Volume of Cones and Pyramids

Geometry

Unit 5, Lesson 8

Mrs. King

Reminder: What is a Pyramid?

Definition:

A shape formed by connecting triangles to a polygon.

Examples:

Reminder: What is a Cone?

Definition:

A shape formed from a circle and a vertex point.

Examples:s

Volume Of A Cone.Consider the cylinder and cone shown below:

The diameter (D) of the top of the cone and the cylinder are equal.

D D

The height (H) of the cone and the cylinder are equal.

H H

If you filled the cone with water and emptied it into the cylinder, how many times would you have to fill the cone to completely fill the cylinder to the top ? 3 times.

This shows that the cylinder has three times the volume of a cone with the same height and radius.

www.ltscotland.org.uk/Images/volumesofsolids_tcm4-123355.ppt

Formulas

Volume of a Cone:

V= 1/3 r2h

Volume of a Cylinder:

V = r2 h

Example #1

Calculate the volume of:

V= 1/3 r2h V= 1/3 ()(7)2(9)

V = 147m3

Example #2

Calculate the volume of:

V= 1/3 r2h V= 1/3 ()(5)2(12)

V = 100cm3

An ice cream cone is 7 cm tall and 4 cm in diameter. About how much ice cream can fit entirely inside the cone? Find the volume to the nearest whole number.

r = = 2d2

V = πr 2h13

V = π(22)(7)13

V ≈ 29.321531

About 29 cm3 of ice cream can fit entirely inside the cone.

Example #3:

Compare

Compare a Prism to a Pyramid.

Make a conjecture to what the formula might be for Volume of a Pyramid.

Formulas

Volume of a Prism: Volume of a Pyramid:

V = 1/3 BhBhV area baseB

Example #4

Calculate the volume of:

10”

15”V = 1/3 BhV = 1/3 (102)(15)

V = 500in3

Find the volume of a square pyramid with base

edges 15 cm and height 22 cm.

Because the base is a square, B = 15 • 15 = 225.

V = Bh13

= (225)(22)13

= 1650

Example #5

Find the volume of a square pyramid with base edges 16 m

and slant height 17 m.

The altitude of a right square pyramid intersects the base at the center of the square.

Example #6

Because each side of the square base is 16 m, the leg of the right triangle along the base is 8 m, as shown below.

Step 1: Find the height of the pyramid.

172 = 82 + h2 Use the Pythagorean Theorem.289 = 64 + h2

225 = h2

h = 15

Example #6, continued

Step 2: Find the volume of the pyramid.

= 1280

V = Bh13

= (16 x 16)1513