Volume of Geometric Solids
Emma is shopping for a fish tank. The box says "20 gallons," and the tank measures 24 in. by 12 in. by 16 in. Does it really hold 20 gallons? To answer that, you need volume: the amount of space a solid takes up. We'll come back to Emma's tank at the end.
Units of volume and capacity
Volume is measured in cubic units. One cubic inch ($1\text{ in.}^3$) is a cube that is 1 inch on every edge. Count how many of those cubes fit inside a solid and you have its volume.
Cubic units grow fast. Since $1\text{ ft} = 12\text{ in.}$, a cube 1 foot on a side holds $12 \times 12 \times 12$ little cubic inches.
Here are the conversions you need.
| System | Equivalences |
|---|---|
| Customary | $1\text{ ft}^3 = 1{,}728\text{ in.}^3$, $\;1\text{ yd}^3 = 27\text{ ft}^3$ |
| Customary capacity | $1\text{ gal} = 231\text{ in.}^3$, so $1\text{ ft}^3 \approx 7.48\text{ gal}$ |
| Metric | $1\text{ cm}^3 = 1\text{ mL}$, $\;1\text{ dm}^3 = 1{,}000\text{ cm}^3 = 1\text{ L}$ |
Capacity is how much liquid a container holds. Volume tells you the space; capacity is that same space measured in gallons, liters, or cups.
A note on notation: some problems write decimals with a comma, like $\pi \approx 3,14$ or $1,5\text{ ft}$. In the US we write a decimal point: $\pi \approx 3.14$ and $1.5\text{ ft}$.
An aquarium has the shape of a rectangular prism with the following measurements: $60\text{ in}$ long, $30\text{ in}$ wide and $40\text{ in}$ high. Calculate how many…
An aquarium has the shape of a rectangular prism with the following measurements: $60\text{ in}$ long, $30\text{ in}$ wide and $40\text{ in}$ high. Calculate how many gallons of water fit in the aquarium if it is filled completely. (Remember that $1\text{ in}^3 = 1\text{ liter}$).
A coffee thermos has a cylindrical shape. The base radius measures $4 \text{ cm}$ and the total height is $25 \text{ cm}$. If the thermos is filled to $80 \%$ of its…
A coffee thermos has a cylindrical shape. The base radius measures $4 \text{ cm}$ and the total height is $25 \text{ cm}$. If the thermos is filled to $80 \%$ of its capacity, how many $150 \text{ ml}$ cups can be completely filled?
(Take $\pi \approx 3,14$ and remember that $1 \text{ dm}^3 = 1 \text{ liter}$).
Prisms and cylinders
A prism or a cylinder is a stack of identical layers. Each layer has the same shape and area as the base. Stack them $h$ high and you get:
$$V = B \cdot h$$Here $B$ is the area of the base and $h$ is the height (the perpendicular distance between the bases). For a rectangular prism, $B = \ell w$, so $V = \ell w h$. For a cylinder, the base is a circle, so $V = \pi r^2 h$.
When you cut a solid into two equal pieces, each piece has half the volume. A cylinder cut in half lengthwise gives two pieces of volume $\tfrac{1}{2}\pi r^2 h$.
A package containing a gaming keyboard has the shape of a rectangular prism. Its dimensions are $45\text{ in}$ long, $15\text{ in}$ wide and $5\text{ in}$ high.…
A package containing a gaming keyboard has the shape of a rectangular prism. Its dimensions are $45\text{ in}$ long, $15\text{ in}$ wide and $5\text{ in}$ high. Calculate the volume of air the box occupies in cubic inches.
A cylindrical soda can has a diameter of $6\text{ cm}$ and a height of $12\text{ cm}$. Calculate its volume using the value of $\pi \approx 3,14$.
A cylindrical soda can has a diameter of $6\text{ cm}$ and a height of $12\text{ cm}$. Calculate its volume using the value of $\pi \approx 3,14$.
