Cylinders, Cones, and Spheres
A soda can, an ice cream cone, a basketball. None of them has flat faces all the way around like a box. How much soda fits in the can? How much aluminum does it take to make it? These are solids of revolution, and by the end of this page you will be able to answer both questions.
What is a solid of revolution?
Take a flat shape and spin it around a straight line called the axis. The space it sweeps out is a solid of revolution. Spin a rectangle around one of its sides and you get a cylinder. Spin a right triangle around one of its legs and you get a cone. Spin a semicircle around its diameter and you get a sphere.
A right trapezoid (two parallel sides and one side that is perpendicular to both) spun around its perpendicular side makes a frustum of a cone: a cone with its tip sliced off, like an upside-down coffee cup.
The cylinder and cone have flat circular bases. The sphere has no flat faces at all. None of these is a polyhedron, because a polyhedron has only flat polygon faces.
Imagine you are a graphic designer creating a 3D logo. To do this, you rotate a rectangle with a base of $5 \text{ cm}$ and a height of $12 \text{ cm}$ around one of its…
Imagine you are a graphic designer creating a 3D logo. To do this, you rotate a rectangle with a base of $5 \text{ cm}$ and a height of $12 \text{ cm}$ around one of its longest sides. Draw the solid of revolution obtained and identify the names of its main elements (base and height), indicating the measurement of each one.
Identify which of the following plane figures, when rotated around the indicated axis, would result in a frustum of a cone (like an inverted coffee cup). Briefly explain…
Identify which of the following plane figures, when rotated around the indicated axis, would result in a frustum of a cone (like an inverted coffee cup). Briefly explain why you have chosen that option.
Radius, height, and slant height
Every round solid has a radius $r$ (the radius of a circular base, or of the sphere) and a height $h$ (the distance between the bases, or from the tip of a cone straight down to the base). A cone has one more measurement: the slant height $g$, which runs along the surface from the tip to the edge of the base.
The radius, the height, and the slant height form a right triangle, and $g$ is the hypotenuse. So the Pythagorean Theorem gives
$$g^2 = r^2 + h^2 \qquad\text{so}\qquad g = \sqrt{r^2 + h^2}.$$Why? The height drops straight down to the center of the base, so it meets the radius at a right angle.
For a costume party, you want to make a conical hat. You have measured the base radius, which is $9 \text{ cm}$, and the height of the hat, which will be…
For a costume party, you want to make a conical hat. You have measured the base radius, which is $9 \text{ cm}$, and the height of the hat, which will be $12 \text{ cm}$. Use the Pythagorean theorem to calculate the slant height ($g$) you will need to draw the net on the cardboard.
Surface area of a cylinder
Imagine peeling the label off a can and cutting off the top and bottom. You get a rectangle and two circles. That flat pattern is the net of the cylinder.
The rectangle is exactly as wide as the circle is long around, so its width is the circumference $2\pi r$. That gives:
A famous brand of potato chips uses cylindrical tubs. If the base radius of the tub is $4 \text{ cm}$ and its height is $20 \text{ cm}$, calculate the area of the paper…
A famous brand of potato chips uses cylindrical tubs. If the base radius of the tub is $4 \text{ cm}$ and its height is $20 \text{ cm}$, calculate the area of the paper label that surrounds the side of the tub (lateral area). Take $\pi \approx 3,14$.
Look at the net of a cylinder. It is known that the area of one of its circular bases is $28,26\text{ cm}^2$ and that the height of the lateral rectangle is…
Look at the net of a cylinder. It is known that the area of one of its circular bases is $28,26\text{ cm}^2$ and that the height of the lateral rectangle is $10\text{ cm}$. Calculate:
- a) The radius of the base.
- b) The lateral area of the cylinder.
- c) The total area.
(Use $\pi \approx 3,14$).
Volume of a cylinder
Volume measures how much a solid holds. For a cylinder, picture stacking identical circular coins. Each coin has area $\pi r^2$, and the stack is $h$ tall, so
Volume is measured in cubic units: in.$^3$, ft$^3$, cm$^3$.
