Similar Shapes: Length, Area and Volume

Summary

This educational script discusses the concept of similar shapes, using cubes and prisms as examples. It explains that similar shapes have proportional dimensions, with the scale factor of lengths multiplied by the same factor. The script elaborates on how this scale factor affects areas and volumes differently: area scales with the square of the length factor, while volume scales with its cube. Practical examples are given to demonstrate calculating the volume of prisms based on their scale factors, emphasizing the mathematical patterns behind geometric similarities.

Takeaways

• 🔍 Two shapes are similar if they have undergone uniform enlargement by the same scale factor.
• 📐 Cubes are always similar shapes because their dimensions are proportionally scaled.
• 📏 The length scale factor is the ratio of corresponding linear dimensions of similar shapes.
• 📈 The area scale factor is the square of the length scale factor.
• 📊 The volume scale factor is the cube of the length scale factor.
• 🧩 For cubes, scaling one dimension differently would result in a shape that is no longer a cube.
• 🔢 The volume of a cube is calculated by multiplying its length, width, and height.
• 📋 To find the volume of a larger similar prism, multiply the volume of the smaller prism by the cube of the length scale factor.
• 🔄 The ratio of surface areas between similar prisms can be used to find the ratio of their volumes by considering the scale factors.
• 📘 When given the volume ratio of similar shapes, you can find the scale factor by dividing the larger volume by the smaller one and then taking the cube root to find the linear scale factor.
• 📌 The surface area of similar shapes scales with the square of the linear dimensions, not the volume.

Q & A

What is the definition of similar shapes?

Two shapes are similar if there has been a uniform enlargement, meaning all lengths of a shape have been multiplied by the same scale factor.

Why are all cubes always similar shapes?

All cubes are similar shapes because if you scale their dimensions by the same factor, the proportions remain constant.

How do you calculate the area scale factor when comparing similar shapes?

The area scale factor is calculated by squaring the length scale factor.

What is the volume scale factor for a shape that has been scaled by a factor of 2?

The volume scale factor is 2 cubed, which equals 8.

If Prism A has a length of 10 cm and Prism B has a length of 20 cm, what is the scale factor for length?

The scale factor for length is 2, as you multiply by 2 to get from 10 cm to 20 cm.

How do you find the volume of a larger prism when given the volume of a smaller similar prism?

You multiply the volume of the smaller prism by the volume scale factor, which is the cube of the length scale factor.

What is the relationship between the surface area ratio and the volume ratio of similar prisms?

The volume ratio is the cube of the surface area ratio's square root.

If the surface area of two similar prisms is in the ratio of 25 to 4, what is the volume ratio?

The volume ratio is the cube of the square root of the surface area ratio, which is 125 to 8.

How can you find the volume of a smaller similar cone if you know the volume of a larger one?

You divide the volume of the larger cone by the cube of the length scale factor to find the volume of the smaller cone.

If two similar cones have volumes in the ratio of 27 to 64, what is the ratio of their surface areas?

The ratio of their surface areas is the square of the ratio of their lengths, which is 9 to 16.

What is the volume of Cone A if Cone B has a volume of 3200 cm³ and the surface area ratio of Cone A to Cone B is 35 cm² to 560 cm²?

First, find the scale factor for the surface area (16), then find the length scale factor (4), and finally calculate the volume scale factor (64). Divide the volume of Cone B by the volume scale factor to get the volume of Cone A, which is 50 cm³.

Outlines

00:00

🔍 Understanding Similar Shapes and Scale Factors

This paragraph introduces the concept of similar shapes and how they relate to uniform enlargement. The speaker uses the example of two cubes, one with a side length of one centimeter and another with a side length of two centimeters, to explain scale factors. It's explained that the length scale factor is 2, but when considering area, the scale factor is 2 squared (4), and for volume, it's 2 cubed (8). The paragraph further applies this concept to find the volume of a larger prism given the volume of a smaller prism and their corresponding length scale factors.

05:04

📐 Calculating Volumes of Similar Prisms

The second paragraph presents a problem-solving approach for finding the volume of a prism when given the volume of a similar, smaller prism. It explains how to calculate the scale factor for length and then apply it to find the volume scale factor by cubing it. The speaker then uses this method to solve a problem where the volume of prism B is known, and the task is to find the volume of prism A. Another example is given where the surface area ratio of two prisms is known, and the goal is to find the volume ratio.

