Download Mod 3 - Aim #11 - Manhasset Public Schools

Aim #11: What is the volume of a pyramid?
Do Now:
Cube
B = Base area
2h = height
Cube
height scaled down by a factor of 2
CC Geometry H
Pyramid
___ = Base area
___ = height
2h
B
B
B
Volume of Cube = __________
How many total pyramids with height h would be needed to equal the volume of the
cube?___ Therefore, one pyramid represents _____ of the volume of the cube.
Volume of Pyramid = ___ ______ or _______
Suppose we wanted to calculate the volume of the cone shown below.
-Since the base, A, is an irregular shape, we could compare this cone to a right
rectangular pyramid that has the same base area A and height h (shown below).
-What does Cavalieri‛s Principle say about the volume of the general cone compared
to the volume of the right rectangular pyramid?
What can we conclude about the formula to find the volume of a general cone?
Volume of a general cone =
1
(area of the base x height)
3
Exercises
3
1. A pyramid has volume 24 in and height 6 in. Find the area of its base.
2. A cone fits inside a cylinder so that their bases are the same and their heights are
the same, as shown in the diagram below. Calculate the volume that is inside the
cylinder, but outside of the cone. Give an exact answer.
12
5
3
3. A square pyramid has a volume of 245 in . The height of the pyramid is 15 in. What
is the area of the base of the pyramid? What is the length of one side of the base?
4. Find the volume of the rectangular pyramid shown.
4m
9m
6m
5. The right circular cone shown has a base with radius of 7. The slant height
of the cone‛s lateral surface is
. Find the volume of the cone, to the
nearest tenth.
3
6. Gold has a density of 19.32 g/cm . If a square pyramid has a base edge length of 5 cm,
height of 6 cm, and a mass of 942 g, is the pyramid made of solid gold? Recall that density can
be calculated with the formula: density = mass
volume
7. Sand falls from a conveyor belt and forms a pile on a flat surface in the shape of a
cone. The diameter of the pile is approximately 12 ft. and the height is approximately 5
ft. Estimate the volume of the pile of sand to the nearest cubic foot.
8. The frustum of a pyramid is formed by cutting off the top part by a plane parallel to the
base. The base of the pyramid and the cross-section where the cut is made are called the
bases of the frustum. The distance between the planes containing the bases is called the
height of the frustum. Find the volume of the frustum shown if the bases are squares of
edge lengths 2 in. and 3 in., and the height of the frustum is 4 in.
Name ______________________
CC Geometry H
Date _________________
HW #11
1. What is the volume formula for a right circular cone with radius r and height h?
2. Identify the solid shown, and find its volume.
4
3
6
3. Find the volume of the right square pyramid shown.
4. Find the volume of the circular cone below. Round your answer to the
nearest hundredth.
5. Find the volume of a pyramid whose base is a square with edge length 3 cm. and whose
height is also 3 cm.
6. Suppose you fill a conical paper cup with a height of 6‛‛ with water. If all the
water is then poured into a cylindrical cup with the same radius and same height as
the conical paper cup, to what height will the water reach in the cylindrical cup? 7. Use the diagram to answer the questions that follow.
a. Determine the volume of the cone shown. Give an exact answer.
b. Could a cone that has a height of 33 and radius 12 be the dimensions of a similar
cone? Why or why not?
c. Calculate the volume of the cone described in part (b) in two ways. (Hint: Use the
volume formula and the scaling principle for volume.)
Mixed Review
1. Two jars of peanut butter by the same brand are sold in a grocery store. The
first jar is twice the height of the second jar, but its diameter is one-half as much
as the shorter jar. The taller jar costs $1.89, and the shorter jar costs $2.79.
Which jar is the better buy? Justify your answer.
2. The diagram shows rectangle ABCD with diagonal BD. What is the exact
perimeter of ABCD and to the nearest tenth?
A
B
14
30
D
o
C