Volume of Geometric Solids
How much water fits in a swimming pool? How much ice cream fits in an ice cream shop cone? Both questions are about volume: the space a solid takes up. At the end of the page we answer them.
Prisms and cylinders
A prism or a cylinder is like a "stack" of identical bases, one on top of another. Its volume is the area of the base times the height. The more layers, the more volume.
$$V = A_{base} \times h \qquad \text{Rectangular prism: } V = a \times b \times c \qquad \text{Cylinder: } V = \pi r^2 h$$You have bought a new smartphone and want to calculate the volume of its box to find out how much space it will take up on your shelf. The box is a rectangular prism…
You have bought a new smartphone and want to calculate the volume of its box to find out how much space it will take up on your shelf. The box is a rectangular prism with the following dimensions: length of $16 \text{ cm}$, width of $9 \text{ cm}$, and height of $5 \text{ cm}$. Calculate its volume in $\text{cm}^3$.
A famous soft drink brand uses cylindrical cans for its drinks. If the radius of the base of the can is $3 \text{ cm}$ and its height is $12 \text{ cm}$, calculate the…
A famous soft drink brand uses cylindrical cans for its drinks. If the radius of the base of the can is $3 \text{ cm}$ and its height is $12 \text{ cm}$, calculate the volume of drink that the can can hold. (Use $\pi \approx 3.14$).
A well-known milk-shake brand has designed a new cylindrical container for its summer limited edition. The can has a diameter of $8\text{ cm}$ and a height of…
A well-known milk-shake brand has designed a new cylindrical container for its summer limited edition. The can has a diameter of $8\text{ cm}$ and a height of $15\text{ cm}$. If the manufacturer indicates that the milk shake occupies $90\%$ of the container's total capacity to leave space for air, calculate how many milliliters ($ml$) of product each can contains.
Other bases: triangle, trapezoid, and hexagon
The rule $V = A_{base} \times h$ works for any right prism. Only the way you calculate the area of the base changes:
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Triangle: $\dfrac{b \times h}{2}$
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Trapezoid: $\dfrac{(B + b) \times h}{2}$
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Regular hexagon with side $l$: it is divided into 6 equilateral triangles, each with area $\dfrac{\sqrt{3}}{4} l^2$.
Cavalieri's principle. If two solids have the same height and all their cross sections parallel to the base have the same area, they have the same volume. That is why a prism and a cylinder with the same height and the same base area have equal volume, even though they look very different.
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 ground) 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 ground) 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?
Cavalieri's principle states that two solids of the same height whose cross-sections at the same distance from the base have the same area also have the same volume.…
Cavalieri's principle states that two solids of the same height whose cross-sections at the same distance from the base have the same area also have the same volume. Compare a regular hexagonal prism with a height of $20\text{ cm}$ and a base area of $30\text{ cm}^2$ to a cylinder with the same height and the same base area. If we pour $0.5\text{ liters}$ of water into each, which one will fill up first? Justify your answer by calculating the volume of both.
A concrete ramp is to be built for a skate park. The ramp is shaped like a right prism whose base is a right trapezoid. The parallel sides (bases) of the trapezoid…
A concrete ramp is to be built for a skate park. The ramp is shaped like a right prism whose base is a right trapezoid. The parallel sides (bases) of the trapezoid measure $3\text{ m}$ and $1\text{ m}$, and its height (length of the base of the ramp on the ground) is $5\text{ m}$. If the width of the ramp is $4\text{ m}$, calculate how many cubic meters of concrete are needed and the total cost if the price is $1\,900\text{ \$/m}^3$.
Pyramids and cones: one third
If you 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 it. The same thing happens with the cone and the cylinder.
$$V_{pyramid} = \frac{A_{base} \times h}{3} \qquad\qquad V_{cone} = \frac{\pi r^2 h}{3}$$You have a glass paperweight in the shape of a regular square pyramid. The area of its square base is $25 \text{ cm}^2$ and the height of the pyramid is $9 \text{ cm}$.…
You have a glass paperweight in the shape of a regular square 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 birthday party, party hats in the shape of a cone have been made out of card. If each hat has a radius of $8 \text{ cm}$ and a height of $15 \text{ cm}$, what…
For a birthday party, party hats in the shape of a cone have been made out of card. 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 $\pi \approx 3.14$).