Volume in fluid ounces
Drinks are sold by liquid volume, such as fluid ounces (fl oz). A fluid ounce and a cubic inch are close in size, so some problems say to use the simplified equivalence
$$1\text{ in.}^3 = 1\text{ fl oz}.$$(The real conversion is about $1\text{ in.}^3 \approx 0.55\text{ fl oz}$, but follow whatever the problem tells you.) With this rule, the number of cubic inches is the number of fluid ounces. Only the unit label changes.
A soda can has a cylindrical shape. Its base radius measures $3 \text{ in}$ and its height is $12 \text{ in}$. Calculate the volume of soda it can hold if it is…
A soda can has a cylindrical shape. Its base radius measures $3 \text{ in}$ and its height is $12 \text{ in}$. Calculate the volume of soda it can hold if it is completely filled. Express the result in cubic inches ($cm^3$).
A soda company wants to design a new cylindrical can. The radius of the base must be $3,5\text{ in}$ and the height of the can $12\text{ in}$. Calculate the total area…
A soda company wants to design a new cylindrical can. The radius of the base must be $3,5\text{ in}$ and the height of the can $12\text{ in}$. Calculate the total area of sheet metal needed to manufacture one can (excluding the sealing tabs) and the volume of soda it can hold in fluid ounces ($1\text{ in}^3 = 1\text{ fl oz}$).
Cones: lateral area and volume
A cone is like a "pointy" cylinder. Two facts:
Why $\pi r g$? Unroll the side of a cone and you get a slice of a circle (a sector) with radius $g$. Its curved edge has length $2\pi r$, the circumference of the base. Its area is about "half the base times the height": $\tfrac{1}{2} \cdot 2\pi r \cdot g = \pi r g$.
Why one third? Fill a cone with sand and pour it into a cylinder with the same base and height. It takes exactly three cones to fill the cylinder.
An ice cream cone has the shape of a cone. If the base area is $25 \text{ cm}^2$ and the height of the cone is $10 \text{ cm}$, calculate its volume. Remember that the…
An ice cream cone has the shape of a cone. If the base area 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 a third of the volume of a cylinder with the same base and height: $V = \frac{1}{3} \cdot A_{base} \cdot h$.
A temporary event tent is being manufactured in the shape of a perfect cone. The height of the tent from the ground to the top vertex is $4\text{ ft}$ and the diameter…
A temporary event tent is being manufactured in the shape of a perfect cone. The height of the tent from the ground to the top vertex is $4\text{ ft}$ and the diameter of the circular base is $6\text{ ft}$. How many square feet of canvas have been needed to cover only the lateral part of the tent? (Fact: First you will need to find the slant height using the Pythagorean theorem).
Spheres
A sphere has only one measurement, its radius $r$: the distance from the center to any point on the surface.
The surface area is exactly four times the area of the sphere's biggest circle (a "great circle"), which has area $\pi r^2$.
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 $\pi \approx 3,14$).
A famous chocolate brand sells its chocolates in a sphere-shaped box with a diameter of $10\text{ in}$. If the box is filled to $75\%$ of its capacity with liquid…
A famous chocolate brand sells its chocolates in a sphere-shaped box with a diameter of $10\text{ in}$. If the box is filled to $75\%$ of its capacity with liquid chocolate, how many cubic inches of chocolate are inside? Express the result rounded to two decimal places.
Working backward to find the radius
Sometimes you know the circumference, the area, or the volume and need the radius. Undo the formula, using inverse operations.
-
From the circumference: $C = 2\pi r$, so $r = \dfrac{C}{2\pi}$.
-
From the volume of a cylinder: $r^2 = \dfrac{V}{\pi h}$, then take the square root.
In a geography class, we observe a globe. If we know that the length of the equator (great circle) is approximately $62,8 \text{ cm}$, what is the radius of that sphere?…
In a geography class, we observe a globe. If we know that the length of the equator (great circle) is approximately $62,8 \text{ cm}$, what is the radius of that sphere? (Use $\pi \approx 3,14$).
A water tank has the shape of a cylinder with a height of $5\text{ m}$. If we want the tank to have a capacity of $100.000\text{ liters}$, what must the radius of the…
A water tank has the shape of a cylinder with a height of $5\text{ m}$. If we want the tank to have a capacity of $100.000\text{ liters}$, what must the radius of the tank's base be? (Remember that $1\text{ m}^3 = 1.000\text{ liters}$). Round the radius to two decimal places.
