8.8: Spheres and Hemispheres: Surface Area and Volume
Learning Objectives
- Find the surface area of a sphere.
- Find the volume of a sphere.
Spheres
A sphere is a three-dimensional figure that has the shape of a ball:
Spheres can be characterized in three ways:
1. A sphere is the set of all points that lie a fixed distance \begin{align*}r\end{align*} from a single center point \begin{align*}O\end{align*}.
2. A sphere is the surface that results when a circle is rotated about any of its diameters:
3. A sphere results when you construct a polyhedron with an infinite number of faces that are infinitely small. To see why this is true, recall regular polyhedra.
As the number of faces on the figure increases, each face gets smaller in area and the figure comes more to resemble a sphere. When you imagine a figure with an infinite number of faces, it would look like a sphere.
- A 3-D figure that looks like a ball is called a _____________________________.
Parts of a Sphere
As described above, a sphere is the surface that is the set of all points a fixed distance from a center point \begin{align*}O\end{align*}. The words used for spheres are similar to those used for circles:
The distance from \begin{align*}O\end{align*} to the surface of the sphere is \begin{align*}r\end{align*}, the radius:
- The ________________________ of a sphere is the distance from the center to the surface of the sphere.
The diameter \begin{align*}d\end{align*}, of a sphere is the length of the segment connecting any two points on the sphere’s surface and passing through the center \begin{align*}O\end{align*}.
You can find a diameter (actually an infinite number of diameters) on any plane within the sphere. Two diameters are shown in each sphere below:
- The _________________________ of a sphere connects two points on the surface of the sphere and goes through the center.
A chord for a sphere is similar to the chord of a circle except that it exists in three dimensions. Keep in mind that a diameter is a kind of chord—a special chord that intersects the center of the circle or sphere.
- A __________________________ connects two points on the surface of a sphere.
A secant is a line, ray, or line segment that intersects a circle or sphere in two places and extends outside of the circle or sphere.
- A _______________________ is a chord that extends in either or both directions.
A tangent intersects the circle or sphere at only one point.
In a circle, a tangent is perpendicular to the radius that meets the point where the tangent intersects with the circle. The same thing is true for the sphere. All tangents are perpendicular to the radii that intersect with them.
- A tangent is _____________________________ to the radius at the point where the two intersect, which is on the edge of the circle or sphere.
Finally, a sphere can be sliced by an infinite number of different planes. Some planes include point \begin{align*}O\end{align*}, the center of the sphere. Other points do not include the center. Look at the diagrams below:
The parts of a sphere are very similar to the parts of a circle: they have the same names and definitions, except the parts of a sphere are in ____________________ dimensions while the parts of a circle are in two dimensions.
Surface Area of a Sphere
You can figure out the formula for the surface area of a sphere by taking measurements of spheres and cylinders. Here we show a sphere with a radius of 3 and a right cylinder with both a radius and a height of 3, and we express the area in terms of \begin{align*}\pi\end{align*}.
Now try a larger pair, expressing the surface area in decimal form:
Look at a third pair:
Is it a coincidence that a sphere and a cylinder whose radius and height are equal to the radius of the sphere have the exact same surface area? Not at all! In fact, the ancient Greeks used a method that showed that the following formula can be used to find the surface area of any sphere (or any cylinder in which \begin{align*}r = h\end{align*} ).
The Surface Area of a Sphere is given by:
\begin{align*}A = 4 \pi r^2\end{align*}
- If a sphere has a radius that is equal to both the radius and the height of a cylinder, then their surface areas are the _________________________!
Example 1
Find the surface area of a sphere with a radius of 14 feet.
Use the formula:
\begin{align*}A &= 4 \pi r^2\\ &= 4 \pi (14^2)\\ &= 784 \pi \ ft^3 \quad \text{or}\\ & \approx (784)(3.14) = 2461.76 \ ft^3\end{align*}
The sphere has an exact surface area of \begin{align*}784 \pi \ ft^3\end{align*}, which is approximately \begin{align*}2461.76 \ ft^3\end{align*}.
Example 2
Find the surface area of the following figure in terms of \begin{align*}\pi\end{align*}.
(This means give an exact answer, not an approximate answer.)
