Solids of revolution
A drinks can, an ice-cream cone, a tennis ball. None of them has flat faces: they are solids of revolution, which are obtained by rotating a plane shape about a line. If you rotate a rectangle about one of its sides, a cylinder "appears".
How they are generated
Move the slider to rotate the plane shape about the axis and you will see how the solid is formed:
| Shape that rotates | Axis of rotation | Solid |
|---|---|---|
| Rectangle | one of its sides | Cylinder |
| Right-angled triangle | one of the shorter sides | Cone |
| Right-angled trapezium | the side perpendicular to the parallel sides | Frustum of a cone |
| Semicircle | its diameter | Sphere |
Elements: the radius $r$ of the base, the height $h$ and the slant height $g$ (the side that, when it rotates, "sweeps out" the curved surface). In the cone, $g$ is the hypotenuse: $g = \sqrt{h^2 + r^2}$.
Imagine you have a rectangular piece of card with a base of $10\text{ cm}$ and a height of $15\text{ cm}$. If you spin it quickly around one of its $15\text{ cm}$ sides…
Imagine you have a rectangular piece of card with a base of $10\text{ cm}$ and a height of $15\text{ cm}$. If you spin it quickly around one of its $15\text{ cm}$ sides (the side marked as the axis of rotation in the figure), what solid of revolution is generated? Draw the resulting solid and state its radius and its height.
You can do it on paper and send a photo with your answer.
A solid of revolution is generated by rotating a rectangle with base $4\text{ cm}$ and height $7\text{ cm}$ about one of its longer sides. 1. What geometric shape is…
A solid of revolution is generated by rotating a rectangle with base $4\text{ cm}$ and height $7\text{ cm}$ about one of its longer sides.
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What geometric shape is obtained?
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Calculate its total surface area.
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If it were rotated about the shorter side, would it have the same volume? Justify your answer by performing the calculations.
The cylinder
If you cut off the label of a can and flatten it out, you get a rectangle: its base is the length of the circumference, $2\pi r$, and its height is $h$.
A standard drink can has a cylindrical shape. If the radius of its base measures $3,25\text{ cm}$ and its height is $12\text{ cm}$, calculate the area of the paper label…
A standard drink can has a cylindrical shape. If the radius of its base measures $3,25\text{ cm}$ and its height is $12\text{ cm}$, calculate the area of the paper label that wraps around the side of the can (lateral area). Use $\pi \approx 3,14$.
A courier company uses cylindrical cardboard tubes to send blueprints. If each tube has a lateral area of $942\text{ cm}^2$ and a height of $50\text{ cm}$: 1. Calculate…
A courier company uses cylindrical cardboard tubes to send blueprints. If each tube has a lateral area of $942\text{ cm}^2$ and a height of $50\text{ cm}$:
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Calculate the radius of the base of the tube.
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Calculate the amount of cardboard needed to make the two circular lids for each tube. (Use $\pi \approx 3,14$).
The cone and the frustum of a cone
The net of the curved surface of a cone is a sector of a circle with radius equal to the slant height $g$:
In the frustum of a cone (like a popcorn bucket), with radii $R$ and $r$: $A_L = \pi\,(R + r)\,g$.
For a wizard costume, a student has made a cone-shaped hat. If the base radius is $10\text{ cm}$ and the height of the hat is $24\text{ cm}$, use Pythagoras' theorem to…
For a wizard costume, a student has made a cone-shaped hat. If the base radius is $10\text{ cm}$ and the height of the hat is $24\text{ cm}$, use Pythagoras' theorem to calculate the slant height ($g$) of the cone.
An ice cream cone has the shape of a cone. The radius of the top opening is $3\text{ cm}$ and its depth (height) is $10\text{ cm}$. What volume of ice cream fits exactly…
An ice cream cone has the shape of a cone. The radius of the top opening is $3\text{ cm}$ and its depth (height) is $10\text{ cm}$. What volume of ice cream fits exactly inside the cone? (Assume the ice cream is level with the rim, without overflowing).
Calculate the lateral surface area and volume of a frustum of a cone (such as a popcorn bucket) whose bases have radii of $10\text{ cm}$ and $6\text{ cm}$, knowing that…
Calculate the lateral surface area and volume of a frustum of a cone (such as a popcorn bucket) whose bases have radii of $10\text{ cm}$ and $6\text{ cm}$, knowing that its height is $15\text{ cm}$. Note: You will need to calculate the slant height first using Pythagoras' theorem.
The sphere
A sphere cannot be flattened onto a plane without distorting it (which is why every world map distorts something). Its surface area is exactly four times the area of its great circle:
$$A = 4\pi r^2$$A tennis ball has an approximate diameter of $6,7\text{ cm}$. Calculate the total surface area of felt needed to cover the ball, treating it as a perfect sphere. Express…
A tennis ball has an approximate diameter of $6,7\text{ cm}$. Calculate the total surface area of felt needed to cover the ball, treating it as a perfect sphere. Express your answer in $\text{cm}^2$.
If we cut an orange in half, we get two hemispheres. If the radius of the orange is $4\text{ cm}$, calculate the volume of one of these halves.
If we cut an orange in half, we get two hemispheres. If the radius of the orange is $4\text{ cm}$, calculate the volume of one of these halves.
A brand of vitamin supplements designs its capsules in the shape of a cylinder with two hemispheres at the ends. The total length of the capsule is $22\text{ mm}$ and…
A brand of vitamin supplements designs its capsules in the shape of a cylinder with two hemispheres at the ends. The total length of the capsule is $22\text{ mm}$ and the diameter of the central cylinder is $8\text{ mm}$. Calculate the total volume of the capsule in cubic millimetres.