A piece of cheese has the shape of a cylinder with a radius of $10\text{ cm}$ and a height of $8\text{ cm}$. If we cut the cheese exactly in half vertically, what will…
A piece of cheese has the shape of a cylinder with a radius of $10\text{ cm}$ and a height of $8\text{ cm}$. If we cut the cheese exactly in half vertically, what will be the volume of each of the two resulting parts? (Use $\pi \approx 3,14$).
When the base takes extra work
The rule $V = B \cdot h$ works for any prism. The only challenge is finding $B$. Here are two bases you will see a lot.
Triangle: $B = \tfrac{1}{2} \cdot \text{base} \cdot \text{height}$. For a right triangle, the two legs are the base and height of the triangle.
Regular hexagon: Six equal triangles meet at the center. Each one is equilateral, so the distance from the center to a vertex equals the side $s$. The apothem $a$ is the distance from the center to the middle of a side. Use the Pythagorean Theorem to find it. Some problems call this Pythagoras' theorem; it is the same thing.
$$B = \tfrac{1}{2} \cdot a \cdot P \qquad (P = \text{perimeter})$$Calculate the volume of a prism whose base is a right triangle with legs $6\text{ cm}$ and $8\text{ cm}$, knowing that the height of the prism is $15\text{ cm}$.
Calculate the volume of a prism whose base is a right triangle with legs $6\text{ cm}$ and $8\text{ cm}$, knowing that the height of the prism is $15\text{ cm}$.
A city council has installed concrete planters in the shape of a regular hexagonal prism. If each side of the hexagonal base measures $30 \text{ in}$ and the height of…
A city council has installed concrete planters in the shape of a regular hexagonal prism. If each side of the hexagonal base measures $30 \text{ in}$ and the height of the planter is $60 \text{ in}$, calculate the volume of soil (in gallons) needed to fill it completely.
(Fact: remember that you must first find the apothem of the base using Pythagoras' theorem).
Pyramids and cones: one third
Fill a pyramid with water and pour it into a prism with the same base and the same height. You will need exactly three pyramids to fill the prism. The same is true for a cone and a cylinder.
$$V_{\text{pyramid}} = \frac{1}{3}Bh \qquad V_{\text{cone}} = \frac{1}{3}\pi r^2 h$$The height $h$ is always the perpendicular distance from the tip to the base. It is not the slanted edge. If a problem gives you the slant height $\ell$ of a cone or a lateral edge of a pyramid, use the Pythagorean Theorem to find $h$.
The same plan works for any regular base: the center-to-corner distance is the leg of the right triangle. For a regular hexagon, that distance is simply the side length.
A decorative glass pyramid has a square base of side $8\text{ cm}$ and a height of $10\text{ cm}$. Calculate the volume of glass needed to manufacture it.
A decorative glass pyramid has a square base of side $8\text{ cm}$ and a height of $10\text{ cm}$. Calculate the volume of glass needed to manufacture it.
An ice cream cone has the shape of a cone. The base radius is $3\text{ in}$ and its height is $12\text{ in}$. Calculate how many cubic inches of ice cream fit inside it…
An ice cream cone has the shape of a cone. The base radius is $3\text{ in}$ and its height is $12\text{ in}$. Calculate how many cubic inches of ice cream fit inside it (without anything protruding from the top).
A decorative glass pyramid has a square base of side $12 \text{ cm}$. If the lateral edge (the segment joining the top vertex to a vertex of the base) measures…
A decorative glass pyramid has a square base of side $12 \text{ cm}$. If the lateral edge (the segment joining the top vertex to a vertex of the base) measures $10 \text{ cm}$, calculate the volume of glass the piece contains.
(Note: you will need to apply Pythagoras' theorem twice: first to find the diagonal of the base and then to find the height of the pyramid).
Spheres
A sphere is the set of all points at distance $r$ from a center. Its volume is
$$V = \frac{4}{3}\pi r^3$$A hemisphere is half a sphere, so $V = \tfrac{2}{3}\pi r^3$.
