Volume of a geometric solid
How much water fits in an aquarium? How much ice cream is there in a cone? How many lead balls can be made by melting down one large sphere? All of these questions are about volume: the space that a solid takes up.
Units of volume and capacity
Volume is measured in cubic units: $1 \text{ cm}^3$ is a small cube with an edge of 1 cm. Each unit is 1 000 times the next one down (because $10 \cdot 10 \cdot 10$).
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…
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 $1\text{ dm}^3 = 1\text{ litre}$).
Prisms and cylinders
A prism or a cylinder is a "stack" of identical bases. Its volume is the area of the base times the height:
$$V = A_{base} \cdot h \qquad \text{Cuboid: } V = a \cdot b \cdot c \qquad \text{Cylinder: } V = \pi r^2 h$$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.…
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 cylindrical drink can has a diameter of $6\text{ cm}$ and a height of $12\text{ cm}$. Calculate its volume using the value of $\pi \approx 3,14$.
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 $15\text{ cm}$.
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 $15\text{ cm}$.
Pyramids and cones: one third
If you fill a pyramid with water and pour it into a prism with the same base and the same height, you need exactly three pyramids to fill it. The same happens with a cone and a cylinder.
$$V_{pyramid} = \frac{A_{base} \cdot h}{3} \qquad\qquad V_{cone} = \frac{\pi r^2 h}{3}$$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.
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.
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…
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).
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…
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?
The sphere
$$V_{sphere} = \frac{4}{3}\pi r^3$$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…
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 $\pi \approx 3,14$).
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…
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.
Composite solids
Many real objects combine several solids: a silo (cylinder + cone), a capsule (cylinder + two hemispheres), a piece with a hole in it. They are broken down, each part is calculated and the results are added (or the hollow is subtracted).
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…
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…
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 city 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…
A city 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).
Back to the start: To find out how much water fits in an aquarium measuring $60 \times 30 \times 40\text{ cm}$, we calculate its volume as a rectangular prism: $V = 60 \cdot 30 \cdot 40 = 72\,000\text{ cm}^3 = 72\text{ dm}^3$, which is exactly $72\text{ L}$ of water.