Before you start, make sure you know: Square root (Math 6)
The Willis Tower in Chicago is far too tall to measure with a tape measure. But with a small mirror on the sidewalk and a tape measure, you can find its height without leaving the ground. The trick is similar figures, and this page shows how to use them.
Scale drawings and maps
A map, a floor plan, or a model is always similar to the real thing. The scale is the scale factor: it tells you how many real units each unit on the drawing stands for.
A scale can be written as a ratio $1 : n$, or as a statement like $1 \text{ in.} = 6 \text{ mi}$ or $\tfrac14 \text{ in.} = 1 \text{ ft}$. In a $1 : n$ scale, 1 unit on the drawing is $n$ of the same unit in real life.
Length units you will need
$12 \text{ in.} = 1 \text{ ft}$
$3 \text{ ft} = 1 \text{ yd}$
$5{,}280 \text{ ft} = 1 \text{ mi}$
$63{,}360 \text{ in.} = 1 \text{ mi}$
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The scale model of a new sports center is built to a scale of $1:200$. If the Olympic swimming pool is $25 \text{ in}$ long in the model, how many feet will the pool…
The scale model of a new sports center is built to a scale of $1:200$. If the Olympic swimming pool is $25 \text{ in}$ long in the model, how many feet will the pool measure in reality?
A road map has a scale of $1:50.000$. If the distance between two neighboring towns on the map is $12 \text{ in}$, how many miles of real distance separate them?
A road map has a scale of $1:50.000$. If the distance between two neighboring towns on the map is $12 \text{ in}$, how many miles of real distance separate them?
Two towns, A and B, are separated by $18$ miles in real life. On a road map, this distance is represented by a segment of $12$ in. 1. Calculate the scale of the map. 2.…
Two towns, A and B, are separated by $18$ miles in real life. On a road map, this distance is represented by a segment of $12$ in.
Calculate the scale of the map.
If two other towns are separated on the same map by $7,5$ in, what is their actual distance in miles?
Two photos, two screens, or two rectangles are similar if they keep the same shape. Width and height must grow by the same factor. Try different values of $k$ below and watch every length change by the same factor.
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We have two similar rectangles. The first one is $8 \text{ cm}$ wide and $12 \text{ cm}$ high. If the second one has a width of $20 \text{ cm}$, what must its height be…
We have two similar rectangles. The first one is $8 \text{ cm}$ wide and $12 \text{ cm}$ high. If the second one has a width of $20 \text{ cm}$, what must its height be so that it keeps the same shape?
A smartphone with a $6 \text{ pulgadas}$ diagonal has a screen that is $13,2 \text{ cm}$ high and $7,4 \text{ cm}$ wide. If we want to view the same content on a tablet…
A smartphone with a $6 \text{ pulgadas}$ diagonal has a screen that is $13,2 \text{ cm}$ high and $7,4 \text{ cm}$ wide. If we want to view the same content on a tablet that is exactly twice as large (scale factor $k=2$), what will the dimensions of the tablet's screen be?
A rectangular photograph of $10 \times 15$ in is to be enlarged to make a poster. If we want the longest side of the enlargement to be $1,2$ feet: 1. How long will the…
A rectangular photograph of $10 \times 15$ in is to be enlarged to make a poster. If we want the longest side of the enlargement to be $1,2$ feet:
How long will the short side of the poster be?
How many times larger is the area of the poster than the area of the original photograph?
What happens to the area when you double every length? Look at the squares.
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On a house plan with a scale of $1:75$, the living room has a rectangular shape with dimensions of $8$ in long and $6$ in wide. Calculate the real area of the living…
On a house plan with a scale of $1:75$, the living room has a rectangular shape with dimensions of $8$ in long and $6$ in wide. Calculate the real area of the living room in square feet ($m^2$).
Two television screens are similar. The first one has a diagonal of $32$ inches and its area is approximately $2800$ $cm^2$. If the second screen has a diagonal of $48$…
Two television screens are similar. The first one has a diagonal of $32$ inches and its area is approximately $2800$ $cm^2$. If the second screen has a diagonal of $48$ inches:
What is the scale factor between the large screen and the small one?
On a map with a scale of $1:25.000$, a nature reserve has the shape of a trapezoid. On paper, its bases measure $4 \text{ in}$ and $6 \text{ in}$, and its height is…
On a map with a scale of $1:25.000$, a nature reserve has the shape of a trapezoid. On paper, its bases measure $4 \text{ in}$ and $6 \text{ in}$, and its height is $3 \text{ in}$.
a) Calculate the area of the reserve on the map in $\text{in}^2$.
b) Using the scale factor of the areas, calculate the real area of the reserve in square miles ($\text{mi}^2$).
c) If the government decides to expand the reserve by increasing all its linear dimensions by $20\%$, what will the new real surface area be?
Now double the edge of a cube. Eight small cubes fit inside the big one.
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An artist has made a bronze sculpture that weighs $12$ lb and is $40$ in tall. If they want to make an exact replica of the same material but with a height of $1$ foot,…
An artist has made a bronze sculpture that weighs $12$ lb and is $40$ in tall. If they want to make an exact replica of the same material but with a height of $1$ foot, how much will the new sculpture weigh? (Keep in mind that the weight is proportional to the volume).
