Solids of Revolution
On the roof of many Dominican houses there is a cylindrical water tank. The orange cones mark road work on the Autopista Duarte, and a baseball is a sphere. What do they have in common? All three are formed by rotating a plane figure around an axis. That is why they are called solids of revolution. On this page you will learn how to generate them, unfold them, and calculate how much material and how much water they need.
How they are generated
Glue a cardboard rectangle to a pencil along one of its sides and spin it very fast: you will see a cylinder appear. Move the slider to see how the plane figure sweeps out space as it rotates.
| Rotating figure | Axis of rotation | Solid formed |
|---|---|---|
| Rectangle | one of its sides | Cylinder |
| Right triangle | one of its legs | Cone |
| Semicircle | its diameter | Sphere |
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.
Elements of the cylinder and the cone
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Radius ($r$): the radius of the circular base.
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Height ($h$): the perpendicular distance between the bases (or from the vertex to the base in the cone).
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Slant height ($g$): the segment that draws the lateral surface as it rotates. In the cylinder $g = h$. In the cone it is the slanted side.
In the cone, the height, the radius, and the slant height form a right triangle, because the height is perpendicular to the base. The slant height is the hypotenuse:
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.
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.
The cylinder: net and area
If you cut a can along a vertical line and unroll it, you get its net (flat pattern): two circles (the bases) and a rectangle (the lateral surface).
The base of the rectangle is equal to the circumference of the base, because the rectangle wraps around it. That gives us the formulas:
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.
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.
The cone: net and area
The net of the cone is a circular sector (a slice of pizza) with radius equal to the slant height $g$, plus the circle of the base.
The arc of the sector wraps around the base, so its length is $2\pi r$. The sector is like a "triangle" with base $2\pi r$ and height $g$, and that is why its area is $\frac{2\pi r \cdot g}{2} = \pi r g$.
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 $18.84\text{ cm}$.
The sphere
The sphere has no flat net. To calculate its surface area we use a direct formula.
Volume of solids of revolution
Remember the equivalences: $1\ \text{m}^3 = 1{,}000$ liters and $1\ \text{cm}^3 = 1$ mL.
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 $1\text{ m}^3 = 1000\text{ liters}$).
Cross sections: what you get when you cut
A cross section is the plane figure that appears when you cut a solid with a plane.
| Solid | Horizontal cut (parallel to the base) | Vertical cut through the axis |
|---|---|---|
| Cylinder | Circle | Rectangle |
| Cone | Circle (smaller as you go up) | Isosceles triangle |
| Sphere | Circle | Circle (if it passes through the center) |
In the sphere, if the plane is at a distance $d$ from the center, the radius of the cutting circle comes from a right triangle:
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?
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…
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).
Frustum of a cone
If you cut a cone with a plane parallel to the base and remove the tip, what remains is a frustum of a cone, like a water tank or a cup. It has two bases, with radii $R$ (larger) and $r$ (smaller).
Two ideas are enough:
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Slant height of the frustum: it forms a right triangle with the height $h$ and the difference of the radii: $g^2 = h^2 + (R - r)^2$.
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Volume: that of the large cone minus that of the small cone you removed. To find the measurements of the small cone, use the fact that the triangles of the complete cone and of the removed cone are similar: they have the same shape, so their sides are proportional.
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).
Hollow solids and combined solids
Many real objects combine several solids. The key is to decompose:
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If a piece is missing (hollow cylinder, space between a cone and a cylinder), subtract volumes.
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If there are parts joined together (two cones joined at their bases), add volumes and areas.
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For mass: $\text{mass} = \text{density} \times \text{volume}$.
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.
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).
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.
Rotating a composite figure
When you rotate a figure made up of several pieces, each piece generates its own solid. You identify the axis, recognize which solid each piece gives, and add the volumes.
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.
Go back to the beginning: the tank on the roof is a cylinder, the orange cone is a cone, and the baseball is a sphere. All three come from rotating a rectangle, a right triangle, and a semicircle. With $\pi r^2 h$, $\frac{1}{3}\pi r^2 h$, and $\frac{4}{3}\pi r^3$ you know how much water fits in them.