Volume of solids and scale
A scale model of a building, two glasses with the same shape but different sizes, an ice sculpture that melts while keeping its shape: these are all similar solids, and volume is the quantity that changes fastest with scale.
Volume and the ratio $k$ cubed
Two Rubik's cubes have different sizes but are geometrically similar. The edge of the small cube measures $2 \text{ cm}$ and the edge of the large cube measures…
Two Rubik's cubes have different sizes but are geometrically similar. The edge of the small cube measures $2 \text{ cm}$ and the edge of the large cube measures $6 \text{ cm}$. If the volume of the small cube is $8 \text{ cm}^3$, calculate the volume of the large cube using the scale factor.
A plastic tetrahedron has a volume of $15 \text{ cm}^3$. A toy factory decides to build a similar tetrahedron, but tripling the length of all its edges. Calculate the…
A plastic tetrahedron has a volume of $15 \text{ cm}^3$. A toy factory decides to build a similar tetrahedron, but tripling the length of all its edges. Calculate the volume of the new tetrahedron.
Scale models and 3D scale plans
In a scale model at a scale of $1:n$, the real volume is not $n$ times that of the model, but $n^3$ times: a small model represents a building that is enormous in volume.
An architect designs the model of a skyscraper to a scale of $1:100$. If the volume of the model in the studio is $5000 \text{ cm}^3$, determine the actual volume that…
An architect designs the model of a skyscraper to a scale of $1:100$. If the volume of the model in the studio is $5000 \text{ cm}^3$, determine the actual volume that the skyscraper will have, expressed in cubic metres ($m^3$).
In a pizzeria, soft drinks are served in two similar cylindrical glasses. The large glass is exactly twice as tall and twice as wide as the small glass. If the capacity…
In a pizzeria, soft drinks are served in two similar cylindrical glasses. The large glass is exactly twice as tall and twice as wide as the small glass. If the capacity of the small glass is $250 \text{ mL}$, how many millilitres of soft drink does the large glass hold?
Mass and density in similar solids
If two objects are made of the same material (same density), their mass is proportional to their volume: it also changes by $k^3$. This explains why a statue twice as tall weighs eight times as much, not twice as much.
A solid bronze miniature of a famous monument weighs $1,5 \text{ kg}$ and has a height of $20 \text{ cm}$. The real monument, built with exactly the same type of bronze…
A solid bronze miniature of a famous monument weighs $1,5 \text{ kg}$ and has a height of $20 \text{ cm}$. The real monument, built with exactly the same type of bronze and being mathematically similar, has a height of $4 \text{ m}$. Calculate the weight of the real monument in tonnes.
An ice sculpture in the shape of a complex polyhedron melts uniformly under the sun such that at all times it maintains a shape similar to the original. If after 2 hours…
An ice sculpture in the shape of a complex polyhedron melts uniformly under the sun such that at all times it maintains a shape similar to the original. If after 2 hours it is observed that its surface area has decreased by $36\%$, what percentage of its original volume has melted up to that point?
Area and volume at the same time
Many problems give the ratio between areas (or between total surface areas) and ask for the volume, or the other way round. You have to go through $k$ first: from area to $k$ with a square root, and from $k$ to volume by cubing.
The ratio between the total surface areas of two similar cones is $\frac{16}{25}$. Determine the ratio between the volumes of these cones.
The ratio between the total surface areas of two similar cones is $\frac{16}{25}$. Determine the ratio between the volumes of these cones.
In a museum there are two similar marble statues. The volume of the first statue is $27$ times the volume of the second. If the height of the second statue is…
In a museum there are two similar marble statues. The volume of the first statue is $27$ times the volume of the second. If the height of the second statue is $10 \text{ cm}$, find the height of the first statue.
A conical funnel of height $H$ is completely full of oil. All the oil is allowed to drip into a conical container similar to the funnel but of height $2H$, which is…
A conical funnel of height $H$ is completely full of oil. All the oil is allowed to drip into a conical container similar to the funnel but of height $2H$, which is initially empty. Once the funnel has emptied completely, what fraction of the height of the large container will be covered by the oil?
Cuts parallel to the base: cones and pyramids with the top removed
If you cut a cone or a pyramid with a plane parallel to the base, the small shape at the top is similar to the complete original. By comparing heights (or radii) you obtain the ratio $k$ needed to calculate its volume, and the resulting frustum is the difference.
A cocktail glass is in the shape of a perfect inverted cone. Juice is poured into the glass until it reaches exactly half of the total height of the glass. What fraction…
A cocktail glass is in the shape of a perfect inverted cone. Juice is poured into the glass until it reaches exactly half of the total height of the glass. What fraction of the total capacity of the glass is occupied by the juice?
The figure illustrates the glass and the level of juice:
A square-based pyramid has a height of $12 \text{ cm}$ and its base edge measures $6 \text{ cm}$. This pyramid is cut by a plane parallel to the base to form a frustum…
A square-based pyramid has a height of $12 \text{ cm}$ and its base edge measures $6 \text{ cm}$. This pyramid is cut by a plane parallel to the base to form a frustum of a pyramid whose top base has an edge of $2 \text{ cm}$. Using the similarity between the original pyramid and the smaller top pyramid, calculate the volume of the resulting frustum of a pyramid.