Similar Shapes: Length, Area and Volume
Summary
TLDRThis 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
🔍 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.
📐 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.
📏 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.
🌟 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.
Mindmap
Keywords
💡Similar Shapes
💡Scale Factor
💡Cube
💡Area Scale Factor
💡Volume Scale Factor
💡Cross Section
💡Prisms
💡Surface Area
💡Volume
💡Cones
💡Ratio
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
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