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.

Volume Formula for Prisms & Cylinders

V = Bh

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

Volume of a Cylinder

7 cm

5 cm

V = 769.7 cm3

hrV 2π=572 ⋅=πV

Round to the nearest tenth. V Bh=

WS 1 – 5

How many cones will it take to fill the cylinder?

 They have the same size base.  They also have the same height.

Volume Formula for Cones & Pyramids

13

V Bh=

8 ft

12 ft

10 ft

Find the volume. 1V Bh3

=

V (12 10 8)3

=• •

V 3= 320 ft

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

=

WS 6 – 8

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

r

Radius of a Sphere

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.

Surface Area of a Sphere

24SA rπ=

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≈

Volume of a Sphere

343π=V r

2 cm

Volume of a Sphere

(round to the nearest hundredths)

343

V rπ=

333.51V cm=

34 23π⎛ ⎞= ⎜ ⎟

⎝ ⎠V

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=

WS 9 – 16