GeometryNovember 13, 2013

Today’s Question:How do we find the volume of cylinders, prisms, cones, and pyramids?Standard: MMC9-12.G.GMD1 and 3

Which of these are volumes?

• A gallon• An acre• A liter• A square meter• A cubic foot

• A barrel full• A truck load• A metric ton• A teaspoon• A lightyear

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

What is a prism?

Definition of a Prism

• A prism is a “polyhedron” with two congruent faces, called bases, that lie in parallel planes. The other faces are called lateral faces.

• A polyhedron is any solid bounded by polygons, referred to as faces.

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) 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 area

2 ofheight of base5. V

2435. V

Volume of a Cylinder

7 cm

5 cm

V = 769.7 cm3

hrV 2572 V

Round to the nearest tenth.V Bh

You try! Find the volume.

• A cylinder has a radius of 5 cm and a height of 8 cm. Sketch the cylinder and label its dimensions, then find the volume. State your answer both in terms of π and as a decimal rounded to the nearest hundredth.

Try Another!

• A cube has a volume of 343 cubic meters. What is the length of a side of the cube?

x

xx

Homeworkp. 746-747nos. 13-18,

25-30

Ticket Out the door• You need to buy a container that holds as

much liquid as possible for water storage. You go to Home Depot and they offer two different containers, one cylindrical and the other a rectangular prism. They are both 3.5’ tall. The rectangular prism’s base is 30” x 30” and the cylinder’s diameter is 30”. They are the same price. Which do you buy? Justify your answer.

GeometryNovember 14, 2013

Today’s Question:How do we find the volume of cylinders, prisms, cones, and pyramids?Standard: MMC9-12.G.GMD1 and 3MMC9-12.G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

MMC9-12.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Opener

What is the volume of the cylinder with a diameter of 40 mm and a height of 112 mm? Use the correct units!

Homework Answers

13. 512 in3

14. 240 cm3

15. 318.26 in3

16. 6785.84 m3

17. 288.40 ft3

18. 381.00 cm3

25. 924 m3

26. 628.32 ft3

27. 116.91 cm3

28. u = 10 ft

29. v = 13.92 yds

30. w = 4 cm

#15- Challenge Problem

Measuring with Millimeters

There are 25.4 mm per inch. Measure the diameter and the height of the cone to the nearest millimeter.

How many cones will it take to fill the

cylinder?If:• They have the same size base.

• They also have the same height.

Volume Formula for Cones & Pyramids

1

3V Bh

8 ft

12 ft

10 ft

Find the volume. 1V Bh

3

V (12 10 8)

3

V 3 320 ft

Find the volume!

Base dimension are 6’ x 8’ and the height is 12’

Volume = ?

Find the volume!

The cone’s radius is 12 mm and its height is 20 mm.

Volume = ?

Find the Missing Dimension!

The pyramid’s base is 20’x 20’. The volume of the pyramid is 8000 cubic feet. What is its height?

2 cm

3 cm

What is the volume of this cone?Round to the nearest tenth.

3.6 cm

V r

3

2

( ) h

V 2

3

2

( ) 3

V 3 12.6 cm

1V Bh

3

Ticket Out the Door!

• A cone and a pyramid have the same volume and the same height. If the pyramid’s base is 12’ x 10’, what is the radius of the base of the cone?

HomeworkHandout from Geometry

book p. 755-756,nos. 8-14 and 17-22

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 r

8 in

Surface Area of a Sphere(round to the nearest hundredths) SA4 r2

SA4 82

SA804.25 in2

10 cm

Surface Area of a Sphere(round to the nearest hundredths) SA4 r2

SA4 52

SA314 16. 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

r

212.5

4

SA

25 2 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.

SA4 r2

24 1.5 SA

29 mSA 228.27m

Volume of a Sphere

34

3V r

2 cm

Volume of a Sphere

(round to the nearest hundredths)

34

3V r

333.51V cm

342

3

V

10 cm

34

3V r

Volume of a Sphere

(round to the nearest hundredths)

3523.60V cm

345

3

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

v

314.14 mV c

WS9 – 16