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.

You have a right cylinder and a right cone. The figure shows a horizontal cut (parallel to the base) with an orange dashed line and a vertical cut passing through the centre of the bases (the axis of the solid) with a red dashed line.

1. a) If we make the horizontal cut (parallel to the base) in a cylinder and in a cone, what plane geometric figure do we obtain as the cross-section in each one?

1. b) Do we obtain the same figure in each solid if the cut is vertical and passes through the centre of the bases? State which figure results in the cylinder and which in the cone.

Cian wants to build a card party hat in the shape of a cone for his school's end-of-year festival. In his notebook, he writes: "Since a cone is generated by rotating a right triangle around one of its legs, to make the lateral surface I will cut out an isosceles triangle whose base has the length of the base circumference and whose height is that of the cone; then I will join the two equal sides together." Explain with reasoning what geometric mistake Cian is making and what would happen if he tried to roll up that triangle. Then, name the 2D geometric shape that actually constitutes the net of a cone's lateral surface and explain which element of the cone (radius, height, or slant height) matches the radius of that 2D shape.

When carefully peeling off the paper label that wraps completely and without overlapping around the lateral wall of a cylindrical can of tomato sauce, it is found to have the shape of a rectangle measuring $31.4\text{ cm}$ along the base and $10\text{ cm}$ in height. Knowing that this cylinder is generated by revolving a rectangle around an axis that coincides with one of its sides, deduce with reasoning the exact dimensions (base and height) of that original generating rectangle (take $\pi \approx 3.14$). If we also wanted to manufacture the two circular metal lids for the can, what must the exact radius of each one be so that they seal the cylinder perfectly?

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, Aoife 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 $2\text{ metres}$ and a height of $5\text{ metres}$. If the tank is filled to three-quarters of its capacity, how many litres of water does it contain? (Remember that $1\text{ m}^3 = 1000\text{ litres}$).

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 centre. 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 Pythagoras' 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 $18.84\text{ cm}$.

You have a rectangular piece of card that measures exactly $10\text{ cm}$ wide by $20\text{ cm}$ long. Imagine that you rotate it $360^\circ$ in space: first around its $10\text{ cm}$ side (obtaining cylinder $A$) and, in a second test, around its $20\text{ cm}$ side (obtaining cylinder $B$). Before performing any calculations, estimate with reasoning: which of the two cylinders do you think will have a larger lateral surface area? And which one do you think will enclose a greater internal volume? Write down your initial intuition. Then, check your estimation by calculating the lateral area of both cylinders (leave the result in terms of $\pi$) and the volume of each one using the formula $V = \pi \times r^2 \times h$. Finally, compare the results with your prediction and explain mathematically why the radius has a greater impact than the height on the volume of a cylinder.

In a technology class, students want to build a scale model of a traffic cone whose circular base has a radius of $9\text{ cm}$ and whose height is $12\text{ cm}$. To cut out the piece of card corresponding to the lateral surface, two students propose different approaches: Aoife first calculates the slant height $g$ using the Pythagorean theorem and directly applies the simplified formula for the lateral area: $A_{lat} = \pi \times r \times g$; meanwhile, Cian also calculates the slant height $g$, but then finds the central angle $\alpha$ that the unfolded circular sector must have by solving the proportion $\frac{\alpha}{360^\circ} = \frac{r}{g}$, and then calculates the area with the general formula for a circular sector: $A_{sector} = \frac{\pi \times g^2 \times \alpha}{360^\circ}$. Carry out the calculations step by step using both methods (take $\pi \approx 3.14$). Prove algebraically why Aoife's direct formula is equivalent to Cian's procedure. If you have to draw the net on card using a ruler, compass, and protractor to assemble it physically, which of the two methods provides you with the indispensable information for drawing it? Justify your answer.

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 centre 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 $2 \text{ metres}$ and the diameter of the central circular cross-section where they meet is $120 \text{ cm}$. Calculate the total volume of the buoy in litres 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 centre. 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 $4 \text{ metres}$ and that of the lower base (bottom) is $2 \text{ metres}$. The total height of the tank is $6 \text{ metres}$. If the tank is filled up to half its height ($3 \text{ metres}$), 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.

An agricultural supplies company in Tomelloso (Ciudad Real) commissions the design of a vertical silo to store grain. The entire structure is a solid of revolution made up of a lower cylinder with an outer diameter of $4\text{ m}$ and a height of $6\text{ m}$, topped by a conical roof with the same circular base and a height of $1.5\text{ m}$. First, geometrically model the cross section: describe precisely what 2D polygonal shape (indicating the measurements of all its sides) generates this entire solid when rotated $360^\circ$ around the vertical axis of symmetry of the silo. Second, to waterproof the exterior, a protective resin will be applied to the vertical cylindrical wall and to the conical roof (the floor and the internal horizontal joint between the cylinder and the cone are not treated). Calculate the total square meters of surface area that must be waterproofed (take $\pi \approx 3.14$). Finally, if each can of resin costs $\text{€}\ 45$, covers exactly $12\text{ m}^2$, and fractional cans are not sold, determine the minimum budget required if applying two full coats to the entire surface is mandatory.

An educational design workshop manufactures models to explain the geometry of planet Earth and cartographic projections. To construct the lateral surface of a protective conical dome, they cut a circular sector with a radius of $20\text{ cm}$ and a central angle of $144^\circ$ from a sheet of plastic. State and solve the inverse problem: when curving the circular sector by joining its straight edges to form the cone, calculate the radius of the resulting circular base and the vertical height that the cone will reach (round the height to the nearest tenth of a centimeter). Furthermore, for the spherical model of the globe, a student on the team proposes: "So that schools can cut out and assemble a perfect sphere from card in class, we can print its flat net on a single continuous rectangular sheet, just as we do with the cylinder and the cone." Analyze and explain in detail why the student's proposal is an insurmountable geometric error, basing your answer on the properties of the net of a sphere and explaining what inevitably happens to any flat map of the world when trying to represent the Earth's surface without distorting it.