10:05

📏 Applying Scale Factors to Surface Areas and Volumes

This section continues the theme of scale factors but focuses on surface areas and volumes of similar prisms. It explains how to find the scale factor for length from the surface area ratio, then use it to determine the volume scale factor by cubing it. The speaker solves a problem involving the surface area and volume of two similar prisms, demonstrating how to calculate the unknown volume of one prism using the known volume and surface area of the other.

15:06

🌟 Solving Problems with Scale Factors in Similar Cones

The final paragraph extends the concept of scale factors to three-dimensional shapes like cones. It shows how to calculate the scale factor for volume from the given volumes of two similar cones and then find the scale factor for surface area by squaring the length scale factor. The speaker provides examples of how to use these scale factors to find unknown volumes and surface areas. The paragraph concludes with a couple of practice problems for the viewer to solve, applying the principles discussed.

Keywords

💡Similar Shapes

Similar shapes are two or more shapes that have the same form, but not necessarily the same size. They can be transformed into one another by a uniform enlargement or reduction, meaning all corresponding lengths are multiplied by the same scale factor. In the video, the concept is introduced with cubes as an example where changing the scale factor uniformly maintains the shape's similarity.

💡Scale Factor

The scale factor is a ratio that describes how much one dimension of a shape is multiplied by to get the corresponding dimension of a similar shape. It is fundamental to understanding how similar shapes relate to each other in terms of size. The video explains that for cubes, the scale factor for length, area, and volume are different, with area being the square of the length scale factor and volume being the cube of the length scale factor.

💡Cube

A cube is a three-dimensional solid object bounded by six square faces, with equal sides. It is used in the video to illustrate the concept of similar shapes, showing that all sides of a cube can be scaled uniformly to create a similar larger or smaller cube.

💡Area Scale Factor

The area scale factor is the square of the linear scale factor and is used to determine how much the area of a shape changes when it is scaled. The video explains that if a cube's side length is scaled by a factor of two, the area scale factor becomes four, as the area of each face of the cube is multiplied by this factor squared.

💡Volume Scale Factor

The volume scale factor is the cube of the linear scale factor and is used to determine how much the volume of a shape changes when it is scaled. The video uses the example of a cube to show that if the side length is scaled by a factor of two, the volume scale factor becomes eight, as the volume is the product of the side lengths, each scaled by the linear scale factor.

💡Cross Section

A cross section is a cut through an object that reveals its internal structure. In the video, the cross section of cubes is used to illustrate how the area of a shape changes with scaling, showing the front face of the cubes and how its area is calculated.

💡Prisms

Prisms are polyhedrons with two parallel, congruent faces called bases, and other faces are parallelograms. The video uses prisms to demonstrate how to calculate the volume of similar shapes when the scale factor is known, and how to find the volume of one prism when the volume of a similar prism is given.

💡Surface Area

Surface area is the sum of the areas of all surfaces of a three-dimensional object. The video discusses how the surface area of similar prisms can be used to find the scale factor for length and subsequently the volume scale factor.

💡Volume

Volume is the measure of the space occupied by a three-dimensional object. The video explains how to calculate the volume of similar shapes, such as prisms and cones, using the scale factor and the concept that volume scales with the cube of the linear dimensions.

💡Cones

Cones are geometric shapes that have a circular base and a single vertex. The video uses cones to demonstrate the concept of similar shapes and how to calculate the volume and surface area of similar cones using the scale factor.

💡Ratio

A ratio is a quantitative relationship between two numbers, showing how many times one number contains or is contained within the other. The video uses ratios to compare the surface areas and volumes of similar shapes, such as prisms and cones, to find the scale factors and solve for unknown dimensions.

Highlights

Two shapes are similar if there's a uniform enlargement by the same scale factor.

Cubes are always similar shapes because all their dimensions are multiplied by the same scale factor.

A one-centimeter cube has dimensions of one centimeter in width, height, and depth.

A two-centimeter cube has dimensions of two centimeters in width, height, and depth.

The length scale factor to get from one to two centimeters is two.

The area scale factor is the square of the length scale factor.

The volume scale factor is the cube of the length scale factor.