Cutting a solid with a plane
When a plane slices through a solid, the flat shape that appears is a cross-section.
-
Cylinder: parallel to the bases gives a circle. Straight through the axis gives a rectangle.
-
Cone: parallel to the base gives a smaller circle. Straight through the tip gives a triangle.
-
Sphere: any slice gives a circle. The biggest circle comes from a cut through the center.
Slicing a sphere
Suppose a sphere of radius $R$ is cut by a plane that is $d$ units from the center. The center, the middle of the cut circle, and a point on the edge of that circle form a right triangle.
By the Pythagorean Theorem, $r^2 + d^2 = R^2$, so $r = \sqrt{R^2 - d^2}$.
The spherical cap
Cutting a sphere gives two pieces called spherical caps. The smaller one has height $h = R - d$.
A wooden sphere of radius $13\text{ cm}$ is cut by a horizontal plane. If the distance from the center of the sphere to the cut is $5\text{ cm}$, what is the radius of…
A wooden sphere of radius $13\text{ cm}$ is cut by a horizontal plane. If the distance from the center of the sphere to the cut is $5\text{ cm}$, what is the radius of the circle formed in the cross-section of the cut? (Apply the Pythagorean theorem in the right triangle that is formed).
A wooden sphere with a radius of $20\text{ cm}$ has been cut by a plane located $12\text{ cm}$ from the center. When the two fragments are separated, two spherical caps…
A wooden sphere with a radius of $20\text{ cm}$ has been cut by a plane located $12\text{ cm}$ from the center. When the two fragments are separated, two spherical caps are obtained. Calculate the total area (curved surface plus the flat circular base of the cut) of the smaller spherical cap.
(Hint: use the Pythagorean theorem to find the radius of the cross-section circle).
Frustum of a cone
A frustum is what is left when you slice the tip off a cone with a plane parallel to the base. It has two circular bases, a small one with radius $r$ and a large one with radius $R$, and a height $h$ between them.
The dashed vertical line on the right makes a right triangle with legs $h$ and $R - r$, so the slant height is
$$g = \sqrt{h^2 + (R - r)^2}.$$Why? The frustum is a big cone minus the small cone that was cut off, and $\pi R G - \pi r g$ simplifies to $\pi(r + R)g$ when you use similar triangles.
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 manufacture 25 identical lampshades (bear in mind that the lampshade has no bases, it is only the lateral surface).
Combined solids
Many real objects are two or more solids glued together: an ice cream cone with a scoop, a pencil, a silo. The plan:
-
Split the object into basic solids.
-
For volume, add the volumes of the pieces.
-
For surface area, add only the surfaces you can actually see or touch. Faces that are glued together are not counted.
Calculate the volume of a piece of wood that has been formed by joining a cylinder and a hemisphere (half a sphere). The cylinder has a height of $10\text{ cm}$ and its…
Calculate the volume of a piece of wood that has been formed by joining a cylinder and a hemisphere (half a sphere). The cylinder has a height of $10\text{ cm}$ and its base radius is $4\text{ cm}$. The hemisphere rests exactly on the top base of the cylinder.
An ice cream cone has the shape of a cone. The wafer has a depth (height) of $12\text{ cm}$ and the radius of the opening is $3\text{ cm}$. If the ice cream fills the…
An ice cream cone has the shape of a cone. The wafer has a depth (height) of $12\text{ cm}$ and the radius of the opening is $3\text{ cm}$. If the ice cream fills the inside of the cone and a perfect scoop with the same radius as the cone protrudes from the top, calculate the total volume of ice cream.
Calculate the surface area of an "ice cream cone" made up of a right cone $12\text{ cm}$ high with a base radius of $5\text{ cm}$, topped by a perfect hemisphere that…
Calculate the surface area of an "ice cream cone" made up of a right cone $12\text{ cm}$ high with a base radius of $5\text{ cm}$, topped by a perfect hemisphere that fits exactly on the base of the cone. If we want to wrap the ice cream (cone and hemisphere) with protective paper, what is the minimum surface area of paper we will need?
Problems that mix solids
Here are two more ideas that show up in harder problems.
