Solids of Revolution: Cylinders, Cones, and Spheres
A can of soup, an ice cream cone, a basketball. None of them has a flat face on its side, and each one can be made by spinning a flat shape around a line. How much label paper wraps the can? How much ice cream fits in the cone? Let's find out.
Spinning Shapes into Solids
A solid of revolution is the 3-D shape you get when you spin a flat figure 360° around a straight line called the axis. Drag the slider to watch the rectangle sweep out a cylinder:
| Flat shape | Spun around | Solid |
|---|---|---|
| Rectangle | one of its sides | Cylinder |
| Right triangle | one of its legs | Cone |
| Right trapezoid | the side perpendicular to the bases | Frustum (a cone with its tip cut off) |
| Semicircle | its diameter | Sphere |
Every solid has the same key measurements:
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the radius $r$ of the circular base (the distance from the axis),
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the height $h$ (measured along the axis),
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for a cone, the slant height $\ell$: the length of the slanted side that sweeps out the lateral surface. Some problems call it $g$ or $s$.
Imagine you have a rectangular piece of cardstock with a base of $10\text{ cm}$ and a height of $15\text{ cm}$. If you rotate it quickly around one of its $15\text{ cm}$…
Imagine you have a rectangular piece of cardstock with a base of $10\text{ cm}$ and a height of $15\text{ cm}$. If you rotate it quickly around one of its $15\text{ cm}$ sides, what solid of revolution is generated? Draw the resulting solid and state what its radius and height measure.
Cylinders
Peel the label off a soup can and flatten it. You get a rectangle. Its width is the distance around the can, $2\pi r$, and its height is $h$. The two lids are circles of radius $r$.
Why the volume formula? The base has area $\pi r^2$, and the cylinder stacks that circle $h$ units high, just like a prism.
Working backward is common too. If the lateral area is $L = 2\pi r h$ and you know $L$ and $h$, divide: $r = \dfrac{L}{2\pi h}$.
A standard soda can has a cylindrical shape. If its base radius measures $3,25\text{ cm}$ and its height is $12\text{ cm}$, calculate the area of the paper label that…
A standard soda can has a cylindrical shape. If its base radius measures $3,25\text{ cm}$ and its height is $12\text{ cm}$, calculate the area of the paper label that wraps around the side of the can (curved surface area). Use $\pi \approx 3,14$.
Look at the following net of a solid of revolution. a) What solid is it? b) If the radius of the circle is $R = 5\text{ cm}$ and the height of the rectangle is…
Look at the following net of a solid of revolution.
- a) What solid is it?
- b) If the radius of the circle is $R = 5\text{ cm}$ and the height of the rectangle is $H = 10\text{ cm}$, calculate the total surface area of the figure.
A courier company uses cylindrical cardboard tubes to send plans. If each tube has a lateral area of $942\text{ cm}^2$ and a height of $50\text{ cm}$: 1. Calculate the…
A courier company uses cylindrical cardboard tubes to send plans. If each tube has a lateral area of $942\text{ cm}^2$ and a height of $50\text{ cm}$:
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Calculate the radius of the base of the tube.
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Calculate the amount of cardboard needed to manufacture the two circular caps of each tube. (Use $\pi \approx 3,14$).
Capacity and Working Backward
Volume tells you how much a container holds. You will often be given the volume and asked for a missing measurement, or asked to convert the answer to everyday units.
Always convert so that all lengths use the same unit before you plug into a formula.
A glass has the shape of a cylinder with an internal radius of $3,5\text{ cm}$. If we pour a soft drink into it that occupies a volume of $250\text{ cm}^3$, what height…
A glass has the shape of a cylinder with an internal radius of $3,5\text{ cm}$. If we pour a soft drink into it that occupies a volume of $250\text{ cm}^3$, what height will the liquid reach inside the glass? Round the result to two decimal places.
Calculate the volume of a cylindrical water tank that has a radius of $2\text{ meters}$ and a height of $5\text{ meters}$. If $1\text{ ft}^3$ is equivalent to…
Calculate the volume of a cylindrical water tank that has a radius of $2\text{ meters}$ and a height of $5\text{ meters}$. If $1\text{ ft}^3$ is equivalent to $1000\text{ liters}$, how many gallons of water can the tank hold?
Cones
An ice cream cone's lateral surface unrolls into a sector (a slice of a circle) whose radius is the slant height $\ell$. The base is one circle.
