Mensuration

A general quadrilateral $ABCD$ has diagonal $BD = 30$ cm. The heights $h_1$ and $h_2$ from $A$ and $C$ to the diagonal are $10$ cm and $12$ cm. Calculate its area.

If the area of a trapezium is $28$ cm$^2$ and its parallel sides are $8$ cm and $6$ cm, find its altitude.

The area of a rhombus is $96$ cm$^2$. If its side is $10$ cm, find its altitude.

Find the area of a quadrilateral where the diagonal is $12$ cm and the sum of perpendiculars from the opposite vertices is $10$ cm.

In the given figure, $ABCD$ is a trapezium where $AD \parallel BC$. The parallel sides are $AD = 12\text{ cm}$ and $BC = 20\text{ cm}$. If the area of the trapezium is $128\text{ cm}^2$, find the perpendicular distance (height) between the parallel sides.

A trapezium has parallel sides of $10$ cm and $22$ cm. The other two sides are $10$ cm each. Find the area of the trapezium.

In a quadrilateral $ABCD$, $AB = 7$ cm, $BC = 12$ cm, $CD = 12$ cm, $DA = 9$ cm and diagonal $AC = 15$ cm. What is the area of this quadrilateral? (Assume $\angle ABC$ is not necessarily $90^\circ$)

The parallel sides of a trapezium are $25$ cm and $11$ cm, while its non-parallel sides are $15$ cm and $13$ cm. Calculate the area of this trapezium.

A floor consists of $2000$ tiles which are in the shape of a rhombus. The diagonals of each tile are $40$ cm and $25$ cm. Find the total area of the floor in square meters.

An industrial component is shaped like a trapezium. The sum of the parallel sides is $60$ cm and the area is $600$ cm$^2$. If the non-parallel sides are $13$ cm and $15$ cm, find the height of the trapezium.

A quadrilateral has a diagonal of length $20$ cm and the offsets from the opposite vertices to this diagonal are $8$ cm and $12$ cm. Find its area.

Find the area of a circle with a radius of $7$ cm. (Take $\pi = \frac{22}{7}$)

The area of a rectangle is $96$ cm$^2$. If its length is $12$ cm, find its width.

Find the area of a right-angled triangle with legs of $5$ cm and $12$ cm.

Calculate the area of a sector of a circle with radius $21$ cm and central angle $60^{\circ}$. (Take $\pi = \frac{22}{7}$)

Find the area of a polygon $ABCDE$ where $ABC$ is an equilateral triangle with side $8$ cm and $ACDE$ is a rectangle with $CD = 5$ cm.

The area of a triangle is equal to the area of a square of side $12$ cm. If the height of the triangle is $16$ cm, find its base.

A rectangular room is $10$ m long, $8$ m wide and $4$ m high. It has one door of $2$ m by $1.5$ m and two windows each of $1.2$ m by $1$ m. Find the cost of painting the four walls at Rs $50$ per square meter.

A field is in the form of a polygon $ABCDE$ where $AC = 18$ m, $AM = 14$ m, $AN = 6$ m. $EN \perp AC$, $BM \perp AC$ and $DP \perp AC$ (with $P$ on $AC$). If $EN = 12$ m, $BM = 8$ m, and $DP = 6$ m where $P$ is at $10$ m from $A$, calculate the total area of the field.

A wire in the shape of an equilateral triangle encloses an area of $121\sqrt{3}$ cm$^2$. If the same wire is bent into the form of a circle, find the area of the circle. (Take $\pi = 22/7$)

Calculate the total surface area of a cuboid with length $5$ cm, width $4$ cm, and height $3$ cm.

If the surface area of one face of a cube is $25 \text{ cm}^2$, what is the total surface area of the cube?

A cylinder has a diameter of $14$ cm and a height of $5$ cm. Find its curved surface area.

Find the total surface area of a cube of side $10$ cm.

A metal cylinder with a radius of $7\text{ cm}$ and a height of $10\text{ cm}$ is shown below. A smaller hole in the shape of a cylinder with radius $3\text{ cm}$ is drilled vertically through the center of the cylinder. Find the total surface area of the resulting hollow cylinder (take $\pi = \frac{22}{7}$).

The lateral surface area of a cube is $256 \text{ cm}^2$. What is its total surface area?

The radius of a cylinder is doubled and the height is halved. What happens to its curved surface area?

The radius of a solid cylinder is tripled and its height is reduced to one-ninth of its original height. What is the ratio of the lateral surface area of the new cylinder to the lateral surface area of the original cylinder?

A room is $10$ m long, $8$ m wide, and $6$ m high. It has one door of dimensions $2$ m $\times$ $1.5$ m and two windows of dimensions $1.5$ m $\times$ $1$ m each. Find the cost of painting the four walls of the room at the rate of Rs $20$ per square metre.

A large cube of side $12$ cm is melted to form $n$ small cubes of side $3$ cm each. What is the ratio of the total surface area of the large cube to the sum of the total surface areas of all the $n$ small cubes?

What is the volume of a cuboid if its length is $l$, breadth is $b$, and height is $h$?

A cube-shaped box has a volume of $1000\text{ cm}^3$. What is the area of one of its faces?

Find the volume of a cylinder whose radius is $10$ cm and height is $14$ cm. (Take $\pi = 3.14$)

A cylindrical pillar has a diameter of $1$ m and a height of $7$ m. What is its volume in $\text{m}^3$? (Use $\pi = \frac{22}{7}$)

Find the volume of wood required to make a closed box of external dimensions $20\text{ cm} \times 12\text{ cm} \times 10\text{ cm}$ if the wood is $1$ cm thick.

If the side of a cube is tripled, its volume becomes how many times the original volume?

The volume of a cylinder is $448\pi\text{ cm}^3$ and its height is $7$ cm. Find its lateral surface area.

A cylindrical metallic pipe is $20\text{ cm}$ long with an outer radius of $10\text{ cm}$ and an inner radius of $8\text{ cm}$. This pipe is melted and recast into a solid cuboid of length $12\text{ cm}$ and height $11\text{ cm}$. If the value of $\pi$ is taken as $\frac{22}{7}$, what is the breadth of the resulting cuboid?

If each edge of a cube is doubled, how many times does its volume become?

A closed cylindrical tank of radius $7 \text{ m}$ and height $3 \text{ m}$ is made from a sheet of metal. How much sheet metal is required? (Use $\pi = \frac{22}{7}$)