Unlocking Area and Volume with Scale Factors: A Guide to Proportional Measures

Area of $K L M N$	Area of $P Q R S$
$A_{1} = K L \cdot L M$	$A_{2} = P Q \cdot Q R$

Area of

K L M N

Area of

P Q R S

A_{1} = K L \cdot L M

A_{2} = P Q \cdot Q R

Volume of Solid $A$	Volume of Solid $B$
$V_{1} = a_{1} \cdot a_{2} \cdot a_{3}$	$V_{2} = b_{1} \cdot b_{2} \cdot b_{3}$

Volume of Solid

A

Volume of Solid

B

V_{1} = a_{1} \cdot a_{2} \cdot a_{3}

V_{2} = b_{1} \cdot b_{2} \cdot b_{3}

Length Scale Factor	Area Scale Factor	Volume Scale Factor
$\frac{a}{b}$	$(\frac{a}{b})^{2}$	$(\frac{a}{b})^{3}$

Length Scale Factor

Area Scale Factor

Volume Scale Factor

\frac{a}{b}

(\frac{a}{b})^{2}

(\frac{a}{b})^{3}

Consider the following cylinder with a volume is $324 π$ cubic inches.

If the cylinder is enlarged by a length scale factor of

3,

what will be the volume of the enlarged cylinder? Answer in exact form.

We are given that the cylinder is enlarged by a scale factor of 3. Therefore, we can write the length scale factor as the ratio of 1 to 3. Length Scale Factor= 1/3 The ratio of the volumes of similar solids is equal to the cube of the ratio of their corresponding linear measures. With this information, we can write the volume scale factor of the cylinders.

Length Scale Factor	Volume Scale Factor
1/3	( 1/3)^3=1/27

We can now equate the volume scale factor to the ratio of the volume of the small cylinder to the volume of the enlarged cylinder. Let V_1 be the volume of the small cylinder and V_2 be the volume of the enlarged cylinder. 127=V_1/V_2 Given that the volume of the small cylinder is 324π, we can substitute this value into the equation and solve it for V_2. By doing so, we will find the volume of the enlarged cylinder.

Therefore, the volume of the enlarged cylinder is 8748π cubic inches.

A solid with a volume of

26

cubic inches was enlarged to create a similar solid with a volume of

702

cubic inches. How many times longer is one side of the larger solid than the corresponding side of the smaller solid?

We need to find the length scale factor to know how many times longer one side is in the bigger solid. Recall that the ratio of volumes of similar solids is equal to the cube of the ratio of their corresponding linear measures.

Length Scale Factor	Volume Scale Factor
a/b	( a/b)^3

Since we have the volume of the two solids, we can write a proportion by using the ratio of the volumes. (a/b)^3=702/26 Now, we can take the cube roots of both sides of the equation to find the length scale factor.

The length scale factor is 3. That means the larger solid side is three times greater than the corresponding side in the smaller solid.

The following polygons are similar. Find the area of polygon $B .$ If necessary, round the answer to one decimal place.

It is a given that the rectangles are similar and two corresponding sides measure 4 centimeters and 6 centimeters. The length scale factor is the ratio of these corresponding sides. Length Scale Factor:4/6= 2/3 Recall that the ratio of the areas of similar figures is equal to the square of the ratio of their corresponding side lengths. We can use this information to write the area scale factor for the given figures.

Length Scale Factor	Area Scale Factor
2/3	( 2/3)^2=4/9

We can now equate the area scale factor to the ratio of the small rectangle to the big rectangle. Let A_A be the area of the small rectangle and A_B be the area of the big rectangle. 4/9=A_A/A_B Given that the area of the small rectangle is 27 square centimeters, we can substitute this value into the equation and solve it for A_B. By doing so, we will find the area of the big rectangle.

Therefore, the rectangle marked with B has an area of 60.8 square centimeters.

In similar fashion, we can now find the area of the small triangle. In this exercise, the two corresponding sides measure 5.5 centimeters and 8 centimeters. Let's calculate the length scale factor. Length Scale Factor:5.5/8= 11/16 Again, the square of the length scale factor is equal to the ratio of the areas of the given figures.

Length Scale Factor	Area Scale Factor
11/16	( 11/16)^2=11^2/16^2

Let A_B be the area of the small triangle and A_A be the area of the big triangle. Then, let's equate the area scale factor to the ratio of the areas of the triangles. 11^2/16^2=A_B/A_A Since the area of the big triangle is 16 square centimeters, let's substitute this value for A_A into the equation and solve it for A_B.

Therefore, the triangle marked with B has an area of about 7.6 square centimeters.

