Exercises: Volume

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 $1 \text{ liter}$.

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 volume of drink that the can can hold. (Use $\pi \approx 3.14$).

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?

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 volume of air is trapped inside the hat when someone is wearing it? (Use $\pi \approx 3.14$).

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 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 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.

You have decided to build a gel ant farm for a science project. The container is shaped like a rectangular prism with dimensions $25\text{ cm}$ 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 $15\text{ cm}$?

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.

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 ($m^3$).

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 ($hl$).

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.

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 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?

A designer has created a trophy for an eSports tournament with a complex geometric shape. The trophy consists of a square-based right prism made of wood, topped by a square-based right pyramid made of metal. The edge of the shared base measures $12\text{ cm}$. The height of the prism is $20\text{ cm}$, and the slant edge of the pyramid measures $10\text{ cm}$. Calculate the total volume of the trophy in cubic decimeters ($dm^3$).

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 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.

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).

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.

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.

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 $1\,900\text{ \$/m}^3$.

A perfume company sells its star fragrance in a bottle consisting of a glass cylinder $4\text{ cm}$ in diameter and $10\text{ cm}$ in height, topped by a cone-shaped lid with the same base and $3\text{ cm}$ in height. If the bottle is filled to only $85\%$ of its total capacity to leave room for air, how many milliliters ($ml$) of perfume does each bottle contain? (Take $\pi \approx 3.1416$).