Volume of Prisms, Cylinders, Pyramids, Cones, and Spheres
How many gallons of water does a backyard swimming pool hold? How much ice cream fits in a cone? Both questions ask for volume: the amount of space inside a solid. By the end of this page you can answer both.
Cubic units and capacity
Area counts squares. Volume counts cubes. A cube that measures 1 inch on every edge has a volume of 1 cubic inch (1 in.³). The volume of a solid is the number of unit cubes that fill it.
A conversion factor for length must be cubed when you convert volume. One foot is 12 inches, so one cubic foot is $12 \times 12 \times 12 = 1{,}728$ cubic inches.
Volume is also used to measure how much a container holds (its capacity). These are the conversions you need:
| Unit | Equals |
|---|---|
| 1 ft³ | 1,728 in.³ |
| 1 gallon (gal) | 231 in.³ |
| 1 ft³ | about 7.48 gal |
| 1 fluid ounce (fl oz) | about 1.8 in.³ (128 fl oz = 1 gal) |
| 1 cm³ | 1 mL |
| 1 dm³ = 1,000 cm³ | 1 liter (L) = 1,000 mL |
| 1 m³ | 1,000 L |
| 1 fl oz | about 29.57 mL |
| 1 m | about 3.28 ft |
| 1 m³ | about 35.31 ft³ (because $3.28^3 \approx 35.3$) |
| 1 pound (lb) | 16 ounces (oz) |
The rule is the same every time: find the volume in cubic units first, then convert.
Pools are usually measured in feet, so cubic feet convert straight to gallons with $1 \text{ ft}^3 \approx 7.48$ gal.
Sometimes a problem gives measurements in meters but asks for feet or gallons. You can convert each length to feet first (1 m $\approx$ 3.28 ft), or convert the finished volume with 1 m³ $\approx$ 35.31 ft³. Just remember that the volume factor is the cube of the length factor.
For a class project, you have to estimate how much water fits in your backpack. If we assume that your bag has a capacity of $18.500 \text{ cm}^3$, how many liters of…
For a class project, you have to estimate how much water fits in your backpack. If we assume that your bag has a capacity of $18.500 \text{ cm}^3$, how many liters of water could it hold if it were completely waterproof?
Remember that $1 \text{ dm}^3$ is equivalent to $1 \text{ liter}$.
In the science laboratory they have a small aquarium in the shape of a rectangular prism. Its measurements are $40 \text{ in}$ long, $20 \text{ in}$ wide and…
In the science laboratory they have a small aquarium in the shape of a rectangular prism. Its measurements are $40 \text{ in}$ long, $20 \text{ in}$ wide and $25 \text{ in}$ high. If we fill it with water to a height of $20 \text{ in}$, how many gallons of water have we poured in?
A municipal swimming pool in the shape of a rectangular prism has a capacity of $1200\text{ ft}^3$. If we know that the length of the pool is $50\text{ meters}$ and the…
A municipal swimming pool in the shape of a rectangular prism has a capacity of $1200\text{ ft}^3$. If we know that the length of the pool is $50\text{ meters}$ and the width is $12\text{ meters}$, calculate the depth (height) of the pool. If the city council decides to fill it only to $4/5$ of its capacity for safety, how many gallons of water will there be in the pool?
Prisms
A prism is like a stack of identical copies of its base. The number of layers is the height, so
$$V = B \cdot h$$where $B$ is the area of the base and $h$ is the distance between the two bases. For a rectangular prism, $B = \ell w$, so $V = \ell \cdot w \cdot h$.
The base can be any polygon. For a triangular prism, $B = \tfrac{1}{2}bh$ (use the triangle's base and height). For a trapezoid, $B = \tfrac{1}{2}(b_1 + b_2)h$. Be careful: a prism lying on its side has its "base" at the end, and the height $h$ of the prism is its length.
You can also work backward to find a missing dimension: $h = \dfrac{V}{\ell w}$.
You have bought a new smartphone and want to calculate the volume of its box to know how much space it will take up on your shelf. The box is a rectangular prism with…
You have bought a new smartphone and want to calculate the volume of its box to know how much space it will take up on your shelf. The box is a rectangular prism with the following dimensions: $16 \text{ cm}$ long, $9 \text{ cm}$ wide and $5 \text{ cm}$ high. Calculate its volume in $\text{cm}^3$.
You are camping with your friends and you have a tent in the shape of a right triangular prism. The base of the tent (the floor) is a triangle with a base of…
You are camping with your friends and you have a tent in the shape of a right triangular prism. The base of the tent (the floor) is a triangle with a base of $2 \text{ m}$ and a height of $1,5 \text{ m}$. If the tent has a length (depth) of $3 \text{ m}$, what is the volume of air inside the tent?
