Volume of Geometric Solids
How many gallons does a fish tank hold? How much ice cream fits in a cone? How many small candles can you make by melting one big one? Every one of these is a question about volume: the amount of space a solid takes up.
Units of volume and capacity
Volume is measured in cubic units. A cube that is $1$ inch on every edge has a volume of $1\text{ in.}^3$.
Since $1\text{ ft} = 12\text{ in.}$, a cubic foot is a $12 \times 12 \times 12$ block of those little cubes:
| Conversion | Why |
|---|---|
| $1\text{ ft}^3 = 1{,}728\text{ in.}^3$ | $12^3 = 1{,}728$ |
| $1\text{ yd}^3 = 27\text{ ft}^3$ | $3^3 = 27$ |
| $1\text{ gal} = 231\text{ in.}^3$ | definition of a US gallon |
| $1\text{ ft}^3 \approx 7.48\text{ gal}$ | $1{,}728 \div 231$ |
In science (and on soda bottles) you will also see metric units. Each cubic unit is 1,000 times the next smaller one, because $10 \cdot 10 \cdot 10 = 1{,}000$.
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 $1\text{ in}^3 = 1\text{ liter}$).
Prisms
A prism is a stack of identical layers, one on top of the other. Its volume is the area of the base times the height:
$$V = B \cdot h \qquad \text{Rectangular prism: } V = \ell \cdot w \cdot h$$Here $B$ is the area of the base and $h$ is the perpendicular distance between the two bases.
The hard part is usually the area of the base. Two bases need a bit of work.
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Triangle: $B = \tfrac{1}{2}bh$. Do not forget the $\tfrac12$.
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Regular hexagon: split it into $6$ equilateral triangles. If the side is $s$, the apothem $a$ (center to the midpoint of a side) comes from the Pythagorean Theorem, and $B = \tfrac{1}{2}aP$ where $P$ is the perimeter.
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.
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 has installed concrete planters in the shape of a regular hexagonal prism. If each side of the hexagonal base measures $30 \text{ cm}$ and the height of the…
A city has installed concrete planters in the shape of a regular hexagonal prism. If each side of the hexagonal base measures $30 \text{ cm}$ and the height of the planter is $60 \text{ cm}$, calculate the volume of soil (in liters) needed to fill it completely.
(Fact: remember that you must first find the apothem of the base using the Pythagorean theorem).
Cylinders
A cylinder is a prism whose base is a circle, so the same rule works: stack the layers.
$$V = B \cdot h = \pi r^2 h$$The radius $r$ is half the diameter. You can leave the answer in terms of $\pi$ or use $\pi \approx 3.14$ (or your calculator's $\pi$ key) when the problem asks for a decimal.
Cutting a solid exactly in half gives two pieces with half the volume each. A cylinder cut along a plane through its axis is the standard example.
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 $\pi \approx 3,14$).
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 $1 \text{ dm}^3 = 1 \text{ liter}$).
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 need exactly three pyramids to fill the prism. The same is true for a cone and a cylinder.
$$V_{\text{pyramid}} = \frac{1}{3}Bh \qquad\qquad V_{\text{cone}} = \frac{1}{3}\pi r^2 h$$This works for any pyramid, whatever the shape of its base (triangle, square, hexagon…). In a pyramid, $B$ is the area of that base. You will prove why the $\tfrac13$ appears in the section on Cavalieri's Principle.
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).
The sphere
$$V_{\text{sphere}} = \frac{4}{3}\pi r^3 \qquad\qquad V_{\text{hemisphere}} = \frac{2}{3}\pi r^3$$The radius is cubed, so if you double the radius the volume grows by $2^3 = 8$ times.
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.
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?
Cavalieri's Principle: why these formulas work
Take two stacks of $100$ coins. One is neat and straight. The other is pushed into a leaning tower. Both use the same coins, so they have the same volume. That is the idea behind Cavalieri's Principle.
This is why an oblique prism has volume $Bh$ too, and it is also how we justify the sphere formula. Compare a sphere of radius $R$ with a cylinder of radius $R$ and height $2R$ that has a double cone removed from it. In the figure, $y$ is the height of the cutting plane above the center.
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.
Similar solids, frustums and hollows
Two solids are similar when one is a scaled copy of the other. If the scale factor is $k$, then lengths are multiplied by $k$, areas by $k^2$ and volumes by $k^3$.
A frustum is a pyramid or cone with its top sliced off by a plane parallel to the base. To find its volume, extend it to the full cone (or pyramid), then subtract the small cone you cut off. The small one is similar to the big one.
There are shortcut formulas for frustums, if you prefer them. For a cone with radii $R$ and $r$: $V = \tfrac{\pi h}{3}(R^2 + Rr + r^2)$. For a pyramid with base areas $B_1$ and $B_2$: $V = \tfrac{h}{3}\left(B_1 + B_2 + \sqrt{B_1 B_2}\right)$. Check the cone formula with the example: $\tfrac{\pi \cdot 6}{3}(16 + 8 + 4) = 56\pi$ ✓.
Walls of constant thickness
A hollow solid with walls of thickness $t$ has an inner cavity that is similar to the outside. The walls shrink the distance from the center to each face by $t$. The inner dimensions are smaller than the outer ones by an amount that depends on the shape:
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A cube with edge $a$ and walls on every face: inner edge $a - 2t$ (wall on both sides).
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A regular octahedron with edge $a$ has an inscribed-sphere radius of $\tfrac{\sqrt6}{6}a$ (center to a face). Walls $t$ thick shrink that distance to $\tfrac{\sqrt6}{6}a - t$. Since the inner octahedron is similar to the outer one, its edge is $\left(\tfrac{\sqrt6}{6}a - t\right) \cdot \tfrac{6}{\sqrt6} = a - t\sqrt6$.
The octahedron volume is $V = \tfrac{\sqrt2}{3}a^3$. Use the same units for everything: $2\text{ mm} = 0.2\text{ cm}$.
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).
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).
Composite solids: adding pieces
Many real objects are made of several simple solids: a silo (cylinder, cone and dome), a pill (cylinder and two hemispheres), a tent (prism and pyramid). Break the object into pieces, find each volume, and add them. When pieces touch, the shared face has no volume, so nothing is counted twice.
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 $4 \text{ cm}$).
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 $750\text{ lb/ft}^3$?
Composite solids: taking pieces away
If a solid has a hole, a hollow inside or a corner cut off, find the volume of the full solid and subtract the part that is missing.
An outdoor furniture company has designed a concrete trash receptacle in the shape of a regular square-based frustum of a pyramid. The external dimensions of the top…
An outdoor furniture company has designed a concrete trash receptacle 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 receptacle 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 tetrahedra 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 tetrahedra 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?
Same volume, new shape
When you melt, pour or submerge something, volume is conserved. Set the volume before equal to the volume after.
A cylindrical glass with an internal diameter of $6 \text{ cm}$ contains water up to a height of $10 \text{ cm}$. An irregularly shaped stone is placed inside and the…
A cylindrical glass with an internal diameter of $6 \text{ cm}$ contains water up to a height of $10 \text{ cm}$. An irregularly shaped stone is placed inside and the water level rises to $12,5 \text{ cm}$ without overflowing. What is the volume of the stone expressed in cubic centimeters?
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 $g/cm^3$?
Back to the start: A fish tank measures $24 \times 12 \times 16$ inches, so $V = 24 \cdot 12 \cdot 16 = 4{,}608\text{ in.}^3$. Divide by $231$ in.³ per gallon: $4{,}608 \div 231 \approx 19.95$. It is sold as a 20-gallon tank.