# 11.7 – Spheres

## Key Terms

• Pythagorean Theorem – The theorem that relates the side lengths of a right triangle.
• The theorem states that the square of the hypotenuse equals the sum of the squares of the legs: $a^{2}+b^{2}=c^{2}$
• Axis – The diameter of a sphere.
• Diameter – A line segment that contains the center of the circle and has endpoints on the circle.
• This term also refers to the length of this line segment; the diameter of a circle is twice the radius.
• Sphere – A three-dimensional figure consisting of all points in space that are the same distance from a given point.

## Review

Remember
• Surface Area
• BA = base area
• LA = lateral area
• p = perimeter of base
• h = height of solid
• s = slant height
• Volume
• B = base area
• h = height
• Surface area measures the total area of all the faces of a solid.
• Volume measures the total amount of space inside a solid.

## Notes

What is a Sphere?
• How are circles and spheres alike?
• They both consist of all the points that lie the same distance from a center point.
• How are circles and spheres different?
• Circles are two-dimensional, but spheres are three-dimensional.
• Like a circle, a sphere has a radius and a diameter (called an axis). A sphere has an infinite number of radii and diameters.
• The radius of a sphere is the distance from the center of the sphere to any point on the sphere.
•  Diagram of a Sphere

• Surface Area of a Sphere

• Volume of a Sphere

• Compare Surface Area & Volume of a Sphere
• Both formulas use $\pi$
• This makes sense because all cross-sections of spheres are circles.
• Both formulas are multiplied by a constant.
• You’ve seen this happen a lot with other solids. In fact, other volume formulas had a fraction constant with 3 on the bottom.
• Both formulas include radius r.
• This makes sense because a sphere is defined by its radius.
• The surface area formula uses $r^{2}$.
• You can remember the exponent of 2 for this formula because surface area answers are always given as $units^{2}$.
• The volume formula uses $r^{3}$.
• You can remember the exponent of 3 for this formula because volume answers are always given as $units^{3}$.

Surface Areas of Cylinders, Cones, and Spheres
• The SA of a cone is less than the SA of a sphere, which is less than the SA of a cylinder.

Volumes of Cylinders, Cones, and Spheres
•  The volume of a cone is less than the volume of a sphere, which is less than the volume of a cylinder.

Surface Area of a Sphere
• How to Find the Surface Area of Any Sphere
• Step 1: Write the surface area formula.
• Step 2: Find the radius of the sphere.
• Step 3: Square the radius.
• Step 4: Multiply the result by $4\pi$.
•  Example

• SA (Sphere) = LA (Cylinder)

Volume of a Sphere
• How to Find the Volume of Any Sphere
• Step 1: Write the volume formula.
• Step 2: Find the radius of the sphere.
• Step 3: Cube the radius.
• Step 4: Multiply the result by $\frac{4}{3}\pi$.
•  Example

Pythagorean and Spheres
• The cross-sectional areas of the cylinder and the sphere are found using the same general formula.
• The Pythagorean theorem can be used to find the radius, a, of any cross-section of the sphere using b and r.
• $a^{2}+b^{2}=r^{2}$
• To isolate “a” (the radius), subtract $b^{2}$ from both sides.
• $a^{2}=r^{2}-b^{2}$

Volume of the Dugout Cylinder
• Try to imagine removing (digging out) the two cones from the inside of the cylinder.
• What’s left over is the dugout cylinder!
• $\pi r^{2}$ is the cross-sectional area of a cylinder, and $\pi b^{2}$ is the cross-sectional area of an hourglass, or double-cone.
• If you then subtract $\pi b^{2}$ (the double-cone) from $\pi r^{2}$ (the cylinder), you can make a new shape, called a dugout cylinder, whose cross-sectional area equals that of the sphere: $\pi r^{2}-\pi b^{2}$
• To find the volume of a dugout cylinder:
• Step 1: Find the volume of each, the cylinder and double-cone.
• For this cylinder: $BA=\pi r^{2}$ and $h=2r$, so V=Bh is $\pi r^{2}\bullet 2r=2\pi r^{3}$
• For this double cone, multiply the volume of a cone by 2: $V=2(\frac{1}{3}\pi r^{2}\bullet r)=\frac{2}{3}\pi r^{3}$
• Note, the height of EACH cone is r, not 2r.

• Step 2: Subtract the volume of the double-cone from the cylinder.

Notice that the cylinder has the formula $2\pi r^{3}$.
We can turn the 2 into a fraction, rewritten as $\frac{6}{3}$; and, $\frac{6}{3}-\frac{2}{3}=\frac{4}{3}$,
which is why the formula for a dugout cylinder is $\frac{4}{3}\pi r^{3}$

Volume of the Dugout Cylinder and the Sphere
• The Pythagorean theorem proves a sphere and dugout cylinder with equal heights have equal cross-sectional areas.
• Cavalieri’s principle proves the sphere and dugout cylinder have equal volumes.

• Volume of a Dugout Cylinder and a Sphere
• Remember: a cross-section IS a circle for both solids below.
• The Pythagorean theorem is $a^{2}+b^{2}=c^{2}$; and so, $r^{2}+r^{2}=c^{2}$ because a = r and b = r for both!
• The area of EACH cross section is $\pi r^{2}$, which can be called $\pi a^{2}$ or $\pi b^{2}$ since they the same thing.
• We’ll use $\pi a^{2}$ to represent the sphere and $\pi b^{2}$ to represent the dugout part of the cylinder just for easy labeling of each solid.

• Sphere + Double Cone = Cylinder

$\frac{4}{3}+\frac{2}{3}=\frac{6}{3}$, and $\frac{6}{3}$ simplifies to 2.

## Examples

 Ex 1. What is the surface area of the sphere below? Answer: $100\pi units^{2}$ Ex 2. The axis of the sphere below is 16 units in length. What is the surface area of the sphere? Answer: $256\pi units^{2}$ Ex 3. What is the volume of the sphere below? Answer: $\frac{500}{3}\pi units^{3}$ Ex 4. Given a sphere with radius r, the formula $4\pi r^{2}$ gives the surface area. Ex 5. Which expression gives the volume of a sphere with radius 5? Answer: $\frac{4}{3}\pi 5^{3}$ Ex 6. It is true that the area of a circle of radius 12 units is equal to the surface area of a sphere of radius 6 units.