Mensuration: Surface Areas and Volumes

Find the area of a circle with radius $7$ cm.

The ratio of the volumes of two spheres is $8:27$. What is the ratio of their surface areas?

Find the total surface area of a hemisphere of radius $10$ cm. (Use $\pi = 3.14$)

The surface area of a solid hemisphere is $462 \text{ cm}^2$. Find its radius.

A solid toy is in the form of a cylinder with hemispherical ends. The total height of the toy is $19 \text{ cm}$ and the diameter of the cylinder is $7 \text{ cm}$. Find the total surface area of the toy. (Use $\pi = \frac{22}{7}$)

A solid iron pole consists of a cylinder of height $220$ cm and base diameter $24$ cm, which is surmounted by another cylinder of height $60$ cm and radius $8$ cm. Find the mass of the pole, given that $1 \text{ cm}^3$ of iron has approximately $8$ g mass. (Use $\pi = 3.14$)

A container shaped like a right circular cylinder having diameter $12$ cm and height $15$ cm is full of ice cream. The ice cream is to be filled into cones of height $12$ cm and diameter $6$ cm, having a hemispherical shape on the top. Find the number of such cones which can be filled.

A hemispherical bowl of internal radius 9 cm is full of liquid. This liquid is to be filled into cylindrical shaped small bottles each of diameter 3 cm and height 4 cm. How many bottles are necessary to empty the bowl?

A cone of height 24 cm and radius of base 6 cm is made up of modeling clay. A child reshapes it in the form of a sphere. Find the surface area of the sphere formed.

A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total surface area of the solid. (Use $\pi = \frac{22}{7}$)

Two cubes each of volume $64$ cm$^3$ are joined end to end. Find the volume of the resulting cuboid.

A cylindrical tank of radius $2$ m and height $7$ m has a hemispherical lid. Find the volume of the tank.

A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to $1$ cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of $\pi$.

Find the volume of a solid which is a combination of a cylinder of height $10$ cm and radius $6$ cm with two cones at the ends each of height $4$ cm.

An engineering student was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminum sheet. The diameter of the model is $3$ cm and its length is $12$ cm. If each cone has a height of $2$ cm, find the volume of air contained in the model.

A solid consists of a right circular cone of height $120$ cm and radius $60$ cm standing on a hemisphere of radius $60$ cm. It is placed upright in a right circular cylinder full of water such that it touches the bottom. If the cylinder's radius is $60$ cm and height is $180$ cm, find the volume of water left in the cylinder.

A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is $14$ mm and the diameter of the capsule is $5$ mm. Find its volume.

A solid toy is in the form of a hemisphere surmounted by a right circular cone of the same radius. The height of the cone is $8$ cm and the radius of its base is $6$ cm. If a cube of side $2$ cm is carved out from the bottom of the hemispherical part, find the volume of the remaining toy. (Take $\pi = 3.14$)

An industrial storage tank is composed of a cylinder of height $10$ m and radius $3$ m, with two identical cones attached to each of its circular ends. If the total volume of the tank is $102\pi$ m$^3$, find the vertical height of each cone.

A decorative pillar is designed by mounting a hemisphere on top of a cylinder. The radius of both the hemisphere and the cylinder is $7$ cm, and the total height of the pillar is $30$ cm. A cylindrical hole of radius $2$ cm is drilled through the center of the entire pillar along its vertical axis of symmetry to accommodate a light fixture. Calculate the volume of the material used in the pillar. (Use $\pi = \frac{22}{7}$)

A solid metallic sphere of radius $3$ cm is melted and recast into a solid right circular cone of radius $3$ cm. Find the height of the cone so formed.

A metallic cylinder of radius 4 cm and height 10 cm is melted and cast into a sphere. What is the radius of the sphere?

If a solid sphere of radius 10 cm is molded into 8 identical small solid spheres, what is the radius of each small sphere?

A cone of height 20 cm and radius of base 5 cm is melted and recast into a sphere. What is the diameter of the sphere?

Find the number of metallic circular discs with 1.5 cm base radius and height 0.2 cm to be melted to form a right circular cylinder of height 10 cm and diameter 4.5 cm.

How many spherical lead shots each 4.2 cm in diameter can be obtained from a rectangular solid of lead with dimensions $66 \text{ cm} \times 42 \text{ cm} \times 21 \text{ cm}$?

A hemispherical tank full of water is emptied by a pipe at the rate of $3\frac{4}{7}$ litres per second. How much time will it take to empty half the tank, if it is 3 m in diameter?

Water is flowing at the rate of 15 km/h through a pipe of diameter 14 cm into a rectangular tank which is 50 m long and 44 m wide. Determine the time (in hours) in which the level of water in the tank will rise by 21 cm.

A solid metallic sphere of radius 10.5 cm is melted and recast into a number of smaller solid cones, each of radius 3.5 cm and height 3 cm. Find the number of cones formed and the total surface area of all these cones combined (Take $\pi = \frac{22}{7}$).

Selvi's house has an overhead tank in the shape of a cylinder. This is filled by pumping water from a sump (an underground tank) which is in the shape of a cuboid. The sump has dimensions $1.57 \text{ m} \times 1.44 \text{ m} \times 95 \text{ cm}$. The overhead tank has its radius $60 \text{ cm}$ and height $95 \text{ cm}$. Find the height of the water left in the sump after the overhead tank has been completely filled with water from the sump which had been full. (Use $\pi = 3.14$)