Volume
How much water fits in a swimming pool? How much ice cream is in a cone? These questions are about volume: the space that a solid takes up.
Prisms and cylinders
A prism or a cylinder is like a "stack" of identical bases, one on top of another. Its volume is the area of the base times the height:
$$V = A_{base} \cdot h \qquad \text{Cuboid: } V = a \cdot b \cdot c \qquad \text{Cylinder: } V = \pi r^2 h$$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 litres…
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 litres of water could it hold if it were completely waterproof?
Remember that $1 \text{ dm}^3$ is equivalent to $1 \text{ litre}$.
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…
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 millilitres ($ml$) of product each can contains.
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 the cone and the cylinder.
$$V_{pyramid} = \frac{A_{base} \cdot h}{3} \qquad\qquad V_{cone} = \frac{\pi r^2 h}{3}$$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…
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.
A designer has created a trophy for an eSports tournament with a complex geometric shape. The trophy consists of a square-based right prism made of wood, topped by a…
A designer has created a trophy for an eSports tournament with a complex geometric shape. The trophy consists of a square-based right prism made of wood, topped by a square-based right pyramid made of metal. The edge of the shared base measures $12\text{ cm}$. The height of the prism is $20\text{ cm}$, and the slant edge of the pyramid measures $10\text{ cm}$. Calculate the total volume of the trophy in cubic decimetres ($dm^3$).
The sphere
$$V_{sphere} = \frac{4}{3}\pi r^3$$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…
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 metres ($m^3$).
Composite solids
Many real objects combine several solids: a silo (cylinder + hemisphere), a trophy (prism + pyramid). They are broken down, the volume of each part is calculated, and the volumes are added (or subtracted, if there is a hollow).
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 $4\text{ m}$,…
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 hectolitres ($hl$).
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…
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 millilitres ($ml$) of perfume does each bottle contain? (Take $\pi \approx 3,1416$).
Back to the beginning: For the swimming pool you calculate the volume of a cuboid by multiplying length, width and depth; for the ice-cream cone you have already seen that it holds a third of the equivalent cylinder (about $94\text{ ml}$).