. (Use $\pi = \frac{22}{7}$ and $1$ litre = $1000 \text{ cm}^3$)

What is the formula for the total surface area of a cube with side length 's'?

State the formula for the volume of a cuboid with length $l$, breadth $b$, and height $h$.

Describe the key properties of the diagonals of a rhombus and state the formula for its area using these diagonals.

A cuboidal water tank is 6 m long, 5 m wide and 4.5 m deep. Solve for the number of litres of water it can hold, given that 1 m³ = 1000 litres.

List the formulas for the following for a cylinder with radius $r$ and height $h$: (i) Area of the circular base (ii) Curved Surface Area (iii) Total Surface Area

List and state the formulas for the area of the following 2-D shapes: (i) A square with side 's' (ii) A rectangle with length 'l' and breadth 'b' (iii) A triangle with base 'b' and height 'h' (iv) A circle with radius 'r' (v) A rhombus with diagonals $d_1$ and $d_2$

Describe the method to find the area of a general quadrilateral using one of its diagonals.

A road roller has a diameter of 84 cm and a length of 1.2 m. Calculate the area of the road it covers in 500 complete revolutions. (Use $\pi = \frac{22}{7}$)

The floor of a building consists of 3000 rhombus-shaped tiles. The diagonals of each tile are 45 cm and 30 cm. Calculate the total cost of polishing the floor, if the cost per m² is ₹4.

A rectangular garden is 90 m long and 75 m broad. A path 5 m wide is to be built around the outside of the garden. Calculate the area of the path. Also, calculate the cost of cementing the path at the rate of ₹15 per m².

A cylindrical road roller has a diameter of $98 \text{ cm}$ and is $1.2 \text{ m}$ long. It takes 800 complete revolutions to level a playground. Evaluate the cost of levelling the playground at a rate of ₹1.50 per $\text{m}^2$.

Explain the meaning of 'Lateral Surface Area' for a solid like a cylinder or a cuboid.

Explain each component of the formula for the total surface area of a cuboid: $TSA = 2(lb + bh + hl)$.

Calculate the lateral surface area of a cylinder with a base radius of 7 cm and a height of 10 cm. (Use $\pi = \frac{22}{7}$)

Solve for the side length of a cube if its total surface area is 294 cm².

The length, breadth and height of a room are 5 m, 4 m and 3 m respectively. Calculate the cost of white washing the four walls and the ceiling of the room at the rate of ₹7.50 per m².

Justify whether the lateral surface area of a cube can ever be numerically equal to its volume. If so, formulate the condition.

Formulate a single, simplified expression for the total surface area of a hollow cylinder with outer radius $R$, inner radius $r$, and height $h$.

Two cubes have volumes in the ratio $1:27$. Evaluate the ratio of their total surface areas and justify your method.

Design a closed cuboidal box using a sheet of metal with a total surface area of $188 \text{ cm}^2$. The length of the box must be $8 \text{ cm}$ and its breadth must be $5 \text{ cm}$. Propose the required height for this box.

A field is in the shape of a rhombus with a perimeter of $200 \text{ m}$ and one diagonal of $80 \text{ m}$. The farmer wants to divide the field equally for his two children. Justify that each child will receive an area of $1200 \text{ m}^2$.

A swimming pool is $50 \text{ m}$ long and $25 \text{ m}$ wide. Its depth varies linearly from $1.5 \text{ m}$ at the shallow end to $3.5 \text{ m}$ at the deep end. Evaluate the capacity of the pool in kiloliters, justifying the shape you use for calculation. (Note: $1 \text{ m}^3 = 1$ kiloliter)

A closed cylindrical tank has a height of 1 m and a base diameter of 140 cm. The tank is made from a sheet of metal. Solve for the area of the metal sheet required to make the tank. If the metal sheet costs ₹80 per m², find the total cost of the sheet for the tank. (Use $\pi = \frac{22}{7}$)

A company designs a cylindrical container for a product with a required volume of $1232 \text{ cm}^3$ and a height of $8 \text{ cm}$. (a) Design the container by finding its radius. (b) A label is placed around the curved surface, leaving a $1 \text{ cm}$ margin from the top and bottom. Evaluate the area of this label. (c) The company considers an alternative cubical container to hold the same volume. Evaluate which container shape (cylinder or cube) is more material-efficient by comparing their total surface areas. (Use $\pi = \frac{22}{7}$)

Propose a practical method to find the area of an irregular quadrilateral field $PQRS$ using only a measuring tape. Formulate the steps and the final calculation.

The floor of a hall is to be covered with 2500 rhombus-shaped tiles. The diagonals of each tile are $40 \text{ cm}$ and $25 \text{ cm}$. (a) Formulate and calculate the total cost of polishing the floor at a rate of ₹22 per $\text{m}^2$. (b) The contractor claims that using square tiles of side $30 \text{ cm}$ would reduce the polishing cost. Critique this claim and justify your conclusion.

A wire is bent into the shape of a square with a side of 11 cm. If the same wire is rebent into a rectangle of length 12 cm, analyze which shape encloses more area and by how much.

A decorative solid is created by joining two identical cubes of edge $6 \text{ cm}$ face-to-face. A cylindrical hole with a diameter of $4.2 \text{ cm}$ is then drilled through the center of the resulting cuboid, from one end to the other. Create a plan to find the surface area of the remaining solid and then calculate it. (Use $\pi = \frac{22}{7}$)

Explain why the unit for volume is a cubic unit (like $cm^3$ or $m^3$) while the unit for area is a square unit (like $cm^2$ or $m^2$).

A student calculates the volume of a cylinder with radius $r=10 \text{ cm}$ and height $h=7 \text{ cm}$ using the incorrect formula $V = 2\pi r h$. Critique the student's formula and evaluate the percentage error in their result compared to the correct volume. (Use $\pi = \frac{22}{7}$)

Describe the surfaces that make up a right circular cylinder. Explain how the formula for its curved surface area, $2\pi rh$, is derived.

Examine two cylindrical containers. The first has a diameter of 14 cm and a height of 10 cm. The second has a diameter of 10 cm and a height of 14 cm. Without calculation, predict which has a greater volume. Then, calculate both volumes to verify. (Use $\pi = \frac{22}{7}$)

A solid metal cube with a side of 12 cm is melted and recast into 8 smaller identical solid cubes. Analyze the relationship between the volumes to solve for the side length of each new cube. Also, compare the total surface area of the original cube with the sum of the total surface areas of the 8 smaller cubes.