Solids of Revolution
Chloe tapes a paper rectangle to a pencil and rolls the pencil quickly between her palms. A blurry can-shaped solid appears. Then she wants to wrap a label around a real can. How long must the paper strip be, and how much metal does the can itself need? By the end of this page you will know how to answer both questions for cylinders, cones, spheres and more.
Spinning plane figures
A solid of revolution is the solid you get when you spin a flat figure all the way around a line called the axis of rotation. The spinning figure sweeps out the space and "builds" the solid.
Move the slider to watch a flat figure sweep out a solid:
| Figure that spins | Axis | Solid formed |
|---|---|---|
| Rectangle | one of its sides | Cylinder |
| Right triangle | one of its legs | Cone |
| Semicircle | its diameter | Sphere |
| Right trapezoid | its side perpendicular to the bases | Frustum (a cone with the tip cut off) |
A frustum has two circular bases of different sizes, a height, and a slant height along its tilted side.
Imagine you stick the following plane figures to a toothpick and spin them quickly on it. Identify which solid of revolution is generated in each case: a) A rectangle…
Imagine you stick the following plane figures to a toothpick and spin them quickly on it. Identify which solid of revolution is generated in each case:
- a) A rectangle rotating around one of its sides.
- b) A right triangle rotating around one of its legs.
- c) A semicircle rotating around its diameter.
Draw in your notebook the solid of revolution generated by rotating the following plane figure around the indicated axis. What is the name of the solid obtained?…
Draw in your notebook the solid of revolution generated by rotating the following plane figure around the indicated axis. What is the name of the solid obtained? Describe its main elements (bases, height and generatrix).
Radius, height and slant height
Every cone and cylinder has the same key measurements:
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the radius $r$ of the circular base (half of the diameter),
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the height $h$, measured straight up, perpendicular to the base,
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for a cone, the slant height $g$, measured along the tilted side from the tip to the edge of the base (many books write $\ell$ or $s$).
Look at the dashed line, the radius and the slanted side. They form a right triangle with legs $r$ and $h$ and hypotenuse $g$. So the Pythagorean Theorem links them:
$$r^2 + h^2 = g^2$$The slant height $g$ is always longer than $h$, because it is the hypotenuse.
In the following drawing of a cone, identify the names of the elements indicated by 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 elements indicated by 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 $8$ in and a base radius of $6$ in, use the Pythagorean theorem to…
To make a cone-shaped party hat, we need to know its slant height ($g$). If the hat has a height of $8$ in and a base radius of $6$ in, use the Pythagorean theorem to calculate how long the slant height must be.
For a costume party, Chloe is making cone-shaped hats. If each hat has a slant height of $25\text{ cm}$ and the base diameter is $14\text{ cm}$, what will the height of…
For a costume party, Chloe is making cone-shaped hats. If each hat has a slant height of $25\text{ cm}$ and the base diameter is $14\text{ cm}$, what will the height of the hat be? Use the Pythagorean theorem to find it.
The net of a cylinder
If you cut a can along a vertical line, cut off the lids and flatten everything, you get its net: one rectangle (the side) and two circles (the bases).
Why is the width $2\pi r$? When you roll the rectangle back up, its bottom edge wraps exactly once around the circular base. So that edge must equal the circumference of the base, $C = 2\pi r$.
A net can fold into a cylinder only if:
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it has exactly 2 circles of the same size and 1 rectangle,
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one side of the rectangle equals the circumference $2\pi r$ of the circles,
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the circles touch the rectangle along the sides of length $2\pi r$.
A cylindrical soda can has a height of $12$ in and its base radius measures $3$ in. Draw a schematic of its net (the figures you would obtain if you cut the can and laid…
A cylindrical soda can has a height of $12$ in and its base radius measures $3$ in. Draw a schematic of its net (the figures you would obtain if you cut the can and laid it flat on a table) and indicate the known measurements on the drawing.
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.
Surface area of a cylinder
The surface area is the total area of the net. We find it by adding the areas of the pieces.
Surface area is measured in square units (in.², cm², ft²).
A tennis ball tube has the shape of a cylinder. The base has a diameter of $7$ in and the height of the tube is $22$ in. Calculate the lateral surface area of the tube…
A tennis ball tube has the shape of a cylinder. The base has a diameter of $7$ in and the height of the tube is $22$ in. Calculate the lateral surface area of the tube (excluding the lids).