A very useful idea: when you melt a solid and recast it, the volume stays the same (if no material is lost). Set the volumes equal and solve.
The PE teacher keeps the handballs in a box. If each ball has a radius of $9\text{ cm}$, calculate the volume of one of them, rounding to two decimal places. (Use…
The PE teacher keeps the handballs in a box. If each ball has a radius of $9\text{ cm}$, calculate the volume of one of them, rounding to two decimal places. (Use $\pi \approx 3,14$).
Three metal balls of radius $3 \text{ cm}$ are melted down to create a single larger sphere. Assuming no material is lost in the process, what will be the radius of the…
Three metal balls of radius $3 \text{ cm}$ are melted down to create a single larger sphere. Assuming no material is lost in the process, what will be the radius of the new resulting sphere? Round the result to two decimal places.
Composite solids
Many real objects are made of several simple solids: a silo (cylinder plus cone plus dome), a propane tank (cylinder plus two hemispheres), a hollow pipe. To find the volume, break the object into pieces, find each volume, then add (or subtract a hole).
An ice cream is to be made in the shape of a cone whose slant height measures $15 \text{ cm}$ and its base radius is $6 \text{ cm}$. Calculate the volume of ice cream…
An ice cream is to be made in the shape of a cone whose slant height measures $15 \text{ cm}$ and its base radius is $6 \text{ cm}$. Calculate the volume of ice cream that fits inside the cone. If a perfect hemisphere of ice cream is added on top of the base (with the same radius) when serving, what will be the total volume of the ice cream?
Calculate the volume of a geometric solid formed by a cylinder of height $10 \text{ cm}$ and radius $4 \text{ cm}$, which has on top a square-based pyramid whose base is…
Calculate the volume of a geometric solid formed by a cylinder of height $10 \text{ cm}$ and radius $4 \text{ cm}$, which has on top a square-based pyramid whose base is inscribed in the circumference of the cylinder and whose height is $6 \text{ cm}$.
(Hint: the side of the square base of the pyramid coincides with the side of a square inscribed in a circle of radius $4 \text{ cm}$).
Displacement and rising water
When you drop an object into a container of water, the object pushes water out of its way, and the level rises. If the object is completely under water, the water that rose has exactly the same volume as the object.
The "extra" water forms a prism or cylinder whose base is the container's base and whose height is the rise $\Delta h$:
$$V_{\text{object}} = B_{\text{container}} \cdot \Delta h$$A cylindrical glass with an internal diameter of $6 \text{ in}$ contains water up to a height of $10 \text{ in}$. An irregularly shaped stone is placed inside and the…
A cylindrical glass with an internal diameter of $6 \text{ in}$ contains water up to a height of $10 \text{ in}$. An irregularly shaped stone is placed inside and the water level rises to $12,5 \text{ in}$ without overflowing. What is the volume of the stone expressed in cubic inches?
A metal piece in the shape of a regular hexagonal pyramid is to be submerged in a cylindrical container of radius $15\text{ in}$ containing water. The pyramid has a base…
A metal piece in the shape of a regular hexagonal pyramid is to be submerged in a cylindrical container of radius $15\text{ in}$ containing water. The pyramid has a base edge of $10\text{ in}$ and a lateral edge of $20\text{ in}$. When the piece is completely inserted, how many inches will the water level in the container rise? (Remember that you must first calculate the height of the pyramid using Pythagoras' theorem twice).
Density and partly filled containers
Density tells you how much mass fits into each unit of volume:
$$\text{density} = \frac{\text{mass}}{\text{volume}} \qquad\text{so}\qquad \text{mass} = \text{density} \times \text{volume}$$Water has a density of $1\text{ g/cm}^3$ (or about $62.4\text{ lb/ft}^3$). Keep the units consistent: if the density is per $\text{cm}^3$, the volume must be in $\text{cm}^3$.
If a container is filled to $75\%$ of its capacity, the volume of contents is $0.75 \times$ the capacity. If a process loses $5\%$ of the material, you keep $95\%$, so multiply by $0.95$.