The figure is made of one-half of a sphere, called a hemisphere, and one cylinder without its top.
\begin{align*}A \text{(hemisphere)} &= \frac{1}{2} A ( \text{sphere})\\ &= \frac{1}{2} (4 \pi r^2)\\ &= 2 \pi (24^2)\\ &= 1152 \pi \ cm^2\end{align*}
Now find the area of the cylinder without its top:
\begin{align*}A \text{(topless cylinder)} &= A (\text{cylinder}) - A (\text{top})\\ &= (2 \pi r^2 + 2 \pi rh) – \pi r^2\\ &= \pi r^2 + 2 \pi rh\\ &= \pi (24^2) + 2 \pi(24)(50)\\ &= 576 \pi + 2400 \pi\\ &= 2976 \pi \ cm^2\end{align*}
So the total surface area is: \begin{align*}1152 \pi \ cm^2 + 2976 \pi \ cm^2 = 4128 \pi \ cm^2\end{align*}
Volume of a Sphere
A sphere can be thought of as a regular polyhedron with an infinite number of congruent polygon faces. A series of polyhedra with an increasing number of faces is shown:
As \begin{align*}n\end{align*}, the number of faces increases to an infinite number, the figure gets closer and closer to becoming a sphere. You can see that the shape in figure a has sharp edges, while the shape in figure f (with more faces) looks rounder.
So a sphere can be thought of as a polyhedron with an infinite number faces. Each of those faces is the base of a pyramid whose vertex is located at \begin{align*}O\end{align*}, the center of the sphere. This is shown below:
Each of the pyramids that make up the sphere would be congruent to the pyramid shown above.
The volume of this pyramid is given by:
\begin{align*}V ( \text{each pyramid}) = \frac{1}{3}Bh\end{align*}
To find the volume of the sphere, you simply need to add up the volumes of an infinite number of infinitely small pyramids:
\begin{align*}V (\text{all pyramids}) &= V_1 + V_2 + V_3 + \ldots + V_n\\ &= \frac{1}{3} B_1 h + \frac{1}{3} B_2 h + \frac{1}{3} B_3 h + \ldots + \frac{1}{3} B_n h\\ &= \frac{1}{3} h ( B_1 + B_2 + B_3+ \ldots + B_n )\end{align*}
The sum of all of the bases of the pyramids is simply the surface area of the sphere.
Since you know that the surface area of the sphere is \begin{align*}4\pi r^2\end{align*}, you can substitute this quantity into the equation above for the sum of the bases:
\begin{align*}V (\text{all pyramids}) &= \frac{1}{3} h ( B_1 + B_2 + B_3+ \ldots +B_n )\\ &= \frac{1}{3} h (4 \pi r^2)\end{align*}
Finally, as \begin{align*}n\end{align*} increases and the surface of the figure becomes more “rounded,” \begin{align*}h\end{align*}, the height of each pyramid becomes equal to \begin{align*}r\end{align*}, the radius of the sphere.
So we can substitute \begin{align*}r\end{align*} for \begin{align*}h\end{align*}. This gives:
\begin{align*}V (\text{all pyramids}) &= \frac{1}{3} r (4 \pi r^2)\\ &= \frac{4}{3} \pi r^3\end{align*}
We can write this as a formula.
Volume of a Sphere
Given a sphere that has radius \begin{align*}r\end{align*}
\begin{align*}V = \frac{4}{3} \pi r^3\end{align*}
Example 3
Find the volume of a sphere with a radius of 6 meters.
Use the volume formula above with \begin{align*}r =\end{align*} _________ :
\begin{align*}V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (6^3) = \frac{864 \pi}{3} = 288 \pi \ m^3\end{align*}
The sphere has an exact volume of \begin{align*}288 \pi \ m^3\end{align*}.
How would you compute the approximate volume? ( __________ )( __________ )
Reading Check:
1. In your own words, describe what a sphere is:
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. What is half of a sphere called?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
3. True/False: The parts of a circle have similar names and definitions to the parts of a sphere.
4. If you answered true for #3 above, provide an example. If you answered false to #3 above, provide a non-example.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
5. In your own words, describe what a tangent is:
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
Graphic Organizer for Lessons 5 – 7: 3-D Surface Area and Volume
3-D Figure | Picture | Surface Area Formula | Volume Formula |
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Cylinder | |||
Cone | |||
Sphere |
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