Cavalieri's Principle and Cross Sections

Cross Section =

• 2D shape formed by cutting a 3D shape.

Perpendicular to the base

straight down.

Parallel to base

staright across.

Multiple Choice

Describe the cross section.

• triangle
• rectangle
• oval
• trapezoid

Multiple Choice

What shape does this cross section make?

• Rectangle
• Circle
• Triangle
• Square

Multiple Choice

What shape will the horizontal cross-section of a sphere be?

• Dome
• Circle
• Ellipse
• Rectangle

Multiple Choice

An oblique shape will always have the same volume as a right shape if they have the same base.

• True

• False - an oblique shape and a right shape can never have the same volume

• False - an oblique shape and a right shape can't have the same base

• False - an oblique shape will always have the same volume as a right shape if they have the same base AND height

Multiple Choice

Based on Cavalieri's Principle, will the two prisms have the same volume?

• No, they will not be same. Although the heights are the same, the cross-sections are different shapes. 

• Yes, the heights of both prisms are the same and they have the same cross-sectional area. Therefore, they will have the same volume.

Multiple Choice

Jenny says that the two prisms DO NOT have the same volume because the cross sections are not the same. Renee disagrees; she says that it isn't the shape that has to be the same but the area. Renee thinks they have the same volume.

Who is correct?

• Jenny

• Renee

• Neither Jenny or Renee is correct.

Multiple Choice

• Only A & B

• Only A & C

• Only B & C

• A, B and C

Multiple Choice

• The base areas are not the same.

• It isn’t true that corresponding cross-sections have the same area.

• The heights are not the same.

Multiple Choice

What solid 3D object is produced by rotating the triangle around line m?

• Triangular Prism

• Cylinder

• Cone

• Triangular Pyramid

Multiple Choice

What 3-D solid is formed when this rectangle is rotated about the vertical line?

• Right, rectangular prism

• Right, circular cylinder

• Right, oval cylinder

• Right, square prism

Multiple Choice

a) Cavalieri’s Principle states that any two objects with the same cross sectional areas and heights must have the same volume.

• True

• False - the cross sectional areas are not relevant

• False - only the slant height is relevant

• False - even if they have the same cross sectional areas and heights, they cannot have the same volume.

Multiple Choice

Describe the cross section.

• circle
• oval
• semi-circle
• trapezoid

Volume

Objective: Find the volume of various geometric shapes

Multiple Choice

What formula could be used to find the volume of this 3D shape?

•  V=1/3Bh
• V=3/4πr²
• V=Bh
• V=2πrh

Multiple Choice

What is the volume of a cylinder with a radius of 3 and a height of 8?

• 226.19
• 150.8
• 28.27
• 1809.56

Multiple Choice

• 55 yd³

• 121 yd³

• 22 yd³

• 330 yd³

Multiple Choice

Find the volume of the figure. Round to the nearest tenth.

• 80 cm³

• 40 cm²

• 80 cm²

• 40 cm³

Multiple Choice

Find the volume of the figure.

• 126 cm³
• 907.5 cm³
• 605 cm³
• 55 cm³

Multiple Choice

What is the volume of a sphere with the radius of 4 inches ? Use 3.14 for pi.

• 561.5 in³
• 267.9 in³
• 201 in³
• 803.8 in³

Conditional Probability

Multiple Choice

How many children prefer vanilla ice cream?

• 78

• 131

• 26

• 105

Multiple Choice

How many total people were surveyed?

• 224

• 448

• 131

• 146

Multiple Choice

P(Pizza|Senior):

• 10/15
• 4/5
• 2/3
• 10/5

Multiple Choice

Consider the following table with information about all of the students taking Statistics at Happy High School.

P( Male | Full-time)

• 28/71

• 3/7

• 12/71

• 3/10

Multiple Choice

A science teacher gave her class two tests. 25% of the class passed both tests and 42% of the class passed the first test. What percent of those who passed the first test also passed the second?

• 59.5%

• 10.5%

• 1.6%

• Not enough information

​Independent Events

If the first event DOES NOT influence the second, we can use the following formula.

​

Probability of Two Independent Events =

Probability of 1st Event × Probability of 2nd Event

Multiple Choice

You roll a die and then flip a coin. What is the probability, as a percent, of getting an even number on the die and then a head on the coin?

• 25%

• 30%

• 10%

• 50%

Multiple Choice

A spinner is pictured.

If you spun the spinner two times, what is the probability of landing on a white space and then a light gray space?

• 5%

• 50%

• 10%

• 0.05%

• 0.01%

​Dependent Events

If the first event DOES influence the second, we can use the following formula.

​

Probability of Two Dependent Events =

Probability of 1st Event × Probability of 2nd Event after the first event

​Example

There are eight shirts in your closet, four blue and four green. You randomly select one to wear on Monday and then a different one on Tuesday. What is the probability you wear blue shirts both days?

​

Probability of Two Dependent Events =

Probability of 1st Event × Probability of 2nd Event after the first event

Multiple Choice

Shareen has to select two students from a class of 23 girls and 25 boys. What is the probability that both students chosen are boys?

• 27%

• 26%

• 25%

• 48%

Fill in the Blanks

A basket contains five apples and seven peaches. You randomly select one piece of fruit and eat it. Then you randomly select another piece of fruit. What is the probability the first piece of fruit was an apple and the second piece was a peach?

Multiple Choice

Ricardo has the spinner pictured here and a bag of marbles filled with 2 red marbles, 3 green marbles, and 3 blue marbles. What is the probability that Ricardo spins red on the spinner and picks a red marble out of the bag?

• 1 ⁄ 16

• 1 ⁄ 2

• 1 ⁄ 4

• 1 ⁄ 8

Multiple Choice

Suppose a number cube is rolled twice. What is the probability that an odd number will show both times?

• 1/3

• 1/2

• 1/6

• 1/4

Multiple Choice

Bag A contains 9 red marbles and 3 green marbles. Bag B contains 9 black marbles and 6 orange marbles. Find the probability of selecting one green marble from bag A and one black marble from bag B.

• 3/20

• 1/4

• 3/31

• 2/27

Multiple Choice

You are playing a game of scrabble. The tiles shown are dropped in a bag. You need to choose two tiles. What is the probability of choosing an "E", keeping it, and then choosing a "H"?

• 4/15

• 1/70

• 1/75

• 1/225