A lead sphere of radius $10\text{ cm}$ is melted down to make small cylinders of diameter $2\text{ cm}$ and height $3\text{ cm}$. Assuming that $5\%$ of the material is…
A lead sphere of radius $10\text{ cm}$ is melted down to make small cylinders of diameter $2\text{ cm}$ and height $3\text{ cm}$. Assuming that $5\%$ of the material is lost in the melting process, how many complete cylinders can be made? If each cylinder has a mass of $107\text{ g}$, what is the density of the lead used in $g/cm^3$?
A grain silo has a composite shape: a central cylindrical part of height $12\text{ ft}$ and diameter $4\text{ ft}$, topped by a hemisphere and with a discharge cone of…
A grain silo has a composite shape: a central cylindrical part of height $12\text{ ft}$ and diameter $4\text{ ft}$, topped by a hemisphere and with a discharge cone of height $3\text{ ft}$ at the bottom. Calculate the total capacity of the silo in cubic feet. If the silo is filled to $85\%$ of its capacity, how many tons of wheat will it contain if the density of the wheat is $750\text{ lb/ft}^3$?
Similar solids and frustums
Two solids are similar if one is an enlarged or reduced copy of the other, with every length multiplied by the same scale factor $k$. Then:
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lengths are multiplied by $k$,
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areas are multiplied by $k^2$,
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volumes are multiplied by $k^3$.
So a cone cut off at half the height (measured from the tip) is a similar cone with $k = \tfrac{1}{2}$. Its volume is $\left(\tfrac{1}{2}\right)^3 = \tfrac{1}{8}$ of the whole cone. This is how you find the radius of the surface of a liquid in a cone-shaped container: the radius is scaled by the same $k$ as the height.
A frustum is what is left of a cone or pyramid when you slice off the top with a plane parallel to the base. To find its volume, think of it as a big cone minus a small cone.
Similar cones also help with sand or liquid in a cone. Volume divided by a flow rate gives the time: $\text{time} = \text{volume} \div \text{rate}$.
A water tank has the shape of a frustum of a cone. The radius of the larger base is $3 \text{ ft}$, the radius of the smaller base is $1,5 \text{ ft}$ and the height of…
A water tank has the shape of a frustum of a cone. The radius of the larger base is $3 \text{ ft}$, the radius of the smaller base is $1,5 \text{ ft}$ and the height of the tank is $4 \text{ ft}$. Calculate the capacity of the tank in cubic feet.
(Hint: you can use the formula for the volume of a frustum of a cone or view it as the difference between two cones).
There is an hourglass formed by two identical cones joined at their vertices, inside a protective glass cylinder. The cones have a base radius of $6\text{ cm}$ and a…
There is an hourglass formed by two identical cones joined at their vertices, inside a protective glass cylinder. The cones have a base radius of $6\text{ cm}$ and a height of $10\text{ cm}$. If the top section is filled with sand up to half its height, calculate the volume of sand in $cm^3$. If the sand falls into the bottom section at a constant rate of $5\text{ cm}^3$ per minute, how long will it take for the top part to empty? (Remember to apply the similarity of cones to find the radius of the sand's surface).
Hollow and carved solids
Sometimes you need the volume of the material, not the space inside. Then:
$$V_{\text{material}} = V_{\text{outer}} - V_{\text{hollow}}$$To find the size of the hollow, subtract the wall thickness from each outside measurement the wall touches: twice for a width that has a wall on both sides, once for the bottom of an open-top container.
The same plan works when the outside is a frustum: find the outer volume with the frustum shortcut, shrink each base side by twice the wall thickness and the height by the bottom thickness to get the hollow, use the shortcut again, and subtract. If the answer is requested in $\text{dm}^3$, convert from $\text{cm}^3$ at the end (divide by $1{,}000$).
You can also carve away corners. If you cut a corner off a cube using the midpoints of the three edges at that corner, the piece you remove is a pyramid. Its three edges are mutually perpendicular, so the base is a right triangle and the third edge is the height.