The slant height is the hypotenuse of the right triangle formed by the height, the radius, and the slant, so you find it with the Pythagorean Theorem.
Why the $\tfrac{1}{3}$? A cone and a cylinder with the same base and the same height: it takes exactly three cones of water to fill the cylinder.
For a wizard costume, a student has made a cone-shaped hat. If the base radius is $10\text{ cm}$ and the height of the hat is $24\text{ cm}$, use the Pythagorean theorem…
For a wizard costume, a student has made a cone-shaped hat. If the base radius is $10\text{ cm}$ and the height of the hat is $24\text{ cm}$, use the Pythagorean theorem to calculate the length of the slant height ($g$) of the cone.
An ice cream cone has the shape of a cone. The radius of the top opening is $3\text{ cm}$ and its depth (height) is $10\text{ cm}$. What volume of ice cream fits exactly…
An ice cream cone has the shape of a cone. The radius of the top opening is $3\text{ cm}$ and its depth (height) is $10\text{ cm}$. What volume of ice cream fits exactly inside the cone? (Assume the ice cream is level with the rim, without protruding).
A photography studio uses a cone-shaped reflector to concentrate light. If the reflector has a height of $12\text{ cm}$ and the diameter of its circular opening is…
A photography studio uses a cone-shaped reflector to concentrate light. If the reflector has a height of $12\text{ cm}$ and the diameter of its circular opening is $10\text{ cm}$, calculate the total surface area of reflective material needed to manufacture it (excluding the base cover).
Frustums and Similar Cones
A paper cup or a popcorn bucket is a frustum: a cone with the tip sliced off by a cut parallel to the base. The trick is to see it as a big cone minus a small cone.
The small cone on top and the big cone are similar: same shape, different size. Every length in the big cone is the same multiple $k$ of the matching length in the small cone. That gives two powerful rules:
So if you cut a cone at half its height, the small cone has half the radius and half the height, and its volume is $\left(\tfrac12\right)^3 = \tfrac18$ of the full cone.
Notice that $\ell$ is the hypotenuse of a right triangle with legs $h$ and $R - r$.
Unrolling a frustum
To make a lampshade or a paper cup from a flat piece, you need its net. The lateral surface of a frustum unrolls into an annular sector: a slice of a ring. Both curved edges are centered at the tip of the full cone.
Call the slant height of the full cone $L$. Then the outer arc has radius $L$ and the inner arc has radius $L - \ell$. Their lengths match the circles of the bases: $2\pi R$ and $2\pi r$.
A sector with central angle $\theta$ has arc length $\dfrac{\theta}{360}\cdot 2\pi L$. Set that equal to $2\pi R$ and solve for $\theta$.
Calculate the lateral area and the volume of a frustum of a cone (like a popcorn bucket) whose bases have radii of $10\text{ cm}$ and $6\text{ cm}$, knowing that its…
Calculate the lateral area and the volume of a frustum of a cone (like a popcorn bucket) whose bases have radii of $10\text{ cm}$ and $6\text{ cm}$, knowing that its height is $15\text{ cm}$. Note: You will first need to calculate the slant height using the Pythagorean theorem.
A lamp designer wants to create a lampshade in the shape of a frustum of a cone. The lampshade has a height of $24 \text{ cm}$, a top diameter of $18 \text{ cm}$ and a…
A lamp designer wants to create a lampshade in the shape of a frustum of a cone. The lampshade has a height of $24 \text{ cm}$, a top diameter of $18 \text{ cm}$ and a bottom diameter of $30 \text{ cm}$. To calculate the cost of the material, they need to know the lateral surface area. However, to manufacture it from a flat piece, they must first determine the height of the original complete cone from which the frustum comes. Calculate this total height and the lateral area of the lampshade (without the bases).
An ice cream glass has the shape of an inverted cone of height $12 \text{ cm}$ and diameter $10 \text{ cm}$ at the rim. It is filled with chocolate milkshake exactly up…
An ice cream glass has the shape of an inverted cone of height $12 \text{ cm}$ and diameter $10 \text{ cm}$ at the rim. It is filled with chocolate milkshake exactly up to half its height. One student claims that the glass is "half full", but another says that there is much less than half the total volume. Calculate what percentage of the total volume of the glass the milkshake actually occupies to check who is right.
Spheres
A sphere can't be flattened without stretching it (that's why every flat world map is distorted). Even so, its surface area has a neat description: it equals four times the area of its biggest circle.
