Exercises: Volume of a geometric solid

A package containing a gaming keyboard is in the shape of a cuboid. 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 centimetres.

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 centimetres 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 cuboid with the following dimensions: $60\text{ cm}$ in length, $30\text{ cm}$ in width and $40\text{ cm}$ in height. Calculate how many litres of water fit in the aquarium if it is completely filled. (Remember that $1\text{ dm}^3 = 1\text{ litre}$).

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 $\pi \approx 3.14$).

Cian wants to calculate the capacity in litres of a cylindrical thermos that he uses to take water to school. Using a ruler, he measures the exterior and obtains a diameter of $8\text{ cm}$ and a height of $20\text{ cm}$. In his notebook, he writes:

$V = \pi \times 8^2 \times 20 \approx 3.14 \times 64 \times 20 = 4\,019,2\text{ cm}^3 \approx 40.2\text{ L}$

and concludes that the thermos holds just over $40\text{ litres}$ of water. Identify the two errors made by Cian (one geometric and one in unit conversion), explain clearly why they are incorrect, and calculate the actual approximate capacity of the thermos in litres, rounding to two decimal places.

At your neighbourhood sports centre, they are going to inflate a giant gymnastics ball (spherical) that has a diameter of $60\text{ cm}$. Before starting to calculate with decimals:

1. a) Estimate with reasoning an upper bound for its capacity: if the ball were fitted exactly inside a cubic box with an edge of $60\text{ cm}$, how many litres of air would that box hold? Based on that bound, reason whether you think it is possible for the ball to exceed $150\text{ L}$.
2. b) Now calculate the exact volume of the ball in cubic decimetres (litres), taking $\pi \approx 3.14$.
3. c) Compare the value obtained with your previous estimate and check whether your intuition was correct by justifying the geometric relationship between the sphere and the cube.

A County Council 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 litres) needed to fill it completely.

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

A coffee flask is cylindrical in shape. The radius of the base measures $4 \text{ cm}$ and the total height is $25 \text{ cm}$. If the flask 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 $1 \text{ dm}^3 = 1 \text{ litre}$).

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 metres.

(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 inner diameter of $6 \text{ cm}$ contains water up to a height of $10 \text{ cm}$. An irregularly shaped stone is dropped into it and the water level rises to $12.5 \text{ cm}$ without overflowing. In the figure, the dashed line marks the initial water level and the solid blue line marks the final level.

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 cosmetics company from Cork wants to market a new perfume in a bottle shaped like a right pyramid with a square base. The design team establishes that the bottle must contain exactly $100\text{ mL}$ of cologne (filled to the vertex) and that, to make it comfortable to hold in one hand, the side of the square base must measure $5\text{ cm}$.

1. a) Express the required volume in cubic centimetres.
2. b) Set up an equation using the formula for the volume of a pyramid and calculate the interior height that the bottle must have.
3. c) If, to lower costs, they decide to keep the same height that you calculated but replace the pyramidal bottle with a cylindrical container whose base has a radius equal to half the side of the original square ($r = 2.5\text{ cm}$), will it have a greater or lesser capacity than the pyramidal bottle? Justify your answer by comparing the formulas without needing to calculate the numerical volume of the cylinder.

Met Éireann, the Irish Meteorological Service, has recorded a precipitation of $45\text{ mm}$ (equivalent to $45\text{ L/m}^2$) at a station in the Midlands during an autumn storm. A farmer has an open-topped rectangular tank on his plot to collect rainwater, measuring $4\text{ m}$ long, $2.5\text{ m}$ wide, and $1.2\text{ m}$ deep, initially empty. Right next to it, he has a homemade cylindrical rain gauge with an inside diameter of $10\text{ cm}$ and a height of $30\text{ cm}$, also open and empty.

1. a) Calculate how many litres of water fell directly into the rectangular tank during the storm.
2. b) What height in centimetres did the water accumulated inside the rectangular tank reach? And what height will it have reached inside the cylindrical rain gauge? Justify your answer based on the definition of "millimetres of rain".
3. c) What percentage of the total capacity of the rectangular tank has been filled with this storm?

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 decimetres ($dm^3$).

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 metres. If the silo is filled to $85\%$ of its capacity, how many tonnes 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 millimetres will the water level rise in the container? (Remember that you must first calculate the height of the pyramid using Pythagoras' Theorem twice).

To redevelop a pedestrian square in Galway, solid concrete planters are planned. Each outer piece is a right prism with a square base of side $80\text{ cm}$ and height $60\text{ cm}$. On the top face, a hemispherical hollow with a radius of $30\text{ cm}$ is hollowed out to hold the soil for the plants. Two site managers propose different methods to order the total volume of concrete for 10 planters:

• Method A: Calculate the volume of the solid prism, calculate the volume of the hemisphere that is removed, subtract both volumes for one planter, and multiply the result by 10.

• Method B: Approximate the hemisphere by a cylinder of radius $30\text{ cm}$ and depth $30\text{ cm}$, and decompose the concrete into four lateral rectangular prisms around the hollow plus a lower quadrangular prism as a support base.

1. a) Analyse the mathematical suitability of both methods. Which of the two methods is exact, and which introduces an approximation error in the volume of concrete? Justify why the discarded method is not rigorous.
2. b) Following the appropriate method, calculate the total volume of concrete needed to manufacture the 10 planters (in $\text{m}^3$, rounding to two decimal places).
3. c) Knowing that the density of the concrete used is $2\,400\text{ kg/m}^3$, determine whether a truck with a maximum payload of $7\,500\text{ kg}$ will be able to transport the 10 planters in a single trip or if it will exceed the authorised limit. Justify your answer with calculations.

In a high school science laboratory, they want to experimentally determine a student's body volume to estimate their body composition, applying the principle of water immersion. A cylindrical tank with an inside diameter of $1\text{ m}$, partially filled with water, is used.

1. a) When the student completely enters the water (submerging their head and expelling the air from their lungs), the water level in the tank rises by exactly $8.5\text{ cm}$. Model the situation mathematically and calculate the volume of the student's body in cubic decimetres (litres), taking $\pi \approx 3.14$.
2. b) If the student's mass is $68\text{ kg}$, calculate their overall average density in $\text{kg/L}$. Considering that muscle and bone tissue has an approximate average density of $1.1\text{ kg/L}$ and fatty tissue of $0.9\text{ kg/L}$, reason whether the obtained value is physiologically coherent and explain whether this student, with empty lungs, would tend to float or sink in freshwater (density of water: $1\text{ kg/L}$).
3. c) Pose and solve the inverse problem: if a solid iron gym weight with a volume of $4.5\text{ dm}^3$ is placed into the same cylindrical tank, how many millimetres will the water level rise? Round to one decimal place.