Solids of Revolution
Glue a cardboard rectangle to a toothpick along one of its sides and spin it very fast: you'll see a cylinder "appear," just like when a spinning top turns and looks like a solid. That's why the cylinder and the cone are called solids of revolution. On this page you'll learn how to generate them and unfold them, and then, as an extension, how to calculate how much material or how much liquid fits in them. For example: how much sheet metal do you need for a soda can?
How they are generated
Move the slider to see how the plane figure, as it rotates, "sweeps" through space and forms the solid:
| Rotating figure | Axis of rotation | Solid formed |
|---|---|---|
| Rectangle | one of its sides | Cylinder |
| Right triangle | one of its legs | Cone |
| Semicircle (extension) | its diameter | Sphere |
The parts of the cylinder and the cone have their own names:
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: a) A…
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:
- a) A rectangle rotating about one of its sides.
- b) A right-angled triangle rotating about one of its legs.
- c) A semicircle rotating about its diameter.
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…
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.
The net of the cylinder
If you cut a can along a vertical line and "unroll" it, you get its net: two circles (the bases) and a rectangle (the lateral surface).
Why is the base of the rectangle $2\pi r$? Because when you roll it up, that base goes all the way around the circumference of the lid.
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…
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.
Identify which of these nets could never form a cylinder and explain why.
Identify which of these nets could never form a cylinder and explain why.
The net of the cone
The net of the cone is a circular sector (a "slice of pizza") whose radius is the slant height $g$, plus the circle of the base.
The arc of the sector, when it closes up, forms the edge of the base. That's why it has the same length: $2\pi r$.
Also, the radius $r$, the height $h$, and the slant height $g$ form a right triangle (the figure that rotated), so the Pythagorean theorem holds:
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.
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.
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…
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.
The sphere (extension)
The sphere is generated by rotating a semicircle about its diameter.
If you cut a solid with a plane, the figure left at the cut is called a cross section:
| Cut | Cylinder | Cone | Sphere |
|---|---|---|---|
| Parallel to the base | circle | circle (smaller) | circle |
| Vertical through the axis | rectangle | triangle | great circle |
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…
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?
Extension: area of the cylinder and the cone
Everything comes from the net: the area is that of the figures that make it up.
In the cone, the area of the circular sector is $\dfrac{\text{arc} \times \text{radius}}{2}$. Here the arc measures $2\pi r$ and the radius is $g$, so we get $\dfrac{2\pi r \cdot g}{2} = \pi r g$.
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…
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).
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…
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…
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.
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).…
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.
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…
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 $18.84\text{ cm}$.
Extension: the sphere and its cross sections
When you cut a sphere of radius $R$ at a distance $d$ from the center, the radius $r$ of the circle of the cut comes from a right triangle with hypotenuse $R$ and legs $d$ and $r$:
The cut divides the sphere into two spherical caps. The height $h$ of a cap is the distance between the cutting plane and the farthest point of the sphere on that side. If the cap does not contain the center, $h = R - d$; if it does contain it, $h = R + d$.
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…
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.
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).…
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.
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…
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).
Extension: volume
Volume measures how much fits inside. It is expressed in cubic units ($\text{cm}^3$, $\text{m}^3$).
Why is the cone one third? If you fill a cone with water and pour it into a cylinder with the same base and height, you need exactly three cones to fill it.
To convert to liters: $1\text{ dm}^3 = 1\text{ L}$, $1\text{ m}^3 = 1{,}000\text{ L}$, and $1\text{ cm}^3 = 1\text{ mL}$. And if you know the density $\rho$, the mass is $m = \rho \cdot V$.
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…
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.
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…
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…
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 $1\text{ m}^3 = 1000\text{ liters}$).
Extension: frustum of a cone
If you cut the tip off a cone with a plane parallel to the base, what's left is a frustum of a cone. A cup or a bucket-shaped water tank has that shape. The trick: complete the large cone and subtract the small cone you removed.
The two cones are similar: their radii and their heights are in the same proportion.
The lateral area is obtained the same way: that of the large cone minus that of the small one.
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…
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.
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…
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).
Extension: composite solids
Many real objects mix solids: an ice cream cone with a scoop, a buoy, a hollow roller. The strategy is always the same:
-
Break the object down into known solids.
-
Calculate the volume or the area of each part.
-
Add or subtract, depending on whether the part is attached or is a hollow.
If the object comes from rotating a figure, identify which piece rotates and about which axis: a rectangle attached to the axis gives a cylinder; a right triangle with one leg on the axis gives a cone. When you join pieces, the volumes are added.
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…
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.
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…
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.
More composite solids with rollers and hollow pieces:
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…
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…
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.
Back to the beginning: the soda can is built from the net, two circles and a rectangle with base $2\pi r$. The sheet metal you need is the sum of their areas, $2\pi r h + 2\pi r^2$, and what fits inside is given by $\pi r^2 h$.