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A N A L Y T I C G E O M E T R Y F O R C O L L E G E G R A D U A T E S
Day 31 – Surface Area & Volume
VOLUME = the number of cubic units contained in its interior
VOLUME has cubic units
Cm3, ft3, units3
The volume of a cube is side3
4 ft
Volume = length•width•height
Volume = s3
Volume = 43
Volume = 64 ft3
In a CUBE they are all the same
When a prism is NOT a cube…
B (area of the BASE) The formula for B will depend on the shape of the base.
B = area of the BASE h = HEIGHT
Volume of a Prism
V = 12 in3
4 in
3 in
2 in
BhV =
hBaseV •= theof area2 ofheight of base5. •Δ⋅Δ⋅=V
2435. •⋅⋅=V
How many cones will it take to fill the cylinder?
They have the same size base. They also have the same height.
2 cm
3 cm
What is the volume of this cone? Round to the nearest tenth.
V r3
2
=( )π h
V 23
2
=( )π 3
V 3= 12.6 cm
1V Bh3
=
2 Types of Answers Rounded
Type the Pi button on your calculator
Toggle your answer
Do NOT write Pi in your answer
Exact Pi will be in
your answer
If you cut a sphere right down the middle you would create two congruent halves called HEMISPHERES.
You can think of the globe. The equator cuts the earth into the northern and southern hemisphere.
Look at the cross section formed when you cut a sphere in half.
What shape is it?
A circle!!! This is called the GREAT CIRCLE of the sphere.
8 in
Surface Area of a Sphere (round to the nearest hundredths) SA = 4π r2
SA = 4π 82
SA ≈ 804.25 in2
10 cm
Surface Area of a Sphere (round to the nearest hundredths) SA = 4π r2
SA = 4π 52
SA ≈ 31416. cm2
25 in
The circumference of a great circle of a sphere is 25 inches. Find the surface area of the sphere. (Round to the nearest tenths.)
25 12.52
= =π π
r
212.5
4 ⎛ ⎞= ⎜ ⎟
⎝ ⎠π
πSA
25 2= π r
2198.9SA in=
A sphere is inscribed in a cube of volume 27 cubic meters. What is the surface area of the sphere? Give an exact answer and an answer rounded to the nearest hundredth.
SA = 4π r2
( )24 1.5= πSA
29 mSA π= 228.27m≈
10 cm
343
V rπ=
Volume of a Sphere
(round to the nearest hundredths)
3523.60V cm=
34 53π⎛ ⎞= ⎜ ⎟
⎝ ⎠V
Volume of a Sphere
A sphere is inside a cube. The cube has a volume of 27 cm3.
Find volume of the sphere. Round to the nearest hundredths.
( )34 1.5
3⎛ ⎞
= ⎜ ⎟⎝ ⎠
πv
314.14 mV c=