Hint: Remember that the lateral area of a cylinder is $A_L = 2 \cdot \pi \cdot r \cdot h$.
A soda 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…
A soda 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 manufacture one of these cans. Round the result to two decimal places.
The cone: net and surface area
The net of a cone is a circular sector (a slice of pizza) plus a circle for the base.
Two facts connect the pieces:
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The sector's radius is the slant height $g$, not the height $h$, because the cone's tilted side becomes the straight edge of the sector.
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The sector's arc wraps around the base, so its length equals the base circumference $2\pi r$.
The sector is the fraction $\dfrac{2\pi r}{2\pi g} = \dfrac{r}{g}$ of a full circle of radius $g$. So its area is $\dfrac{r}{g} \cdot \pi g^2 = \pi r g$.
A traffic cone has a height of $60\text{ cm}$ and a base radius of $15\text{ cm}$. Calculate its lateral area (the surface that would need to be painted orange). To do…
A traffic cone has a height of $60\text{ cm}$ and a base radius of $15\text{ cm}$. Calculate its lateral area (the surface that would need to be painted orange). To do this, you will first need to find the slant height of the cone using the Pythagorean theorem.
We have a piece of cardboard in the shape of a circular sector like the one in the image. By joining the straight edges, we form the lateral surface of a cone. Calculate…
We have a piece of cardboard in the shape of a circular sector like the one in the image. By joining the straight edges, we form the lateral surface of a cone. Calculate the total 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 $18,84\text{ cm}$.
Volume of cylinders and cones
Volume is how much space a solid fills (cubic units: in.³, cm³, ft³). For a cylinder, it is the area of the base times the height. A cone with the same base and height holds exactly one third as much.
For capacity in gallons or liters, you will often need to convert units.
A water tank has the shape of 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,…
A water tank has the shape of 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 gallons of water does it contain? (Remember that $1\text{ ft}^3 = 1000\text{ liters}$).
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 the same base…
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 the same base radius ($5 \text{ cm}$). The cone is inverted, so that its vertex touches the center of the bottom base of the cylinder. Calculate the volume of water that would be needed to fill the empty space left between the walls of the cylinder and the cone.
The sphere
A sphere is the solid you get by spinning a semicircle about its diameter. Every point on it is the same distance $r$ from the center.
Calculate the volume of a bowling ball (which is a perfect sphere) knowing that its radius measures $11$ in. Use the formula $V = \frac{4}{3} \cdot \pi \cdot r^3$ and…
Calculate the volume of a bowling ball (which is a perfect sphere) knowing that its radius measures $11$ in. Use the formula $V = \frac{4}{3} \cdot \pi \cdot r^3$ and take the value of $\pi \approx 3,14$.
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 the volume does not change when melting).
Cross sections
A cross section is the flat shape you see when a plane slices through a solid, like cutting an orange.
| Solid | Cut parallel to the base | Vertical cut through the axis |
|---|---|---|
| Cylinder | circle (same size as the bases) | rectangle |
| Cone | circle (smaller higher up) | isosceles triangle |
| Sphere | circle | circle (the largest one, radius $r$) |
Every slice of a sphere is a circle. How big is it? Draw the right triangle: sphere radius $R$ as the hypotenuse, the distance $d$ from the center to the plane, and the circle's radius $a$.
By the Pythagorean Theorem, $a^2 + d^2 = R^2$, so $a = \sqrt{R^2 - d^2}$.
If we make a horizontal cut (parallel to the base) through a cylinder and a cone, what plane geometric figure will we obtain in the cross-section of the cut? Would it be…
If we make a horizontal cut (parallel to the base) through a cylinder and a cone, what plane geometric figure will we obtain in the cross-section of the cut? Would it be the same if the cut were vertical passing through the center of the bases?
A spherical orange with a diameter of $10\text{ cm}$ is cut by a plane passing at a distance of $3\text{ cm}$ from the center. When separating the two pieces, the…
A spherical orange with a diameter of $10\text{ cm}$ is cut by a plane passing at a distance of $3\text{ cm}$ from the center. When separating the two pieces, the cross-section of the cut is a circle. Calculate the radius of that circle of cut by applying the Pythagorean theorem in the shaded triangle.