For a length scale factor of five, the area scale factor is 25 and the volume scale factor is 125.

In similar prisms, the volume of prism B can be found by multiplying the volume of prism A by the cube of the length scale factor.

To find the volume of prism A, divide the volume of prism B by the cube of the length scale factor.

The ratio of the volumes of two similar prisms is equal to the cube of the ratio of their lengths.

The ratio of the surface areas of two similar prisms is equal to the square of the ratio of their lengths.

To find the volume ratio when the surface area ratio is known, cube the square root of the surface area ratio.

To find the volume of a smaller prism, divide the volume of a larger prism by the cube of the length scale factor.

For similar cones, the volume scale factor can be found by dividing the volumes of the two cones.

The surface area scale factor for similar cones is the square of the length scale factor.

To find the volume of a smaller cone, divide the volume of a larger cone by the cube of the length scale factor.

When given the surface area and volume ratios of similar prisms, you can find the volume of one prism using the cube of the length scale factor.

Transcripts

we can say that two shapes are similar

if

there's been a uniform enlargement so

all of the lengths of a shape have been

multiplied by the same scale factor

here i've got two cubes

and cubes are always similar shapes

because if you didn't multiply them by

the same scale factor so if you multiply

the width by five

but the height by two

it would no longer be a cube

so let's look at these two cubes

so we've got a one centimeter cube so

it's got a width of one a height of one

and a depth of one

and my two centimeter cubes has got a

width of two a height of two and a depth

of two

so the length scale factor

we can see is two to get from one to two

you multiply by two

how about the scale factor for area is

it times two

so let's look at a cross section so the

the square on the front of the shape

square on the front of both of these

shapes

so for the one centimeter cubed it's got

a height of one centimeter a width of

one centimeter

so its area

is one times one

which is one centimeter squared

for the two centimeter cubes

it's got a width of two and a height of

two

so it's got an area of two times two

which is four centimeters squared

so the area scale factor

isn't

times two

it's actually times four

or i'm going to write that as times two

squared

in brackets times four

and how about the volume

so the volume of my one centimeter cube

is one times one

times one

one times one times one is one

so it's got a volume of one centimeter

cubed

for the two centimeter cube

it's got a volume of two times two times

2

which is 8

centimeters cubed

so the volume

scale factor

is times 2 cubed

or times eight

and this is the pattern that we're going

to see when we're using area and volume

scale factors

so the area scale factor is length scale

factor squared

and the volume scale factor is going to

be length scale factor cubed

so if we had a length scale factor

of

five

so if we had a length scale factor of 5

our area scale factor will be times 5

squared

or times 25

and our volume scale factor will be

times 5 cubed

or times 125

okay let's look at an example

the diagram shows two similar prisms

prism a

our smaller prism

has a length of 10 centimeters and prism

b

the larger one

has a length of 20 centimeters

prism a

has a volume of 80 centimeters cubed

find the volume of prism b

so we can find our scale factor for

length first

so what do we do to get from little

shape to big shape

to get from 10 to 20

so scale factor for length

is multiplied by two

so to go from little shape to big shape

we multiply by two

for

volume

we're gonna have scale factor cubed

so for area we've got scale factor

squared for volume scale factor cubed

so two cubed is eight

so the volume scale factor is going to

be times by eight

so the little shape has a volume of

eighty

the scale factor is times by eight for

volume

so we've got eighty

times eight

eight eighths of 64. so 88 640

and it's in centimeters cubed

so that is the volume of prism b

okay one for you to try so

pause the video and give it a go

so again we've got two similar prisms

we've got prism a our small one with a

length of two

and prism b with a length of 6.

this time we're given prism b's volume

so 270 meters cubed

find the volume of prism a

so the scale factor for length

what do i do to get from two to six so

six divided by two

times by three

scale factor for volume

is scale factor cubed

so three cubes

is twenty-seven

three times three times three

so to go from the big shape to a little

shape we're going to divide by the scale

factor

so it's going to be 270

divided by 27

which is 10 so it's 10

meters cubed

okay a different type of question

so in two similar prisms

the surface area of prism a

and the surface area of prism b

are in the ratio 25 to four

what's the ratio

of the volume of prism a to the volume

of prism b

so this is the scale factor for surface

areas

so area scale factor is scale factor

squared

so for area

we have 25 to 4.