A concrete ramp is to be built for a skatepark. The ramp has the shape of a right prism whose base is a right trapezoid. The bases of the trapezoid measure $3\text{ ft}$…
A concrete ramp is to be built for a skatepark. The ramp has the shape of a right prism whose base is a right trapezoid. The bases of the trapezoid measure $3\text{ ft}$ and $1\text{ ft}$, and its height (length of the base of the ramp on the ground) is $5\text{ ft}$. If the width of the ramp is $4\text{ ft}$, calculate how many cubic feet of concrete are needed and the total cost if the price is $95\text{ \$/ft}^3$.
Cylinders and Cavalieri's principle
A cylinder is a stack of circles (like a stack of coins), so the same formula applies. The base is a circle with area $\pi r^2$:
$$V = B \cdot h = \pi r^2 h$$Cavalieri's principle
What if the stack of coins leans to one side? It still has the same coins, so it has the same volume.
So a prism and a cylinder with the same base area and the same height have the same volume, even if the bases have different shapes.
A famous soft drink brand uses cylindrical cans for its drinks. If the base radius of the can is $3 \text{ cm}$ and its height is $12 \text{ cm}$, calculate the volume…
A famous soft drink brand uses cylindrical cans for its drinks. If the base radius of the can is $3 \text{ cm}$ and its height is $12 \text{ cm}$, calculate the volume of soft drink that the can can hold. (Use $\pi \approx 3,14$).
Cavalieri's principle states that two solids with the same height and whose cross-sections at the same distance from the base have the same area, possess the same…
Cavalieri's principle states that two solids with the same height and whose cross-sections at the same distance from the base have the same area, possess the same volume. Compare a regular hexagonal prism with a height of $20\text{ cm}$ and a base area of $30\text{ cm}^2$ with a cylinder of the same height and the same base area. If we pour $0,5\text{ liters}$ of water into each one, which will fill up first? Justify your answer by calculating the volume of both.
Containers that are partly full
Sometimes a container is not full. There are two situations:
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It is filled to a percent of its capacity: find the full volume, then take that percent of it.
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It is filled to a certain height: use that liquid height as $h$ in the formula, not the height of the container.
A well-known milkshake brand has designed a new cylindrical container for its limited summer edition. The container has a diameter of $8\text{ in}$ and a height of…
A well-known milkshake brand has designed a new cylindrical container for its limited summer edition. The container has a diameter of $8\text{ in}$ and a height of $15\text{ in}$. If the manufacturer indicates that the milkshake occupies $90\%$ of the total capacity of the container to leave space for air, calculate how many fluid ounces ($ml$) of product each container contains.
You have decided to build a gel ant farm for a science project. The container has the shape of a rectangular prism with dimensions of $25\text{ cm}$ long, $10\text{ cm}$…
You have decided to build a gel ant farm for a science project. The container has the shape of a rectangular prism with dimensions of $25\text{ cm}$ long, $10\text{ cm}$ wide and $18\text{ cm}$ high. If the gel is sold in tubs of $0,5\text{ liters}$, how many full tubs will you need to buy to fill the ant farm to a height of $15\text{ cm}$?
Pyramids
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.
$$V = \frac{1}{3}\,B \cdot h$$Finding the height
The height $h$ of a pyramid is the perpendicular distance from the apex to the base. Problems often give the lateral edge $e$ (from the apex to a corner of the base) instead. For a regular pyramid, the height, the lateral edge, and the distance from the center of the base to a corner form a right triangle, so use the Pythagorean Theorem:
$$h^2 + (\text{center to corner})^2 = e^2$$For a square base with side $s$, the distance from the center to a corner is half of the diagonal: $\tfrac{1}{2}s\sqrt{2}$, and its square is $\tfrac{s^2}{2}$.
For a regular hexagon with side $s$, the distance from the center to a corner is equal to $s$ (the hexagon is made of six equilateral triangles). Each triangle has area $\tfrac{\sqrt{3}}{4}s^2$, so $B = 6 \cdot \tfrac{\sqrt{3}}{4}s^2 = \tfrac{3\sqrt{3}}{2}s^2$.
You have a glass paperweight in the shape of a regular square-based pyramid. The area of its square base is $25 \text{ cm}^2$ and the height of the pyramid is…
You have a glass paperweight in the shape of a regular square-based pyramid. The area of its square base is $25 \text{ cm}^2$ and the height of the pyramid is $9 \text{ cm}$. Calculate the volume of glass needed to make this paperweight.
