Exercises: Volume of a geometric solid

A package containing a gaming keyboard is in the shape of a rectangular prism. Its dimensions are $45\text{ cm}$ in length, $15\text{ cm}$ in width and $5\text{ cm}$ in height. Calculate the volume of air inside the box in cubic centimeters.

A cylindrical drink 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 stores handball balls 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 radius of the base is $3\text{ cm}$ and its height is $12\text{ cm}$. Calculate how many cubic centimeters of ice cream fit inside it (without overflowing at the top).

Calculate the volume of a prism whose base is a right-angled triangle with legs $6\text{ cm}$ and $8\text{ cm}$, knowing that the height of the prism is $15\text{ cm}$.

An aquarium is in the shape of a rectangular prism with the following dimensions: $60\text{ cm}$ in length, $30\text{ cm}$ in width and $40\text{ cm}$ in height. Calculate how many liters of water fit in the aquarium if it is completely filled. (Remember that $1\text{ dm}^3 = 1\text{ liter}$).

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

(Given: remember that you must first find the apothem of the base using the Pythagorean Theorem).

A coffee thermos is cylindrical in shape. The radius of the base 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 apex to a vertex of the base) measures $10 \text{ cm}$, calculate the volume of glass contained in the piece.

(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 cone is to be made whose slant height measures $15 \text{ cm}$ and whose base radius is $6 \text{ cm}$. Calculate the volume of ice cream that fits inside the cone. If a perfect hemisphere of ice cream of the same radius is added on top of the base when served, what will be the total volume of the ice cream?

A water tank is in the shape of a frustum of a cone. The radius of the larger base is $3 \text{ m}$, the radius of the smaller base is $1.5 \text{ m}$ and the height of the tank is $4 \text{ m}$. Calculate the capacity of the tank in cubic meters.

(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 resulting new sphere? Round the answer to two decimal places.

A cylindrical glass with an internal diameter of $6 \text{ cm}$ contains water up to a height of $10 \text{ cm}$. An irregularly shaped stone is placed inside it and the water level rises to $12.5 \text{ cm}$ without overflowing. What is the volume of the stone expressed in cubic centimeters?

Calculate the volume of a geometric solid formed by a cylinder of height $10 \text{ cm}$ and radius $4 \text{ cm}$, topped by a square pyramid whose base is inscribed in the circle of the cylinder and whose height is $6 \text{ cm}$.

(Hint: the side of the square base of the pyramid is equal to the side of a square inscribed in a circle of radius $4 \text{ cm}$).

A street furniture design company has designed a concrete litter bin in the shape of a frustum of a regular square pyramid. The external dimensions of the upper base are $40\text{ cm}$ on each side, those of the lower base are $30\text{ cm}$ on each side, and the total height is $80\text{ cm}$. To make it functional, the interior 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 litter bin in cubic decimeters ($dm^3$).

A sculptor wishes to create a stainless steel piece consisting of a cube with an edge length of $20\text{ cm}$ from which eight tetrahedra have been removed at its corners. Each tetrahedron is formed by connecting a vertex of the cube to the midpoints of the three edges that meet at that vertex (a midpoint truncation). Determine the final volume of the sculpture. What percentage of the original cube's volume has been removed?

An hourglass consists of 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 upper chamber is filled with sand up to half its height, calculate the volume of sand in $cm^3$. If the sand flows into the lower chamber at a constant rate of $5\text{ cm}^3$ per minute, how long will it take for the upper part to empty? (Remember to use similar cones to find the radius of the sand's surface).

A lead sphere with a radius of $10\text{ cm}$ is melted down to manufacture small cylinders with a diameter of $2\text{ cm}$ and a height of $3\text{ cm}$. Assuming that $5\%$ of the material is lost in the smelting 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 section with a height of $12\text{ m}$ and a diameter of $4\text{ m}$, topped at the upper part by a hemisphere and at the lower part by a discharge cone with a height of $3\text{ m}$. Calculate the total capacity of the silo in cubic meters. If the silo is filled to $85\%$ of its capacity, how many metric tons of wheat will it contain if the density of wheat is $750\text{ kg/m}^3$?

Cavalieri's Principle states that if two solids have the same height and the area of their cross-sections at any 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 contained between a cylinder of radius $R$ and height $2R$, and a double cone inscribed in it. Show 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 that of the two cones.

A perfume bottle has the shape of a regular octahedron with an edge length of $6\text{ cm}$. However, the glass has a thickness of $2\text{ mm}$ on all its faces. The manufacturer claims that the container can hold $80\text{ ml}$ of fragrance ($1\text{ ml} = 1\text{ cm}^3$). Perform the necessary calculations to check whether the claim is true. (Given: The volume of an octahedron with edge length $a$ is $V = \frac{\sqrt{2}}{3} a^3$. You must consider how the thickness affects the edge length of the interior cavity).

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