if we want to find

the scale factor for length

we've got scale factor squared at the

moment

so to find scale factor we square root

so for length the scale factor

will be square root 25 to square root 4.

so 5 to 2.

so

the scale factor for volume is scale

factor cubed

so we're going to take our length scale

factor and cube it

so 5 cubed 5 times 5 times 5

is 125

and 2 cubed

is 8.

so the ratio of the volume of prism a

to the volume of prism b

is 125 to 8.

okay one for you to try so pause the

video give it a go

this time in two similar prisms the

volume of a to the volume of b

is in the ratio

27 to 64.

so this is volume this time

and we want to work out

the ratio of area

so scale factor of volume is scale

factor cubed

so to find the scale factor for length

we're going to cube root

so the cube root of 27 is 3

and the cube root of 64 is 4.

and then to find the scale factor for

area

we're going to square the scale factor

so 3 squared is 9

4 squared is 16.

and one more example

so this time we've got two similar

prisms

again prism a has a surface area

of 32 centimeters squared

prism b has a surface area of 200

centimeters squared

and we're given prism b's volume

of 1

250

centimeters cubed

and we need to find the volume of prism

a

so let's look at the scale factor for

areas and we can work that out so scale

factor

for area

so what do we multiply 32 by to get 200

so if we grab a calculator and do 200

divided by 32

that's 25 over 4 and i'll leave it as a

fraction

so times 25 over 4.

so that's our scale factor for area

if i want scale factor for length

i'm going to

square root

so if i square root 25 over 4

i don't have to use a calculator i will

so square root the answer

that's five over two

the scale factor for length

is five over two

and to get scale effects for volume

that scale factor for length cubed

so this answer

cubed

and that's 125

over eight

so our scale factor for volume

is times 125 over eight

so to go from little shape to big shape

we times by 125 over eight

to go from big shape back to little

shape

we will divide

by 125 over eight

so we're going to get 1250

divided by our scale factor for volume

and that will tell us the volume of the

smaller shape

which is 80

so that's 80

centimeters cubes

okay one for you to try so pause the

video and give it a go

so we've got two similar cones

we've got volume of kone

270

centimeters cubed

and the volume of cone b

is 640 centimeters cubed

so we can work out the scale factor of

volume

and if we do 640 divided by 270

we will get scale factor for volume

and i'm just going to write that as 64

over 27.

so that's our scale factor for volume

to find the scale factor for length

we are going to cube root so volume

scale factor is scale factor cubed

so if we cube root this which is 4 over

3

we get our scale factor for length

and we want to work with surface area

so we're going to square the scale

factor

4 over 3 times 4 over 3

is 16 over 9.

so our scale factor

is times 16 over 9 to go from little to

big

and that's what we want to do

so

cone a is a surface area of 90

centimeters squared

so we're going to do 90 centimeters

squared

times the scale factor

so 90

times 16 over 9

so we can say

that's 10 times 16

which is 160 centimeters squared

okay two questions to finish up so pause

the video and give them a go

question one says the diagram shows two

similar cones

cone a has a surface area

of 35 centimeters squared

and cone b has a surface area

of

560 centimeters squared

and coin b's volume is three thousand

two hundred centimeters cubed

and we need to find

cone a's volume

so let's work out the scale factor for

areas

so if i do 560

divided by 35

which is 16

so the scale factor for

area is times 16.

the scale factor for length

is going to be the square root of that

so scale factor for area is scale factor

squared

so to find scale factor with square root

which will be 4

and scale factor for volume

is our scale factor cubed

so 4 cubed

is 64.

so if i want to go from the little shape

to the big shape i'll times by 64.

but we're going from big shape back to

little shape

so it's divided by 64.

so 3200

divided by 64.

and that is 50.

so it's 50

centimeters cubed

question two

two similar prisms the surface area of

prism a and the surface area of prism b

are in the ratio 36 to 49

so area

is in the ratio 36 to 49

and we want the ratio of volumes

so let's find lengths first

so the scale factor for area is scale

factor squared

so to get from area back to the length

scale factor we square root

so square root 36 is 6.

square root 49 is 7.

and for volume

scale factor is cubed

so we're going to have 6 cubed

which is 216

and 7 cubes

which is 343.

so for volume it's going to be 216

to 343

you