Volume of Geometric Solids

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 $\pi \approx 3.14$).

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 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 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 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 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 $5\,700\text{ \text{RD\$}\ /m}^3$.

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 $9 \text{ cm}$. 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 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 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 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.

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 volume of gas inside it? Express the answer in cubic meters ($m^3$).

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 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 ($hl$).

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.

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.

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 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 $\pi \approx 3.1416$).

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).