A tennis ball has an approximate diameter of $6,7\text{ cm}$. Calculate the total surface area of felt needed to cover the ball, treating it as a perfect sphere. Express…
A tennis ball has an approximate diameter of $6,7\text{ cm}$. Calculate the total surface area of felt needed to cover the ball, treating it as a perfect sphere. Express the result in $\text{cm}^2$.
If we cut an orange in half, we obtain two hemispheres. If the radius of the orange is $4\text{ cm}$, calculate the volume of one of these halves. Hint: Calculate the…
If we cut an orange in half, we obtain two hemispheres. If the radius of the orange is $4\text{ cm}$, calculate the volume of one of these halves.
Hint: Calculate the volume of the whole sphere first and then divide by two.
A spherical gas tank has an outer diameter of $6\text{ meters}$. The engineers want to apply a coat of anti-rust paint that costs $14,50\text{ \$}…
A spherical gas tank has an outer diameter of $6\text{ meters}$. The engineers want to apply a coat of anti-rust paint that costs $14,50\text{ \$}$ per square foot. How much will it cost to paint the total surface of the tank? (Use $\pi \approx 3,14$).
Composite Solids
Many real objects are a cylinder, a cone, and/or a hemisphere stuck together: a pill, a silo, a water tank, an ice cream cone with a scoop.
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Volume: add the volumes of the pieces.
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Surface area: add only the faces you can touch from the outside. Where two pieces are joined, those faces are hidden and do not count.
When there is a price per square foot, find the area first and multiply by the price afterward. For example, 400 ft² at $\$45$ per ft² costs $400 \times 45 = \$18{,}000$. If there is also a charge per foot of edge (for example, sealing a seam), the seam length is a circumference $2\pi r$.
A vitamin supplement brand designs its capsules in the shape of a cylinder with two hemispheres at the ends. The total length of the capsule is $22\text{ mm}$ and the…
A vitamin supplement brand designs its capsules in the shape of a cylinder with two hemispheres at the ends. The total length of the capsule is $22\text{ mm}$ and the diameter of the central cylinder is $8\text{ mm}$. Calculate the total volume of the capsule in cubic millimeters.
In an ice cream parlor they serve a perfectly spherical scoop of ice cream of diameter $6\text{ cm}$ on a cone of height $10\text{ cm}$ and the same diameter as the…
In an ice cream parlor they serve a perfectly spherical scoop of ice cream of diameter $6\text{ cm}$ on a cone of height $10\text{ cm}$ and the same diameter as the scoop. If the ice cream melts and is contained exactly within the cone, will it overflow or will there be free space? Answer by comparing the volumes of the sphere and the cone.
A steel water tank is to be built combining a cylinder of height $4 \text{ ft}$ and radius $3 \text{ ft}$ with a perfect hemisphere on top. If the cost of the steel is…
A steel water tank is to be built combining a cylinder of height $4 \text{ ft}$ and radius $3 \text{ ft}$ with a perfect hemisphere on top. If the cost of the steel is $45 \text{ \$}$ per square foot of outer surface and sealing the joint between the hemisphere and the cylinder costs $15 \text{ \$}$ per linear foot of circumference, calculate the total budget for the materials of the outer surface of the tank (including the flat bottom base).
Scaling, Comparing, and Melting Solids
Three ideas solve a whole family of "what if?" problems.
1. Scaling. Change a measurement and see what happens to the formula. Doubling $r$ in $\pi r^2 h$ multiplies the volume by $2^2 = 4$.
| If every length is multiplied by $k$ | Length | Area | Volume |
|---|---|---|---|
| Multiplier | $k$ | $k^2$ | $k^3$ |
| $k = 2$ | ×2 | ×4 | ×8 |
2. Same volume. When you melt a solid and recast it, or fill one container from another, the volume stays the same. Set the two volume formulas equal to each other.
3. Displacement. A solid submerged in water pushes water out of the way. In a cylindrical container, the water rises by a layer whose volume equals the solid's volume: $\text{rise} = \dfrac{V_{\text{solid}}}{\pi r^2}$.
A stone rolling pin has a cylindrical shape with a radius of $4 \text{ cm}$ and a length of $25 \text{ cm}$. It is to be stored in a cylindrical box so that the rolling…
A stone rolling pin has a cylindrical shape with a radius of $4 \text{ cm}$ and a length of $25 \text{ cm}$. It is to be stored in a cylindrical box so that the rolling pin fits exactly (tangent to the walls and bases). If it is decided to replace that rolling pin with another one that has half the radius but twice the length:
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Will the volume of stone used change? Justify your answer with calculations.