A street furniture design company has designed a concrete trash can in the shape of a regular square-based frustum of a pyramid. The external dimensions of the top base…
A street furniture design company has designed a concrete trash can in the shape of a regular square-based frustum of a pyramid. The external dimensions of the top base are $40\text{ cm}$ per side, those of the bottom base are $30\text{ cm}$ per side, and the total height is $80\text{ cm}$. To be functional, the inside is hollow with the same shape, but leaving walls of a constant thickness of $5\text{ cm}$ on the sides and on the bottom base (the top is open). Calculate the volume of concrete needed to manufacture each trash can in cubic decimeters ($dm^3$).
A sculptor wishes to create a stainless steel piece consisting of a cube with an edge of $20\text{ cm}$ from which eight tetrahedrons have been removed at its corners.…
A sculptor wishes to create a stainless steel piece consisting of a cube with an edge of $20\text{ cm}$ from which eight tetrahedrons have been removed at its corners. Each tetrahedron is formed by joining the vertex of the cube with the midpoints of the three edges that meet at that vertex (a mid-truncation). Determine the final volume of the sculpture. What percentage of the original cube's volume has been removed?
A perfume bottle has the shape of a regular octahedron with an edge of $6\text{ cm}$. However, the glass has a thickness of $2\text{ mm}$ on all its faces. The…
A perfume bottle has the shape of a regular octahedron with an edge of $6\text{ cm}$. However, the glass has a thickness of $2\text{ mm}$ on all its faces. The manufacturer claims that the bottle can hold $80\text{ ml}$ of fragrance ($1\text{ ml} = 1\text{ cm}^3$). Carry out the necessary calculations to check if the claim is true. (Fact: The volume of an octahedron with edge $a$ is $V = \frac{\sqrt{2}}{3} a^3$. You will need to consider how the thickness affects the edge of the inner hollow).
Cavalieri's Principle and the volume of a sphere
Take a stack of coins and push it so it leans. The coins are the same, and the stack is the same height, so the volume cannot have changed.
This is how mathematicians compare a sphere with solids they already know. Slice a sphere of radius $R$ at distance $y$ from its center. The slice is a circle. Its radius $s$, the distance $y$, and $R$ form a right triangle, so by the Pythagorean Theorem $s = \sqrt{R^2 - y^2}$ and the slice has area $\pi(R^2 - y^2)$.
Now compare the sphere with a second solid: a cylinder of radius $R$ and height $2R$ with a double cone removed from the inside. The two cones touch at the center of the cylinder, and each has height $R$ and base radius $R$. Slice both solids at distance $y$ from the center.
The cylinder's slice is a full circle of area $\pi R^2$. The double cone is a cone whose radius equals its distance from the center, so its slice has radius $y$ and area $\pi y^2$. What is left, the ring, has area $\pi R^2 - \pi y^2 = \pi(R^2 - y^2)$. That is exactly the area of the sphere's slice at the same height. The heights are the same ($2R$), so Cavalieri's Principle says the two solids have the same volume.
Cavalieri's Principle states that if two solids have the same height and the area of their cross-sections at any given height is equal, then they have the same volume.…
Cavalieri's Principle states that if two solids have the same height and the area of their cross-sections at any given height is equal, then they have the same volume. Use this idea to compare the volume of a sphere of radius $R$ with the volume enclosed between a cylinder of radius $R$ and height $2R$, and a double cone inscribed in it. Prove mathematically that the volume of the sphere is indeed $V = \frac{4}{3}\pi R^3$ by first calculating the volume of the cylinder and subtracting the volume of the two cones.
Back to the start: Emma's tank has a volume of $24 \times 12 \times 16 = 4{,}608\text{ in.}^3$. Since $1\text{ gal} = 231\text{ in.}^3$, that is $4{,}608 \div 231 \approx 19.95$ gallons. The label "20 gallons" checks out.