Lesson 4 Madison’s Round Garden Develop Understanding

Ready

Find the volume and surface area for each 3-D shape.

1.

a.

Volume

b.

Surface Area

2.

a.

Volume

b.

Surface Area

3.

a.

Volume

b.

Surface Area

4.

a.

Volume

b.

Surface Area

Set

5.

Circle $A$ has a radius of $15 feet$ and a circumference of $30 π feet$ or about $94.25 feet$ . Segment $C D$ is the diameter of the circle.

a.

What is the degree measure of arc $C B D$ ?

b.

What is the length of arc $C B D$ ?

c.

What is the ratio of the length of $\overset{⌢}{C B D}$ to the length of the radius?

6.

The circles have a radius of $1$ , $2$ , $3$ , and $4$ units, respectively. The diameter divides each circle into two equal sectors. Find the arc length of one half of each of these circles. Then fill in the table with the indicated values.

Radius	Arc Length	Radian Measure of Central Angle
$1$	$m \overset{⌢}{B C D} =$	$m ∠ B A D =$
$2$	$m \overset{⌢}{F G H} =$	$m ∠ F E H =$
$3$	$m \overset{⌢}{K L M} =$	$m ∠ K J M =$
$4$	$m \overset{⌢}{O P Q} =$	$m ∠ O N Q =$

7.

Why does the arc length increase as the radius increases, but the ratio of arc length to the radius remains the same?

8.

Each circle has a different radius. Fill in the chart with the indicated values.

Radius	Arc Length	$\frac{Arc Length}{Radius}$	Area of Sector
$10$
$20$
$30$

9.

What is the radian measure of an angle that measures $45 °$ ?

10.

Refer to $\overset{⌢}{E F}$ in $⨀ D$ (problem 8).

a.

How many multiples of this arc would be needed to be equal to the length of the entire circumference of circle $D$ ?

b.

Would this be true for the other arcs and circles in problem 8? Explain your thinking.

Go

Use the diagram for problems 11–13.

11.

Explain why $⨀ F$ is similar to $T$ . (Include center of dilation and scale in your explanation.)

12.

If $⨀ T$ is the pre-image and $⨀ F$ is the image, how would the scale factor be different?

13.

Find the indicated ratios.

a.

$\frac{radius ⨀ F}{radius ⨀ T}$

b.

$\frac{circumference ⨀ F}{circumference ⨀ T}$

c.

$\frac{m \overset{⌢}{G H}}{m \overset{⌢}{R X}}$

d.

$\frac{area ⨀ F}{area ⨀ T}$

e.

$\frac{area sector G F H}{area sector R T X}$

14.

The corresponding linear parts of two circles have the ratio of $1$ to the scale factor or $\frac{1}{scale factor}$ .

Fill in the blanks.

The corresponding areas of two circles have the ratio of $1$ to the or $\frac{1}{}$ .