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Which of the two rolling pins would require a box with a larger total surface area?
A perfume bottle has the shape of a sphere of radius $R = 5 \text{ cm}$. The bottle is contained within a box in the shape of a right cylinder that has the same radius…
A perfume bottle has the shape of a sphere of radius $R = 5 \text{ cm}$. The bottle is contained within a box in the shape of a right cylinder that has the same radius and the same height as the diameter of the sphere.
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Calculate the difference between the volume of the cylindrical box and the volume of the sphere.
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Show that the lateral area of the cylinder is equal to the surface area of the sphere.
A spherical lead ball of diameter $6 \text{ cm}$ has been completely submerged in a cylindrical container of radius $10 \text{ cm}$ containing water. 1. How much has the…
A spherical lead ball of diameter $6 \text{ cm}$ has been completely submerged in a cylindrical container of radius $10 \text{ cm}$ containing water.
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How much has the water level in the container risen after introducing the ball?
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If after this, we melt the lead ball and manufacture a cone with it whose base has the same radius as the original sphere ($3 \text{ cm}$), what height will the cone have?
Solids from Spinning Plane Figures
To find the solid made by a spinning figure, answer three questions:
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Which line is the axis?
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How far is each point of the figure from the axis? The farthest distance is the radius.
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How long is the figure along the axis? That is the height.
The same figure gives different solids when you change the axis:
A solid of revolution is generated by rotating a rectangle of base $4\text{ cm}$ and height $7\text{ cm}$ around one of its long sides. 1. What geometric figure is…
A solid of revolution is generated by rotating a rectangle of base $4\text{ cm}$ and height $7\text{ cm}$ around one of its long sides.
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What geometric figure is obtained?
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Calculate its total area.
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If it rotated around the short side, would it have the same volume? Justify your answer by performing the calculations.
Calculate the volume of the figure generated by rotating $360^\circ$ the following right trapezoid around side $AB$. Side $AD$ measures $6 \text{ cm}$ (smaller base),…
Calculate the volume of the figure generated by rotating $360^\circ$ the following right trapezoid around side $AB$. Side $AD$ measures $6 \text{ cm}$ (smaller base), side $BC$ measures $10 \text{ cm}$ (larger base) and the height of the trapezoid (side $AB$) measures $8 \text{ cm}$. Identify the resulting solid and calculate its total area.
Distances Along a Parallel of Latitude
A parallel is a circle around the Earth at a fixed latitude. Only the equator is as big as the whole Earth. A parallel at latitude $\varphi$ has a smaller radius, found in the right triangle of a cross-section:
In a right triangle, $\cos\varphi = \dfrac{\text{adjacent leg}}{\text{hypotenuse}}$. Here the hypotenuse is the Earth's radius $R$ and the leg next to the angle is the radius of the parallel $r$, so $r = R\cos\varphi$. Use the cosine button on your calculator in degree mode. Useful values: $\cos 0^\circ = 1$, $\cos 30^\circ \approx 0.866$, $\cos 60^\circ = 0.5$.
Two cities on the same parallel, with longitudes that differ by $\Delta\lambda$ degrees, are separated by a fraction $\dfrac{\Delta\lambda}{360}$ of that circle. Remember the circumference formula $C = 2\pi r$ from Math 6.
Two cities, $P$ and $Q$, are located on the same parallel at a latitude of $60^\circ \text{ N}$. City $P$ has a longitude of $15^\circ \text{ E}$ and city $Q$ of…
Two cities, $P$ and $Q$, are located on the same parallel at a latitude of $60^\circ \text{ N}$. City $P$ has a longitude of $15^\circ \text{ E}$ and city $Q$ of $45^\circ \text{ E}$. Knowing that the radius of the Earth is approximately $6371 \text{ km}$:
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Calculate the radius of the parallel $60^\circ \text{ N}$ (Hint: use trigonometry or remember that in the triangle formed, the radius of the parallel is $R \cdot \cos(60^\circ)$).
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Find the minimum distance between the two cities measured along the arc of that parallel.
Back to the start: the label of a soup can is a rectangle of area $2\pi r h$, and its volume is $\pi r^2 h$. An ice cream cone holds one third of that cylinder: $\tfrac13\pi r^2 h$. The basketball's skin has area $4\pi r^2$ and it holds $\tfrac43\pi r^3$. Each solid comes from spinning a flat shape, and its formulas come from that shape.