Again, let's calculate the ratio between the corresponding sides of the given figures. The sides measure 3 centimeters and 5 centimeters each. Length Scale Factor: 3/5 Now, let's square the ratio to find the area scale factor.

Length Scale Factor	Area Scale Factor
3/5	( 3/5)^2=9/25

Again, let A_B be the area of the small figure and A_A be the area of the big figure. Let's equate the area scale factor to the ratio of the areas of the figures 9/25=A_B/A_A Since the area of the big figure is 19.88 square centimeters, we can substitute this value for A_A into the expression and solve it for A_B.

Therefore, the figure marked with B has an area of about 7.2 square centimeters.

The following solids are similar. Find the volume of the solid labeled $A .$ If necessary, round the answer to two decimals places.

We are given that the solids are similar and two corresponding sides measure 2 centimeters and 4 centimeters each. Therefore, the length scale factor is the ratio of these corresponding sides. Length Scale Factor:2/4= 1/2 Recall that the ratio of the volumes of similar solids is equal to the cube of the ratio of their corresponding linear lengths. With this information, we can write the volume scale factor for the given solids.

Length Scale Factor	Volume Scale Factor
1/2	( 1/2)^3=1/8

We can know equate the volume scale factor to the ratio of volume of the small solid to the volume of the big solid. Let V_1 be the volume of the small solid and V_2 be the volume of the big solid. 1/8=V_1/V_2 Given that the volume of the big solid is 6 cubic centimeters, we can substitute this value into the equation and solve it for V_1.

Therefore, the solid labeled as A has a volume of 0.75 cubic centimeters.

In a similar way, we can find the volume of the big pyramid. In this case, the corresponding sides measure 8 centimeters and 10 centimeters. Let's calculate the ratio to find the length scale factor. Length Scale Factor:8/10= 4/5 Now, let's cube the ratio to find the volume scale factor.

Length Scale Factor	Volume Scale Factor
4/5	4/5=64/125

Again, let V_1 the volume of the small pyramid and V_2 the volume of the big pyramid. Let's equate the volume scale factor to the ratio of the volumes of the solids. 64/125=V_1/V_2 Since the volume of the small pyramid is 9 cubic centimeters, we can substitute this value for V_1 into the expression and solve it for V_2.

Therefore, the solid marked with A has a volume of about 17.58 cubic centimeters.

Mark has drawn two similar rectangles on a piece of paper. One has a length of $12$ and the corresponding length on the other rectangle is $20$ inches. Mark attempted to calculate the smaller rectangle's area by using the area of the big rectangle. However, he was unable to solve it. Help Mark calculate the area of the smaller rectangle.

a calculation of an area trying to use the similarity between two rectangles

We want to find the area of the rectangle labeled x. We are given two corresponding sides of measures 20 and 12 inches each. By calculating the ratio of these corresponding sides, we will get the length scale factor. Length Scale Factor:20/12 Now, let's recall that the ratio of the areas of two similar figures is equal to the square of the ratio of their corresponding side lengths. Looking at the procedure, we can see that the ratio was written correctly. Where is the issue? It seems that the ratio was not squared. Let's correct this mistake by squaring the left-hand side of the equation.

Davontay has taken a photograph of his new sailboat, developed it, framed it and put it on the wall.

a picture of a sailboat that is 10 cm high and 15 cm wide

However, the picture is so small he thinks it looks like a sailboat for ants. The picture has to be at least three times bigger than this. He figures that he will print a new picture where the width and length are three times longer compared to the original picture. How many times bigger is this photo compared to the smaller photo?

We know that the Davontay want's to make the width and height three times longer than the original photo. This means the linear scale factor is 3. To determine how many times bigger the new photo is, we can square the linear scale factor. This gives us the area scale factor.

The area scale factor is 9 which means the photo is 9 times bigger than the original.

Area and Volume Scale Factors

Catch-Up and Review

Using Scale Factor of Similar Pyramids

Area Scale Factor

Proof

Using Area Scale Factor to Determine an Unknown Area

Hint

Solution

Using Area Scale Factor to Determine an Unknown Side

Hint

Solution

Practice Finding Linear Scale Factor Given Areas

Volume Scale Factor

Proof

Using Volume Scale Factor to Determine an Unknown Volume

Hint

Solution

Using Volume Scale Factor to Determine an Unknown Side

Hint

Solution

Practice Finding Linear Scale Factor Given Volumes

Finding Area Given Volume of Similar Solids

Hint

Solution

Finding Volume of Similar Composite Solids

Hint

Solution

Relationship Between Length, Area, and Volume Scale Factor

Hint

Answer

Area and Volume Scale Factors

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