Water displacement. A solid that is completely submerged pushes aside exactly its own volume of water. So the volume of the object equals the volume of the "extra" layer of water that rises: $V_{\text{object}} = \pi r_{\text{container}}^2 \times (\text{rise})$.
Fractions and percents of a volume. To find what part of one solid another fills, divide the two volumes and simplify. Often $\pi$ and the measurements cancel (as with the cone in a cylinder: always $\tfrac{1}{3}$).
We want to build a grain silo consisting of a cylindrical body $4\text{ m}$ high with a radius of $3\text{ m}$, finished at the bottom with a cone of the same radius and…
We want to build a grain silo consisting of a cylindrical body $4\text{ m}$ high with a radius of $3\text{ m}$, finished at the bottom with a cone of the same radius and a slant height of $5\text{ m}$ to facilitate unloading. Calculate the total storage volume of the silo and the total surface area of sheet metal needed to manufacture it (including the flat top lid of the cylinder).
A glass sphere with a radius of $15\text{ in}$ is placed into a cylindrical container with a diameter of $32\text{ in}$ containing water. If the sphere is completely…
A glass sphere with a radius of $15\text{ in}$ is placed into a cylindrical container with a diameter of $32\text{ in}$ containing water. If the sphere is completely submerged to the bottom, how many inches will the water level in the container rise? Give your answer rounded to two decimal places.
A perfume bottle has the shape of a cylinder $10\text{ cm}$ high. Inside, the glass has an inverted conical shape that reaches exactly to the bottom and whose base…
A perfume bottle has the shape of a cylinder $10\text{ cm}$ high. Inside, the glass has an inverted conical shape that reaches exactly to the bottom and whose base coincides with the lid of the cylinder. If the radius of the bottle is $3\text{ cm}$, calculate what percentage of the total volume of the cylinder is occupied by the perfume when the bottle is full. Does this percentage depend on the chosen radius and height? Justify your answer.
Hollow solids and spinning a triangle
Hollow objects and mass
For a hollow object, such as a pipe, subtract the empty space from the whole: $V = V_{\text{outer}} - V_{\text{inner}}$. To find mass, use density:
$$\text{mass} = \text{density} \times \text{volume}.$$Make sure the units match. For example, density in kg/m$^3$ needs volume in m$^3$. Science problems often use metric units, and that is fine.
Spinning a right triangle around its hypotenuse
Spinning around the hypotenuse gives two cones glued base to base. Both share the same base radius: the altitude $a$ from the right angle to the hypotenuse. Find it with area: the triangle's area is $\tfrac{1}{2}\,(\text{leg}_1)(\text{leg}_2)$ and also $\tfrac{1}{2}\,(\text{hypotenuse})\,a$, so
$$a = \frac{\text{leg}_1 \cdot \text{leg}_2}{\text{hypotenuse}}.$$The slant heights of the two cones are the triangle's two legs.
A steel steamroller cylinder has the shape of a cylinder with a diameter of $1,2\text{ meters}$ and a length of $2,5\text{ meters}$. Steel has a density of…
A steel steamroller cylinder has the shape of a cylinder with a diameter of $1,2\text{ meters}$ and a length of $2,5\text{ meters}$. Steel has a density of $7850\text{ kg/m}^3$. If the roller is hollow and the thickness of the steel is $10\text{ cm}$ all around its contour (except for the lateral faces which are considered negligible), calculate the total mass of the roller in tons.
A solid of revolution is generated by rotating a right triangle with legs $6\text{ cm}$ and $8\text{ cm}$ around its hypotenuse. a) Describe the resulting solid (you can…
A solid of revolution is generated by rotating a right triangle with legs $6\text{ cm}$ and $8\text{ cm}$ around its hypotenuse.
- a) Describe the resulting solid (you can draw it).
- b) Calculate the total surface area of the generated solid.
(Hint: The solid is formed by two cones joined at their base. You will need to calculate the height of the triangle relative to the hypotenuse).
Back to the soda can: with a radius of $1.5$ in. and a height of $5$ in., it takes $2\pi(1.5)(5) + 2\pi(1.5)^2 = 19.5\pi \approx 61.26$ in.$^2$ of metal and holds $\pi(1.5)^2(5) = 11.25\pi \approx 35.34$ in.$^3$.