Solids of revolution
A tin can, an ice-cream cone, a ball. None of them has flat faces like a polyhedron: they are solids of revolution. The three most important ones are the cylinder, the cone and the sphere.
The three solids of revolution
None of them is a polyhedron: they all have at least one curved surface. The cylinder and the cone also have flat faces (their bases, which are circles); the sphere has no flat faces at all.
An ice cream cone is shaped like a cone. If the area of the base is $25 \text{ cm}^2$ and the height of the cone is $10 \text{ cm}$, calculate its volume. Remember that…
An ice cream cone is shaped like a cone. If the area of the base is $25 \text{ cm}^2$ and the height of the cone is $10 \text{ cm}$, calculate its volume. Remember that the volume of a cone is one-third of the volume of a cylinder with the same base and height: $V = \frac{1}{3} \cdot A_{base} \cdot h$.
Elements of the cylinder and the cone
For a fancy-dress party, you want to make a conical hat. You have measured the radius of the base, which is $9 \text{ cm}$, and the height of the hat, which will be…
For a fancy-dress party, you want to make a conical hat. You have measured the radius of the base, which is $9 \text{ cm}$, and the height of the hat, which will be $12 \text{ cm}$. Use Pythagoras' theorem to calculate the length of the slant height ($g$) that you will need to draw the net on card.
A soft drinks company wants to design a new cylindrical can. The base radius must be $3,5\text{ cm}$ and the height of the can $12\text{ cm}$. Calculate the total…
A soft drinks company wants to design a new cylindrical can. The base radius must be $3,5\text{ cm}$ and the height of the can $12\text{ cm}$. Calculate the total surface area of sheet metal needed to manufacture one can (not including the seals) and the volume of soft drink it can hold in millilitres ($1\text{ cm}^3 = 1\text{ ml}$).
The sphere
The sphere has only one element: the radius $r$, the distance from the centre to any point on the surface.
A basketball has a diameter of $24 \text{ cm}$. What is the total surface area of leather needed to manufacture it? Remember that the radius is half the diameter and use…
A basketball has a diameter of $24 \text{ cm}$. What is the total surface area of leather needed to manufacture it? Remember that the radius is half the diameter and use the formula for the surface area of a sphere $A = 4 \cdot \pi \cdot r^2$. (Use $\pi \approx 3,14$).
Cutting a solid of revolution
If you cut a solid of revolution with a plane, the shape that appears depends on how you cut it.
A famous chocolate brand sells its chocolates in a box shaped like a sphere with a diameter of $10\text{ cm}$. If the box is filled to $75\%$ of its capacity with liquid…
A famous chocolate brand sells its chocolates in a box shaped like a sphere with a diameter of $10\text{ cm}$. If the box is filled to $75\%$ of its capacity with liquid chocolate, how many cubic centimetres of chocolate are inside? Express the result rounded to two decimal places.
An interior design company manufactures lampshades in the shape of a frustum of a cone. The radius of the top base is $12\text{ cm}$, that of the bottom base is…
An interior design company manufactures lampshades in the shape of a frustum of a cone. The radius of the top base is $12\text{ cm}$, that of the bottom base is $20\text{ cm}$, and the height of the lampshade is $15\text{ cm}$. If the special fabric material costs $18,50\text{ €/m}^2$, calculate the cost of the material needed to make 25 identical lampshades (note that the lampshade has no top or bottom bases, only the lateral surface).