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