Exercises: Volume of a geometric solid

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.

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 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.

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 $\pi \approx 3,14$).

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).

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 $15\text{ cm}$.

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 $1\text{ in}^3 = 1\text{ liter}$).

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 city council has installed concrete planters in the shape of a regular hexagonal prism. If each side of the hexagonal base measures $30 \text{ in}$ and the height of the planter is $60 \text{ in}$, calculate the volume of soil (in gallons) needed to fill it completely.

(Fact: remember that you must first find the apothem of the base using Pythagoras' theorem).

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 $1 \text{ dm}^3 = 1 \text{ liter}$).

A decorative glass pyramid has a square base of side $12 \text{ cm}$. If the lateral edge (the segment joining the top vertex to a vertex of the base) measures $10 \text{ cm}$, calculate the volume of glass the piece contains.

(Note: you will need to apply Pythagoras' theorem twice: first to find the diagonal of the base and then to find the height of the pyramid).

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?

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).

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.

A cylindrical glass with an internal diameter of $6 \text{ in}$ contains water up to a height of $10 \text{ in}$. An irregularly shaped stone is placed inside and the water level rises to $12,5 \text{ in}$ without overflowing. What is the volume of the stone expressed in cubic inches?

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 street furniture design company has designed a concrete trash can 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 can 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 tetrahedrons 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?

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 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 $g/cm^3$?

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 $750\text{ lb/ft}^3$?

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.

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).

A metal piece in the shape of a regular hexagonal pyramid is to be submerged in a cylindrical container of radius $15\text{ in}$ containing water. The pyramid has a base edge of $10\text{ in}$ and a lateral edge of $20\text{ in}$. When the piece is completely inserted, how many inches will the water level in the container rise? (Remember that you must first calculate the height of the pyramid using Pythagoras' theorem twice).