Solids of Revolution
A soda can, a waffle cone, a tennis ball. None of them has a flat side, and all of them are solids of revolution: you get them by spinning a flat figure around a line. Two questions come up with every one: how much material does it take to make it, and how much does it hold? Here is a teaser. A can and a cone cup with the same radius and the same height: how many cone cups of soda does it take to fill the can? We'll answer that before the end.
How solids are generated
Spin a flat figure around a fixed line, called the axis of revolution, and it sweeps out a solid. Drag the slider to watch it happen:
| Flat figure | Spins around | Solid |
|---|---|---|
| Rectangle | one of its sides | Cylinder |
| Right triangle | one of its legs | Cone |
| Right trapezoid | the leg perpendicular to the bases | Frustum of a cone |
| Semicircle | its diameter | Sphere |
Every solid of revolution has these parts:
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the radius $r$ of the circular base (the distance from the axis to the farthest point of the figure);
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the height $h$ (measured along the axis);
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for a cone, the slant height $g$, the side that sweeps out the curved surface. In a cone, $g$ is the hypotenuse of the right triangle with legs $r$ and $h$, so $g = \sqrt{h^2 + r^2}$.
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.
Cylinder: surface area
Cut the label off a soda can and flatten it: you get a rectangle. Its width is the distance around the can, $2\pi r$, and its height is $h$. Add the two circular lids and you have the whole net of the cylinder.
The rectangle wraps around the circle, so its width must equal the circumference $2\pi r$. That is why $2\pi r$ shows up in the formula.
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 blueprints. If each tube has a lateral area of $942\text{ cm}^2$ and a height of $50\text{ cm}$: 1. Calculate…
A courier company uses cylindrical cardboard tubes to send blueprints. 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 $\pi \approx 3,14$).
Cylinder: volume
Stack coins and you get a cylinder. Each coin has area $\pi r^2$, and the stack is $h$ tall. So the volume is the base area times the height.
Volume is measured in cubic units (in.³, ft³, cm³), because you multiply three lengths.
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?
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.
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.
Cone: slant height, area and volume
Spin a right triangle with legs $r$ and $h$ around the leg $h$. The hypotenuse $g$ sweeps out the curved surface, and the Pythagorean Theorem links all three lengths.
Unroll the curved surface and you get a sector of a circle with radius $g$ and arc length $2\pi r$. Its area is a fraction of the full circle, and that works out to $\pi r g$.
The volume is exactly one-third of the cylinder with the same base and height. Fill the cone with water and pour it into the matching cylinder: it takes three cones to fill it.
Back to the opening question: a can and a cone with the same radius and height. The can's volume is $\pi r^2 h$ and the cone's is $\tfrac{1}{3}\pi r^2 h$, so it takes 3 cone cups to fill the can.
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).
Frustum of a cone
Slice the tip off a cone with a cut parallel to the base. The part left over is a frustum: a popcorn bucket, a lampshade, a flower pot. Its two bases are circles with radii $R$ (bottom) and $r$ (top), and its height is $h$.
The slanted side of the frustum is the hypotenuse of a right triangle whose legs are $h$ and the difference of the radii, $R - r$. So $g = \sqrt{h^2 + (R - r)^2}$.
In the figure, $x$ is the height of the small cone that was cut off. There are two ways to work with a frustum. The first is the formulas:
The second way is to think of the frustum as a big cone minus a small cone. This is the method to use when the problem doesn't give you the height of the full cone, because the small cone is similar to the big one (the cut is parallel to the base), so their lengths are in proportion.
A right trapezoid spun around its perpendicular leg also gives a frustum. The leg on the axis is the height $h$, and the two parallel sides are the radii.
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).
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.
Sphere
A sphere can't be flattened without stretching it, which is why every flat map of the world distorts something. Its surface area is exactly four times the area of its great circle (the circle you get by cutting through the center).
When the size is in meters but the price is per square foot
Some problems give a length in meters and a cost per square foot. Convert the length first, using $1 \text{ m} \approx 3.28 \text{ ft}$, and then find the area. You can also find the area in m² and convert it, but then you must use $1 \text{ m}^2 \approx 10.76 \text{ ft}^2$, because a conversion factor is squared when it is applied to an area.
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 $\pi \approx 3,14$).
Combining solids
Silos, capsules, and water tanks are made of several solids stuck together. The plan is always the same:
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Break the shape into familiar pieces.
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Volume: add (or subtract) the volumes of the pieces.
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Surface area: add only the faces you can actually see from outside. The circle where two pieces join is hidden, so it does not count.
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.
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).
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.
Scaling solids
Two solids of the same shape but different size are similar. If every length is multiplied by the scale factor $k$, then:
| Quantity | Multiplied by |
|---|---|
| Lengths (radius, height, slant height) | $k$ |
| Surface area | $k^2$ |
| Volume | $k^3$ |
Areas have two dimensions and volumes have three, so each one gets one more factor of $k$.
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.
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?
Displacement and recasting
Two ideas let you use volume formulas in problems about water and metal.
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Displacement. A solid completely submerged pushes aside (displaces) a volume of water equal to its own volume. In a cylindrical container, that volume is a thin cylinder: $V_{\text{solid}} = \pi R^2 \cdot (\text{rise})$.
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Recasting. When you melt a solid and mold a new one, the volume stays the same. The shape, the surface area and the dimensions all change, but not the amount of material.
In an ice cream parlor they serve a perfectly spherical scoop of ice cream of diameter $6\text{ cm}$ on a cone-shaped ice cream cone of height $10\text{ cm}$ and the…
In an ice cream parlor they serve a perfectly spherical scoop of ice cream of diameter $6\text{ cm}$ on a cone-shaped ice cream 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 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?
Circles of latitude
The Earth is close to a sphere, and a circle of latitude (a parallel) is a circle on that sphere that is smaller than the equator. To measure distances along a parallel, you need its radius.
Cut the Earth through its axis. At latitude $\theta$, the radius $R$ from the center to a point on the parallel makes an angle $\theta$ with the equator. The radius of the parallel is the horizontal side of the right triangle, so it is the side adjacent to $\theta$:
$$r_{\text{parallel}} = R \cos\theta$$To find the distance along the parallel between two points, find what fraction of the full circle lies between them. That fraction is the difference in longitude divided by $360^\circ$.
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.