Solids of revolution
A soda can, an ice cream cone, a tennis ball. None of them has flat faces: they are solids of revolution, which are formed by rotating a plane figure around a line. If you spin a rectangle around one of its sides, a cylinder "appears."
How they are generated
Move the slider to rotate the plane figure around the axis and watch how the solid forms:
| Rotating figure | Axis of rotation | Solid |
|---|---|---|
| Rectangle | one of its sides | Cylinder |
| Right triangle | one leg | Cone |
| Right trapezoid | the side perpendicular to the bases | Truncated cone (frustum) |
| Semicircle | its diameter | Sphere |
Elements: the radius $r$ of the base, the height $h$, and the slant height $g$ (the side that "sweeps out" the lateral surface as it rotates). 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.
-
What geometric shape is obtained?
-
Calculate its total surface area.
-
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 $50\text{ cm}$:
-
Calculate the radius of the base of the tube.
-
Calculate the amount of cardboard needed to make the two circular lids for each tube. (Use $\pi \approx 3.14$).
The cone and the truncated cone
The lateral net of a cone is a circular sector whose radius is the slant height $g$:
In the truncated cone (like a movie theater 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 is obtained by rotating a semicircle around its diameter. 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 millimeters.
The sphere has no flat net
The cylinder and the cone unfold without wrinkles: a rectangle and a sector. The sphere does not. Try flattening the peel of half an orange: it tears or wrinkles. No sheet of paper can wrap around a ball without folds or cuts.
What you can do is make an approximate net: cut the sphere into gores (sections running from pole to pole, like the segments of an orange or the panels of a beach ball) and lay them out side by side.
That is why every world map distorts something. On the most widely used map (the Mercator projection), Greenland looks much bigger than Mexico, when in reality their areas are similar (about 2.2 and 2 million km²). Paper globes are made using exactly this kind of gores.
A spherical gas storage tank has an outer diameter of $6\text{ meters}$. Engineers want to apply a coat of anti-rust paint costing 290 pesos per square meter. How much…
A spherical gas storage tank has an outer diameter of $6\text{ meters}$. Engineers want to apply a coat of anti-rust paint costing 290 pesos per square meter. How much will it cost to paint the total surface area of the tank? (Use $\pi \approx 3.14$).