For a video game event, a trophy has been designed consisting of a regular square-based bronze pyramid. The side of the square base measures $12\text{ cm}$ and the…
For a video game event, a trophy has been designed consisting of a regular square-based bronze pyramid. The side of the square base measures $12\text{ cm}$ and the lateral edge (the one joining the vertex to a corner of the base) measures $10\text{ cm}$. Calculate the volume of bronze needed to make one trophy.
Calculate the volume of a regular hexagonal pyramid whose lateral edge measures $13\text{ cm}$ and the side of its base measures $10\text{ cm}$. Remember that in a…
Calculate the volume of a regular hexagonal pyramid whose lateral edge measures $13\text{ cm}$ and the side of its base measures $10\text{ cm}$. Remember that in a regular hexagon, the radius of the circumscribed circle is equal to the side. Express the result in $cm^3$ rounding to the nearest whole number.
Cones
A cone is to a cylinder what a pyramid is to a prism: it has exactly one third of the volume of the cylinder with the same base and height.
$$V = \frac{1}{3}\pi r^2 h$$The slant height $\ell$ is the distance along the surface from the apex to the edge of the base. The height $h$, the radius $r$, and the slant height form a right triangle with $\ell$ as the hypotenuse:
$$h^2 + r^2 = \ell^2$$For a birthday party, cardboard hats have been made in the shape of a cone. If each hat has a radius of $8 \text{ cm}$ and a height of $15 \text{ cm}$, what volume of…
For a birthday party, cardboard hats have been made in the shape of a cone. If each hat has a radius of $8 \text{ cm}$ and a height of $15 \text{ cm}$, what volume of air is "trapped" inside the hat when someone is wearing it? (Use $\pi \approx 3,14$).
A perfume company launches a fragrance in a bottle shaped like a right cone. The base radius is $3\text{ in}$ and the slant height measures $5\text{ in}$. Calculate the…
A perfume company launches a fragrance in a bottle shaped like a right cone. The base radius is $3\text{ in}$ and the slant height measures $5\text{ in}$. Calculate the volume of perfume that fits in the bottle expressed in cubic inches. Hint: you will first need to find the height of the cone using the Pythagorean theorem.
Spheres
A sphere has only a radius, and its volume depends on the cube of the radius:
$$V = \frac{4}{3}\pi r^3$$A hemisphere is half of a sphere, so its volume is $\tfrac{2}{3}\pi r^3$.
A table tennis ball has a radius of $2 \text{ cm}$. Calculate the volume of the table tennis ball in $\text{cm}^3$. (Use the formula for the volume of a sphere…
A table tennis ball has a radius of $2 \text{ cm}$. Calculate the volume of the table tennis ball in $\text{cm}^3$. (Use the formula for the volume of a sphere $V = \frac{4}{3} \pi r^3$ and take $\pi \approx 3,14$).
In a science museum there is an interactive exhibit with a glass sphere filled with a noble gas. If the diameter of the sphere is $60\text{ in}$, what is the volume of…
In a science museum there is an interactive exhibit with a glass sphere filled with a noble gas. If the diameter of the sphere is $60\text{ in}$, what is the volume of gas inside it? Express the result in cubic feet ($m^3$).
Composite solids
Many real objects combine several simple solids: a silo (cylinder plus hemisphere), a trophy (prism plus pyramid), a bottle (cylinder plus cone). To find the volume:
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Break the object into simple solids.
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Find the dimensions each part needs (radius, base, height).
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Compute each volume, then add them.
A grain storage silo has a composite shape: a cylindrical body with a height of $10\text{ ft}$ and a hemispherical dome on top. If the base radius is $4\text{ ft}$,…
A grain storage silo has a composite shape: a cylindrical body with a height of $10\text{ ft}$ and a hemispherical dome on top. If the base radius is $4\text{ ft}$, calculate the total capacity of the silo in gallons ($hl$).
A designer has created a trophy for an eSports* tournament with a complex geometric shape. The trophy consists of a regular square-based wooden prism on which a regular…
A designer has created a trophy for an eSports tournament with a complex geometric shape. The trophy consists of a regular square-based wooden prism on which a regular square-based metal pyramid rests. The edge of the common base measures $12\text{ cm}$. The height of the prism is $20\text{ cm}$, and we know that the lateral edge of the pyramid measures $10\text{ cm}$. Calculate the total volume of the trophy in cubic decimeters ($dm^3$).
