FPC MATH 10
NAME:______Key_______
Date:___________________
Lesson Notes 1.6: Surface Area and Volumes of a Sphere.
LESSON FOCUS: Solve problems involving the surface area and volume of a sphere.
A sphere is the set of points in a 3-D space that are the same
distance from a fixed point, which is the centre. A line segment that
joins the centre to any point on the sphere is a radius. A line segment
that joins two points on a sphere and passes through the centre is a
diameter.
When the cylinder has the same diameter as the sphere, and a height equal to its diameter,
we can see the surface area of a sphere is closely related to the curved surface area of a cylinder
that encloses it. Look at the following diagram:
If the curved surface of the cylinder is
made from paper, it can be cut and pasted
on the surface of the sphere to cover it.
The curved surface area, SAC, of a cylinder with base radius r and height h is:
SAC = 2πrh
When a cylinder has base radius r and height 2r:
SAC = 2πrh = 2πr(2r)
SAC = 4πr2
So, this is also the formula for the surface area of a sphere with radius r.
SA = 4πr2
Example 1) The diameter of a baseball is approximately 3 in. Determine the surface area of a
baseball to the nearest square inch.
CHECK YOUR UNDERSTANDING
The diameter of a softball is approximately 4 in. Determine the surface area of a softball to the
nearest square inch. [Answer: approximately 50 in.2 ]
Example 2) The surface area of a lacrosse ball is approximately 20 in2.What is the diameter of the
lacrosse ball to the nearest tenth of an inch?
CHECK YOUR UNDERSTANDING
The surface area of a soccer ball is approximately 250 in2. What is the diameter of a soccer ball to
the nearest tenth of an inch?
We can use the formula for the surface area of a sphere to develop a formula for the
volume of a sphere. Visualize a sphere covered with very small congruent squares, and each square
is joined by line segments to the centre of the sphere to form a square pyramid. The volume of the
sphere is the sum of the volumes of the square pyramids.
Volume of sphere = sum of volumes of pyramids
1
Volume of sphere = sum of all the [ (base area)(height)]
3
The height of a pyramid is the radius of the sphere.
1
(sum of all the base area)(r)
3
The sum of all the base areas is the surface area of the sphere: 4πr2.
Volume of sphere =
Example 3) The sun approximates a sphere with diameter 870 000 mi. What is the approximate
volume of the sun?
CHECK YOUR UNDERSTANDING
The moon approximates a sphere with diameter 2160 mi. What is the approximate volume of the
moon? [Answer: approximately 5.3 x 109 mi.3]
When a sphere is cut in half, two hemispheres are formed.
Why is a globe constructed from two hemispheres?
It is impossible to create a hallow sphere. Molds are used to create two congruent hemispheres,
which are then glued together.
Example 4) A hemisphere has radius 8.0 cm.
a) What is the surface area of the hemisphere to the nearest tenth of a cm2?
b)What is the volume of the hemisphere to the nearest tenth of a cm3?
CHECK YOUR UNDERSTANDING
A hemisphere has radius 5.0 cm.
a) What is the surface area of the hemisphere to the nearest tenth of a cm2?
b) What is the volume of the hemisphere to the nearest tenth of a cm3?
[Answers: a) approximately 235.6 cm2 b) approximately 261.8 cm3]