When the height is missing: Pythagoras
Sometimes you are given the slant height of a cone or the lateral edge of a pyramid, not its height. The height, the radius (or the distance to the center of the base), and that edge form a right triangle, so you use the Pythagorean theorem: $h^2 + d^2 = \text{edge}^2$.
In a cone: $h = \sqrt{g^2 - r^2}$. In a regular square pyramid with side $l$, the distance from the center to a corner is half the diagonal, $\dfrac{l\sqrt{2}}{2}$. In a regular hexagonal pyramid, that distance equals the side $l$.
A perfume company launches a fragrance in a bottle shaped like a right cone. The radius of the base is $3\text{ cm}$ and the slant height is $5\text{ cm}$. Calculate the…
A perfume company launches a fragrance in a bottle shaped like a right cone. The radius of the base is $3\text{ cm}$ and the slant height is $5\text{ cm}$. Calculate the volume of perfume that fits in the bottle, expressed in cubic centimeters. Hint: first you will need to find the height of the cone using the Pythagorean theorem.
For a video game event, a trophy has been designed consisting of a square-based right pyramid made of bronze. The side length of the square base is $12\text{ cm}$ and…
For a video game event, a trophy has been designed consisting of a square-based right pyramid made of bronze. The side length of the square base is $12\text{ cm}$ and the slant edge (joining the apex to a corner of the base) measures $10\text{ cm}$. Calculate the volume of bronze needed to manufacture one trophy.
Calculate the volume of a regular hexagonal pyramid whose slant edge measures $13\text{ cm}$ and whose base edge measures $10\text{ cm}$. Remember that in a regular…
Calculate the volume of a regular hexagonal pyramid whose slant edge measures $13\text{ cm}$ and whose base edge measures $10\text{ cm}$. Remember that in a regular hexagon, the radius of the circumscribed circle is equal to the side length. Express the result in $cm^3$, rounding to the nearest whole unit.
The sphere
$$V_{sphere} = \frac{4}{3}\pi r^3$$A ping-pong ball has a radius of $2 \text{ cm}$. Calculate the volume of the ping-pong ball in $\text{cm}^3$. (Use the formula for the volume of a sphere…
A ping-pong ball has a radius of $2 \text{ cm}$. Calculate the volume of the ping-pong 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 display featuring a glass sphere filled with a noble gas. If the diameter of the sphere is $60\text{ cm}$, what is the…
In a science museum, there is an interactive display featuring a glass sphere filled with a noble gas. If the diameter of the sphere is $60\text{ cm}$, what is the volume of gas inside it? Express the answer in cubic meters ($m^3$).
Composite solids
Many real objects combine several solids: a silo (cylinder plus hemisphere), a trophy (prism plus pyramid). You break them down, calculate the volume of each part, and add them. If there is a hole, you subtract.
A hemisphere is half of a sphere, so its volume is half:
$$V_{hemisphere} = \frac{1}{2} \cdot \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$$If you are given the density $\rho$, the mass is $m = \rho \times V$. Make sure the units match: if $\rho$ is in g/cm³, the volume goes in cm³ and the mass comes out in grams.
Submerged objects. A submerged object displaces a volume of water equal to its own. That volume is a cylinder with the same radius as the container, with a height equal to how much the water level rises.
A grain storage silo has a composite shape: a cylindrical main body $10\text{ m}$ high and a hemispherical dome on top. If the radius of the base is $4\text{ m}$,…
A grain storage silo has a composite shape: a cylindrical main body $10\text{ m}$ high and a hemispherical dome on top. If the radius of the base is $4\text{ m}$, calculate the total capacity of the silo in hectoliters ($hl$).
