# Lesson 8Cavalieri to the Rescue Solidify Understanding

Define each figure by answering each question.

### 1.

parallelogram

How many sides?

#### b.

Which sides are ?

#### c.

Which sides are ?

#### d.

How many lines of symmetry?

#### e.

What is the measure of the smallest angle of rotational symmetry?

### 2.

octagon (regular)

How many sides?

#### b.

Which sides are ?

#### c.

Which sides are ?

#### d.

How many lines of symmetry?

#### e.

What is the measure of the smallest angle of rotational symmetry?

### 3.

trapezoid

How many sides?

#### b.

Which sides are ?

#### c.

Which sides are ?

#### d.

How many lines of symmetry?

#### e.

What is the measure of the smallest angle of rotational symmetry?

### 4.

rhombus

How many sides?

#### b.

Which sides are ?

#### c.

Which sides are ?

#### d.

How many lines of symmetry?

#### e.

What is the measure of the smallest angle of rotational symmetry?

### 5.

pentagon (regular)

How many sides?

#### b.

Which sides are ?

#### c.

Which sides are ?

#### d.

How many lines of symmetry?

#### e.

What is the measure of the smallest angle of rotational symmetry?

### 6.

rectangle

How many sides?

#### b.

Which sides are ?

#### c.

Which sides are ?

#### d.

How many lines of symmetry?

#### e.

What is the measure of the smallest angle of rotational symmetry?

### 7.

square

How many sides?

#### b.

Which sides are ?

#### c.

Which sides are ?

#### d.

How many lines of symmetry?

#### e.

What is the measure of the smallest angle of rotational symmetry?

### 8.

hexagon (regular)

How many sides?

#### b.

Which sides are ?

#### c.

Which sides are ?

#### d.

How many lines of symmetry?

#### e.

What is the measure of the smallest angle of rotational symmetry?

## Set

### 9.

Calculate the perimeters and the areas of each quadrilateral.

perimeter:

area:

perimeter:

area:

perimeter:

area:

### 11.

The figure contains several triangles. Consider , , , , and .

Given that , which triangles have the same area? Justify your answer.

### 12.

The figure shows a cube with edges of length . The cube has been sliced into pieces by three planes parallel to its faces.

#### a.

Write an expression for the volume of the entire cube in terms of and .

#### b.

Into how many pieces is the cube cut?

#### c.

How many of these pieces are also cubes? Write an expression in terms of and for the volume of each perfect cube you find.

#### d.

How many pieces have a volume of ?

#### e.

How many pieces have a volume of ?

#### f.

Write the volume of the figure as the sum of the volumes of its pieces. (Count them in the diagram.)

### 13.

If the cubes and rectangular prisms in problem 12 were rearranged, would the new figure have the same volume? Explain your reasoning.

### 14.

Use Cavalieri’s principle to explain why a perfect stack of playing cards has the same volume as one that has been bumped sideways.

## Go

Determine if each pair of solids is similar, congruent, or neither. Justify your answers.

A.

Similar

B.

Congruent

C.

Neither

A.

Similar

B.

Congruent

C.

Neither

A.

Similar

B.

Congruent

C.

Neither

A.

Similar

B.

Congruent

C.

Neither

A.

Similar

B.

Congruent

C.

Neither

A.

Similar

B.

Congruent

C.

Neither

A.

Similar

B.

Congruent

C.

Neither