Stick a cardboard rectangle to a pencil along one of its sides and spin it very fast: you will see a cylinder "appear". Think of a tin of beans or a sliotar spinning on a hurley. That is why the cylinder, the cone and the sphere are called solids of revolution.
How they are generated
Move the slider to see how the plane shape, as it rotates, "sweeps" through space and forms the solid:
Shape that rotates
Axis of rotation
Solid that forms
Rectangle
one of its sides
Cylinder
Right-angled triangle
one of the two sides at the right angle
Cone
Semicircle
its diameter
Sphere
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Imagine sticking the following 2D shapes to a cocktail stick and spinning them quickly around it. Identify which solid of revolution is generated in each case: a) A…
Imagine sticking the following 2D shapes to a cocktail stick and spinning them quickly around it. Identify which solid of revolution is generated in each case:
a) A rectangle rotating about one of its sides.
b) A right-angled triangle rotating about one of its legs.
Draw in your notebook the solid of revolution generated by rotating the following 2D shape around the indicated axis. What is the name of the solid obtained? Describe…
Draw in your notebook the solid of revolution generated by rotating the following 2D shape around the indicated axis. What is the name of the solid obtained? Describe its main elements (bases, height and generatrix).
You can do it on paper and send a photo with your answer.
Calculate the volume of the solid generated by rotating $360^\circ$ the following irregular polygon around the indicated vertical axis. The polygon is formed by a…
Calculate the volume of the solid generated by rotating $360^\circ$ the following irregular polygon around the indicated vertical axis. The polygon is formed by a rectangle of $2 \times 4 \text{ cm}$ and a right-angled triangle of base $3 \text{ cm}$ attached to its side.
If you cut a tin along a vertical line and "unroll" it, you get its flat net: two circles (the bases) and a rectangle (the lateral surface).
The curved surface area is the area of that rectangle: $2\pi r h$. For the total surface area, add the two circles: $2\pi r h + 2\pi r^2$. The volume is $\pi r^2 h$.
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A tin of tennis balls is cylindrical in shape. The base has a diameter of $7$ cm and the height of the tin is $22$ cm. Calculate the curved surface area of the tin…
A tin of tennis balls is cylindrical in shape. The base has a diameter of $7$ cm and the height of the tin is $22$ cm. Calculate the curved surface area of the tin (excluding the caps).
A soft drinks brand wants to design a new cylindrical can. The radius of the base must be $3.5\text{ cm}$ and the height of the can $12\text{ cm}$. Calculate the amount…
A soft drinks brand wants to design a new cylindrical can. The radius of the base must be $3.5\text{ cm}$ and the height of the can $12\text{ cm}$. Calculate the amount of metal (total surface area) needed to make one of these cans. Round the result to two decimal places.
A water tank is shaped like a cylinder with a radius of $2\text{ metres}$ and a height of $5\text{ metres}$. If the tank is filled to three-quarters of its capacity, how…
A water tank is shaped like a cylinder with a radius of $2\text{ metres}$ and a height of $5\text{ metres}$. If the tank is filled to three-quarters of its capacity, how many litres of water does it contain? (Remember that $1\text{ m}^3 = 1000\text{ litres}$).
The net of the cone is a circular sector (a "slice of pizza") with radius equal to the slant height $g$, plus the circle of the base.
The arc of the sector is the same length as the circumference of the base, $2\pi r$. The whole circle of radius $g$ has circumference $2\pi g$, so the sector is the fraction $\frac{r}{g}$ of a full circle. That gives the central angle: $\text{angle} = 360^\circ \times \frac{r}{g}$.
The slant height, the height and the radius form a right-angled triangle, so by Pythagoras' theorem $g^2 = h^2 + r^2$. The curved surface area is $\pi r g$ and the total surface area is $\pi r g + \pi r^2$.
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In the following drawing of a cone, identify the names of the parts labelled with the letters $r$, $h$ and $g$. Briefly explain what each of them represents.
In the following drawing of a cone, identify the names of the parts labelled with the letters $r$, $h$ and $g$. Briefly explain what each of them represents.
To make a cone-shaped party hat, we need to know its slant height ($g$). If the hat has a height of $8$ cm and a base radius of $6$ cm, use Pythagoras' theorem to…
To make a cone-shaped party hat, we need to know its slant height ($g$). If the hat has a height of $8$ cm and a base radius of $6$ cm, use Pythagoras' theorem to calculate how long the slant height must be.
A traffic cone has a height of $60\text{ cm}$ and a base radius of $15\text{ cm}$. Calculate its curved surface area (the surface that would need to be painted orange).…
A traffic cone has a height of $60\text{ cm}$ and a base radius of $15\text{ cm}$. Calculate its curved surface area (the surface that would need to be painted orange). To do this, you must first find the slant height of the cone using Pythagoras' theorem.
Cutting a cylinder or a cone parallel to its base gives a circle; cutting either through its axis gives a rectangle (cylinder) or a triangle (cone).
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Calculate the volume of a bowling ball (which is a perfect sphere) knowing that its radius is $11$ cm. Use the formula $V = \frac{4}{3} \times \pi \times r^3$ and take…
Calculate the volume of a bowling ball (which is a perfect sphere) knowing that its radius is $11$ cm. Use the formula $V = \frac{4}{3} \times \pi \times r^3$ and take $\pi \approx 3.14$ as the value of pi.
You have a right cylinder and a right cone. The figure shows a horizontal cut (parallel to the base) with an orange dashed line and a vertical cut passing through the…
You have a right cylinder and a right cone. The figure shows a horizontal cut (parallel to the base) with an orange dashed line and a vertical cut passing through the centre of the bases (the axis of the solid) with a red dashed line.
a) If we make the horizontal cut (parallel to the base) in a cylinder and in a cone, what plane geometric figure do we obtain as the cross-section in each one?
b) Do we obtain the same figure in each solid if the cut is vertical and passes through the centre of the bases? State which figure results in the cylinder and which in the cone.
A spherical orange of diameter $10\text{ cm}$ is cut by a plane passing at a distance of $3\text{ cm}$ from the centre. When the two pieces are separated, the…
A spherical orange of diameter $10\text{ cm}$ is cut by a plane passing at a distance of $3\text{ cm}$ from the centre. When the two pieces are separated, the cross-section of the cut is a circle. Calculate the radius of that circular cross-section by applying Pythagoras' theorem in the shaded triangle.