A perfume company sells its star fragrance in a bottle consisting of a glass cylinder with a diameter of $4\text{ in}$ and a height of $10\text{ in}$, topped by a…
A perfume company sells its star fragrance in a bottle consisting of a glass cylinder with a diameter of $4\text{ in}$ and a height of $10\text{ in}$, topped by a cone-shaped cap with the same base and a height of $3\text{ in}$. If the bottle is only filled to $85\%$ of its total capacity to leave space for air, how many fluid ounces ($ml$) of perfume does each bottle contain? (Take $\pi \approx 3,1416$).
Holes, density, and displacement
Subtracting a hole
When a piece is drilled, subtract the volume of the hole from the volume of the original solid. Mass comes from density:
$$\text{mass} = \text{density} \times \text{volume}$$Water displacement
An object that is completely underwater pushes up the water level. The volume of the water that rises equals the volume of the object.
The extra water forms a cylinder with the container's radius and height $\Delta h$.
Finding a radius from a volume
To find the radius of a sphere from its volume, solve $V = \tfrac{4}{3}\pi r^3$ for $r^3$, then take the cube root (use the $\sqrt[3]{\ }$ function on a calculator if the number is not a perfect cube). For $V = 288\pi$ in.³: $r^3 = \tfrac{3 \cdot 288\pi}{4\pi} = 216$, so $r = \sqrt[3]{216} = 6$ in.
Comparing containers
To compare two containers, compute the volumes and compare the numbers. For a container with a solid part inside, the space left for the contents is the container's volume minus the volume of the solid part.
An industrial steel part has the shape of a rectangular prism with dimensions $10\text{ in} \times 8\text{ in} \times 6\text{ in}$. To lighten its weight, a cylindrical…
An industrial steel part has the shape of a rectangular prism with dimensions $10\text{ in} \times 8\text{ in} \times 6\text{ in}$. To lighten its weight, a cylindrical hole has been made through the part from top to bottom through its largest face. The radius of the cylinder is $2\text{ in}$. Knowing that the density of steel is $7,85\text{ oz/in}^3$, calculate the total mass of the part in pounds. (Take $\pi \approx 3,14$).
In the physics laboratory, the students completely submerge a lead sphere in a cylindrical container half-filled with water. The radius of the container is…
In the physics laboratory, the students completely submerge a lead sphere in a cylindrical container half-filled with water. The radius of the container is $10\text{ in}$. When the sphere is introduced, the water level rises exactly $4,5\text{ in}$ without overflowing. What is the radius of the lead sphere? Express the result in inches to two decimal places.
An ice cream manufacturer wants to compare two containers. The first has the shape of a cone with a radius of $6\text{ cm}$ and a height of $15\text{ cm}$. The second is…
An ice cream manufacturer wants to compare two containers. The first has the shape of a cone with a radius of $6\text{ cm}$ and a height of $15\text{ cm}$. The second is a cylinder with the same radius and the same height, but inside it contains a chocolate hemisphere that occupies the bottom part. If the radius of the hemisphere is also $6\text{ cm}$, which of the two containers has a greater volume available for the vanilla ice cream? Justify your answer by carrying out the calculations.
Frustums and similarity
A frustum is what is left of a pyramid after you cut off the top with a plane parallel to the base. The volume is
$$V_{\text{frustum}} = V_{\text{big pyramid}} - V_{\text{small pyramid (the top)}}$$To find these volumes you need the height of the full pyramid. Use similar triangles: the small top pyramid is a scaled copy of the big one, so $\dfrac{\text{side of top}}{\text{side of bottom}} = \dfrac{\text{height of top pyramid}}{\text{height of big pyramid}}$.
When the frustum is only partly full, the water also forms a smaller frustum. Its top surface is a square whose side you find with similarity: the side of any horizontal slice is proportional to its distance from the apex.
A water tank has the shape of a frustum of a regular square-based pyramid. The edges of the bases measure $4\text{ ft}$ and $2\text{ ft}$ respectively. The height of the…
A water tank has the shape of a frustum of a regular square-based pyramid. The edges of the bases measure $4\text{ ft}$ and $2\text{ ft}$ respectively. The height of the tank is $3\text{ ft}$. If the tank is filled to three-quarters of its height (that is, the water reaches $2,25\text{ ft}$ from the base), how many gallons of water does it contain? (Note: You will have to use similarity to find the side of the water's surface).
Back to the start: A pool 30 ft long, 15 ft wide, and 5 ft deep has $V = 30 \cdot 15 \cdot 5 = 2{,}250$ ft³, which is $2{,}250 \cdot 7.48 = 16{,}830$ gallons. A cone-shaped ice cream cup with a 3-in. diameter and a height of 4 in. has $V = \tfrac{1}{3}\pi \cdot 1.5^2 \cdot 4 = 3\pi \approx 9.4$ in.³, which is about 5.2 fluid ounces.