Caps and scoops
When a plane cuts a sphere, the smaller piece is a spherical cap. Its height is $h = R - d$. The curved part of its surface has area
$$S = 2\pi R h$$Some problems give you this formula. To get the whole outside of the cap, add the flat circular face $\pi a^2$.
The same right triangle solves "scoop on a cone" problems. If a ball of radius $R$ rests on a circular rim of radius $a$, the ball's center is $d = \sqrt{R^2 - a^2}$ above the rim's plane.
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 generates 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 generates a circle (cross-section). Calculate the area of that circle of cut and the area of the resulting upper spherical cap, knowing that the surface area of a spherical cap is calculated as $S = 2 \cdot \pi \cdot R \cdot h$, where $h$ is the height of the cap.
An ice cream cone has the shape of a cone with a height of $15 \text{ cm}$ and a radius of $3 \text{ cm}$ at the top opening. A scoop of ice cream is placed which is a…
An ice cream cone has the shape of a cone with a height of $15 \text{ cm}$ and a radius of $3 \text{ cm}$ at the top opening. A scoop of ice cream is placed which is a perfect sphere with a radius of $3,5 \text{ cm}$. Due to its size, part of the scoop remains outside the cone and part remains inside. If the scoop rests exactly on the circular edge of the cone, what is the distance from the lowest point of the ice cream scoop to the bottom vertex of the cone? (Use the Pythagorean theorem).
Frustums
A frustum is a cone with its tip sliced off by a plane parallel to the base. Water tanks, buckets and lampshades are often frustums. Extend the slanted sides until they meet in a tip. Then the frustum is a big cone minus a small cone, and the two cones are similar.
The lateral area of a frustum with radii $R$ (big) and $r$ (small) and slant height $g$ is also "big cone minus small cone." It simplifies to a neat formula.
A water tank has the shape of an inverted frustum of a cone. The radius of the top base (open) is $4 \text{ meters}$ and that of the bottom base (bottom) is…
A water tank has the shape of an inverted frustum of a cone. The radius of the top base (open) is $4 \text{ meters}$ and that of the bottom base (bottom) is $2 \text{ meters}$. The total height of the tank is $6 \text{ meters}$. If the tank is filled 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 accumulated water.
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 this base rests a…
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 this base rests a frustum of a cone whose top base has a radius of $1 \text{ cm}$ and its slant height measures $5 \text{ cm}$. Knowing that the cap is a perfect hemisphere that fits exactly into the top opening with a radius of $1 \text{ cm}$, calculate the total outer surface area of the bottle (excluding the supporting base).
Composite and hollow solids
Real objects combine several solids. Three strategies cover almost everything:
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Add pieces: spin a polygon made of several shapes, and each shape makes its own solid.
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Subtract a hole: for a hollow solid, take the outer volume minus the inner volume.
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Count only what you can touch: for surface area, leave out faces that are glued together or hidden.
Two identical cones glued at their bases (like a buoy) work the same way. Each cone's height is half the total length, so you double the volume and lateral area of one cone. Remember to convert lengths to the same unit first.
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 the 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 the outer one).
A maritime buoy is formed by joining two identical cones at their bases. The total length of the buoy from tip to tip is $2 \text{ meters}$ and the diameter of the…
A maritime buoy is formed by joining two identical cones at their bases. The total length of the buoy from tip to tip is $2 \text{ meters}$ and the diameter of the central circular cross-section where they join is $120 \text{ in}$. Calculate the total volume of the buoy in gallons and its total outer surface area.
Calculate the volume of the solid generated by rotating the following irregular polygon $360^\circ$ around the indicated vertical axis. The polygon is formed by a…
Calculate the volume of the solid generated by rotating the following irregular polygon $360^\circ$ around the indicated vertical axis. The polygon is formed by a rectangle of $2 \times 4 \text{ cm}$ and a right triangle with a base of $3 \text{ cm}$ joined to its side.
Back to the start: Chloe's strip of paper must be as long as the can's circumference, $2\pi r$, because it wraps around the base exactly once. The metal for the can is the lateral rectangle plus two circles, $2\pi r h + 2\pi r^2$.