An industrial steel part has the shape of a rectangular prism measuring $10\text{ cm} \times 8\text{ cm} \times 6\text{ cm}$. To lighten its weight, a cylindrical hole…
An industrial steel part has the shape of a rectangular prism measuring $10\text{ cm} \times 8\text{ cm} \times 6\text{ cm}$. To lighten its weight, a cylindrical hole has been drilled straight through from top to bottom through its largest face. The radius of the cylinder is $2\text{ cm}$. Given that the density of steel is $7.85\text{ g/cm}^3$, calculate the total mass of the part in kilograms. (Take $\pi \approx 3.14$).
In physics lab, students completely submerge a lead sphere in a cylindrical container half full of water. The radius of the container is $10\text{ cm}$. When the sphere…
In physics lab, students completely submerge a lead sphere in a cylindrical container half full of water. The radius of the container is $10\text{ cm}$. When the sphere is inserted, the water level rises by exactly $4.5\text{ cm}$ without overflowing. What is the radius of the lead sphere? Express the answer in centimeters to two decimal places.
Frustum of a pyramid
A frustum of a pyramid is what is left when you cut off the tip of a pyramid with a plane parallel to the base. Its volume is the large pyramid minus the small pyramid you removed. The small pyramid is similar to the large one, so their measurements are in the same proportion.
A water tank is shaped like a frustum of a square-based pyramid. The edges of the bases measure $4\text{ m}$ and $2\text{ m}$ respectively. The height of the tank is…
A water tank is shaped like a frustum of a square-based pyramid. The edges of the bases measure $4\text{ m}$ and $2\text{ m}$ respectively. The height of the tank is $3\text{ m}$. If the tank is filled to three quarters of its height (that is, the water reaches $2.25\text{ m}$ from the base), how many liters of water does it contain? (Note: You will need to use similarity to find the side length of the surface of the water).
An ice cream manufacturer wants to compare two containers. The first is shaped like a cone with a radius of $6\text{ cm}$ and a height of $15\text{ cm}$. The second is a…
An ice cream manufacturer wants to compare two containers. The first is shaped like 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 height, but contains a chocolate hemisphere filling 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 vanilla ice cream? Give reasons for your answer by showing your calculations.
From volume to liters
In problems you are asked for capacity in liters or milliliters. The equivalences you need:
| Volume | Capacity |
|---|---|
| $1$ cm³ | $1$ mL |
| $1$ dm³ | $1$ L |
| $1$ m³ | $1{,}000$ L |
| $1$ hL (hectoliter) | $100$ L |
If a container is not completely full, first calculate the volume of the liquid (using the height of the liquid, not the height of the container) or take the percentage of the capacity.
For a class project, you have to estimate how much water fits into your school backpack. Assuming your backpack has a capacity of $18,500 \text{ cm}^3$, how many liters…
For a class project, you have to estimate how much water fits into your school backpack. Assuming your backpack 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 lab, they have a small aquarium in the shape of a rectangular prism. Its dimensions are $40 \text{ cm}$ long, $20 \text{ cm}$ wide and $25 \text{ cm}$…
In the science lab, they have a small aquarium in the shape of a rectangular prism. Its dimensions are $40 \text{ cm}$ long, $20 \text{ cm}$ wide and $25 \text{ cm}$ high. If we fill it with water up to a height of $20 \text{ cm}$, how many liters of water have we poured in?
A municipal swimming pool in the shape of a rectangular prism has a capacity of $1200\text{ m}^3$. If the length of the pool is $50\text{ meters}$ and the width is…
A municipal swimming pool in the shape of a rectangular prism has a capacity of $1200\text{ m}^3$. If 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 municipality decides to fill it to only $4/5$ of its capacity for safety reasons, how many liters of water will be in the pool?
Back to the beginning: the swimming pool measuring $10 \times 5 \times 2$ m holds $100{,}000$ L of water. And the ice cream shop cone with a radius of $3$ cm and a height of $10$ cm holds about $94$ mL of ice cream, one third of the cylinder with the same base and height.