Exercises: Solids of revolution

Imagine sticking the following 2D shapes to a cocktail stick and spinning them quickly around it. Identify which solid of revolution is generated in each case:

1. a) A rectangle rotating about one of its sides.
2. b) A right-angled triangle rotating about one of its legs.
3. c) A semicircle rotating about its diameter.

In the following drawing of a cone, identify the names of the parts labelled with the letters $r$, $h$ and $g$. Briefly explain what each of them represents.

A cylindrical soda can has a height of $12$ cm and the radius of its base measures $3$ cm. Draw a schematic of its net (the shapes you would get if you cut the can and laid it flat on a table) and mark the known measurements on the drawing.

You can do it on paper and send a photo with your answer.

To make a cone-shaped party hat, we need to know its slant height ($g$). If the hat has a height of $8$ cm and a base radius of $6$ cm, use Pythagoras' theorem to calculate how long the slant height must be.

Calculate the volume of a bowling ball (which is a perfect sphere) knowing that its radius is $11$ cm. Use the formula $V = \frac{4}{3} \times \pi \times r^3$ and take $\pi \approx 3.14$ as the value of pi.

A tin of tennis balls is cylindrical in shape. The base has a diameter of $7$ cm and the height of the tin is $22$ cm. Calculate the curved surface area of the tin (excluding the caps).

Identify which of these nets could never form a cylinder and explain why.

If we make a horizontal cut (parallel to the base) through a cylinder and a cone, what 2D geometric shape will we get in the cross-section? Would it be the same if the cut were vertical, passing through the center of the bases?

Draw in your notebook the solid of revolution generated by rotating the following 2D shape around the indicated axis. What is the name of the solid obtained? Describe its main elements (bases, height and generatrix).

You can do it on paper and send a photo with your answer.

A soft drinks brand wants to design a new cylindrical can. The radius of the base must be $3.5\text{ cm}$ and the height of the can $12\text{ cm}$. Calculate the amount of metal (total surface area) needed to make one of these cans. Round the result to two decimal places.

For a fancy-dress party, Lucía is making cone-shaped hats. If each hat has a slant height of $25\text{ cm}$ and the diameter of the base is $14\text{ cm}$, what will the height of the hat be? Use Pythagoras' theorem to find it.

Calculate the volume of a perfectly spherical scoop of ice cream that has a radius of $3\text{ cm}$. If the ice cream melts inside a cylindrical container with the same base and the same height as the sphere, what height will the liquid ice cream reach? (Assume that the volume does not change when melting).

A water tank is shaped like a cylinder with a radius of $2\text{ meters}$ and a height of $5\text{ meters}$. If the tank is filled to three-quarters of its capacity, how many liters of water does it contain? (Remember that $1\text{ m}^3 = 1000\text{ liters}$).

A traffic cone has a height of $60\text{ cm}$ and a base radius of $15\text{ cm}$. Calculate its curved surface area (the surface that would need to be painted orange). To do this, you must first find the slant height of the cone using Pythagoras' theorem.

A spherical orange of diameter $10\text{ cm}$ is cut by a plane passing at a distance of $3\text{ cm}$ from the center. When the two pieces are separated, the cross-section of the cut is a circle. Calculate the radius of that circular cross-section by applying the Pythagorean theorem in the shaded triangle.

We have a piece of card in the shape of a sector of a circle like the one in the picture. By joining the straight edges, we form the curved surface of a cone. Calculate the total surface area of this cone, knowing that the radius of the sector (which will be the slant height) is $10\text{ cm}$ and that the arc length is $18.84\text{ cm}$.

A perfume company is designing a new glass bottle. The bottle has a cylindrical base with a radius of $3 \text{ cm}$ and a height of $6 \text{ cm}$. On top of this base rests a frustum of a cone whose top base has a radius of $1 \text{ cm}$ and whose slant height measures $5 \text{ cm}$. Knowing that the stopper is a perfect hemisphere that fits exactly into the top opening of radius $1 \text{ cm}$, calculate the total exterior surface area of the bottle (not including the base it rests on).

An industrial steel roller has the shape of a hollow cylinder. The outer diameter of the roller is $20 \text{ cm}$, the inner diameter is $16 \text{ cm}$, and its total length is $50 \text{ cm}$. If the density of steel is $7.85 \text{ g/cm}^3$, calculate the total mass of the roller in kilograms. (Remember that to find the volume of the hollow cylinder you must subtract the volume of the inner cylinder from that of the outer one).

For an art exhibition, a metal piece has been built consisting of a cone that fits perfectly inside a cylinder of the same height ($12 \text{ cm}$) and same base radius ($5 \text{ cm}$). The cone is inverted, so that its vertex touches the center of the lower base of the cylinder. Calculate the volume of water needed to fill the empty space remaining between the walls of the cylinder and the cone.

An ice cream cone is shaped like a cone of height $15 \text{ cm}$ and radius $3 \text{ cm}$ at the top opening. A scoop of ice cream, which is a perfect sphere of radius $3.5 \text{ cm}$, is placed on top. Due to its size, part of the scoop remains outside the cone and part inside. If the scoop rests right on the circular rim of the cone, what is the distance from the lowest point of the scoop of ice cream to the bottom vertex of the cone? (Use Pythagoras' theorem).

A sea buoy is formed by joining two identical cones at their bases. The total length of the buoy from end to end is $2 \text{ meters}$ and the diameter of the central circular cross-section where they meet is $120 \text{ cm}$. Calculate the total volume of the buoy in liters and its total outer surface area.

A sphere of radius $13 \text{ cm}$ is cut by a horizontal plane passing at a distance of $5 \text{ cm}$ from the center. This cut produces a circle (cross-section). Calculate the area of that circular section and the area of the resulting upper spherical cap, knowing that the surface area of a spherical cap is calculated as $S = 2 \times \pi \times R \times h$, where $h$ is the height of the cap.

A water tank is shaped like an inverted frustum of a cone. The radius of the upper (open) base is $4 \text{ meters}$ and that of the lower base (bottom) is $2 \text{ meters}$. The total height of the tank is $6 \text{ meters}$. If the tank is filled up to half its height ($3 \text{ meters}$), use similar triangles to find the radius of the water surface at that moment and calculate the volume of water collected.

Calculate the volume of the solid generated by rotating $360^\circ$ the following irregular polygon around the indicated vertical axis. The polygon is formed by a rectangle of $2 \times 4 \text{ cm}$ and a right-angled triangle of base $3 \text{ cm}$ attached to its side.