Lesson 8 Cavalieri to the Rescue Solidify Understanding

Ready

Define each figure by answering each question.

1.

parallelogram

a.

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)

a.

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

a.

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

a.

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)

a.

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

a.

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

a.

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)

a.

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.

a.

Parallelogram ABCD with height 6ft, one side 10 ft and one side 15 ft.

perimeter:

area:

b.

Rectangle EFGH with width 10 ft, length 15 ft.

perimeter:

area:

c.

Rectangle RSTV with width 8 ft and length 15 ft.

perimeter:

area:

10.

Compare and contrast your answers in problem 9. What do you notice about the areas and perimeters? Write a statement about your observations.

11.

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

Trapezoid ADGH with diagonals AG and DH and line segments AF and DF.

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 cube with edges of length a b

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.)

g.

Show that your answers in and are equivalent.

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.

A vertical stack of cards and a deck of cards bumped sideways. deck of cardsdeck of cards

Go

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

15.

A small cone with radius 1 ft, height 4 ft and slant height 7 ft. Larger cone with radius 4 ft, height 12 ft, and slant height 28 ft.
A.

Similar

B.

Congruent

C.

Neither

16.

Rectangular prism with sides 4 in, 5 in, and 8 in; another rectangular prism with sides 4 in, 5 in, and 8 inc.
A.

Similar

B.

Congruent

C.

Neither

17.

Triangular prism with sides 2 cm, 5 cm, and 7 cm. another triangular prism with 2 cm, 5 cm, and 7 cm.
A.

Similar

B.

Congruent

C.

Neither

18.

Rectangular prism with sides 6 m, 6 m, 10 m; another rectangular prism 9 m, 15 m, and 9 m.
A.

Similar

B.

Congruent

C.

Neither

19.

Triangular Prism with base sides 10 in and 6 in and volume 120 inches cubed. Triangular Prism with base sides 5 in and 3 in and volume 60 inches cubed.
A.

Similar

B.

Congruent

C.

Neither

20.

Semi-Sphere D with arc ABC = 6pi ft and Semi-Sphere S with arc PQR =6pi yards
A.

Similar

B.

Congruent

C.

Neither

21.

Sphere A with volume 12 inch cubed and Sphere S with Volume 125 cm cubed.
A.

Similar

B.

Congruent

C.

Neither