A video game designer is creating two models of a defensive tower. The small model has a lateral surface area of $150 \text{ in}^2$ and a volume of $250 \text{ in}^3$.…
A video game designer is creating two models of a defensive tower. The small model has a lateral surface area of $150 \text{ in}^2$ and a volume of $250 \text{ in}^3$. If the large model must have a lateral surface area of $600 \text{ in}^2$ so that the details look good:
a) What is the scale factor $k$ between the lengths of both models?
b) What will the volume of the large model be?
c) If 40 ounces of resin were needed to manufacture the small model, how many pounds of resin will be needed to manufacture 10 large models?
At the same time of day, the Sun's rays hit everything at the same angle. A pole and its shadow, and a tree and its shadow, form two similar right triangles. This is the first way to measure something you can't reach.
Because the triangles are similar, the ratio $\dfrac{\text{height}}{\text{shadow}}$ is the same for both objects. (This is a proportional relationship, and that ratio is its constant of proportionality.)
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To measure the height of a tree in the schoolyard, we take advantage of a sunny day. A batten driven vertically into the ground that is $1,5 \text{ m}$ high casts a…
To measure the height of a tree in the schoolyard, we take advantage of a sunny day. A batten driven vertically into the ground that is $1,5 \text{ m}$ high casts a shadow of $0,9 \text{ m}$. If at that same moment the tree's shadow measures $6 \text{ m}$, what is the height of the tree?
A content creator is recording a video outside. To calculate the height of a modern sculpture, they place their tripod, which is $1,20$ feet high, so that it casts a…
A content creator is recording a video outside. To calculate the height of a modern sculpture, they place their tripod, which is $1,20$ feet high, so that it casts a shadow of $90$ in on the ground. At that same moment, the shadow of the sculpture measures $4,5$ feet. What is the height of the sculpture?
Sometimes there is no sun, but there is still a pair of similar triangles. When light bounces off a flat mirror, the angle it comes in at equals the angle it leaves at. A person and a building are both vertical, so each makes a right angle with the flat ground. Two right triangles with an equal acute angle are similar by AA.
Back to the start: this is exactly how you can find the height of the Willis Tower from the sidewalk. The tower is $1{,}450$ ft tall.
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Calculate the value of the width of the building's base ($x$) using the similarity of the triangles formed in the figure. The person's eyes are $1,60$ ft above the…
Calculate the value of the width of the building's base ($x$) using the similarity of the triangles formed in the figure. The person's eyes are $1,60$ ft above the ground, the distance from the person to the fence is $2$ ft and the distance from the fence to the building is $10$ ft. The height of the fence is $2,40$ ft.
To measure the height of the Willis Tower in Chicago without using complex instruments, a group of students uses a mirror placed on the ground. The tallest student, who…
To measure the height of the Willis Tower in Chicago without using complex instruments, a group of students uses a mirror placed on the ground. The tallest student, who is $1,85 \text{ m}$ tall, stands at such a distance that he sees the top of the tower reflected in the center of the mirror. The mirror is $120 \text{ m}$ from the base of the tower and the student is $2,3 \text{ m}$ from the mirror. If the student's eyes are $10 \text{ cm}$ from the top of his head:
a) Draw a diagram of the situation identifying the similar triangles.
b) Calculate the height of the tower given that the angle of incidence of the light is equal to the angle of reflection.
c) If another student of height $1,60 \text{ m}$, whose eyes are also $10 \text{ cm}$ below the top of their head, wanted to see the same thing without moving the mirror, at what distance from the mirror should they stand?
A pinhole camera is a closed box with a small hole through which light enters, projecting the image of an external object onto the opposite wall. If we have a box of…
A pinhole camera is a closed box with a small hole through which light enters, projecting the image of an external object onto the opposite wall. If we have a box of $25 \text{ cm}$ depth and we want to photograph a tree of $12 \text{ m}$ height that is at a distance of $30 \text{ m}$ from the camera:
a) What height will the image of the tree projected inside the box have?
b) If we move the camera closer to the tree until the projected image measures twice as much, at what distance from the tree will we be?
c) What is the scale factor between the real object and the image in this latter case?
Three parallel lines cut any two transversals into proportional segments.
The same idea works inside a triangle. A line parallel to one side cuts off a smaller triangle that is similar to the whole one. Some problems show two triangles placed this way: they share the top vertex, and the cut line is parallel to the base.
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In the following figure, the lines $r$ and $s$ are intersected by three mutually parallel lines. Knowing that the segments on the line $r$ measure $AB = 3 \text{ cm}$…
In the following figure, the lines $r$ and $s$ are intersected by three mutually parallel lines. Knowing that the segments on the line $r$ measure $AB = 3 \text{ cm}$ and $BC = 4 \text{ cm}$, and that the corresponding segment on the line $s$ is $A'B' = 4,5 \text{ cm}$, calculate the length of the segment $B'C'$.
To renew the design of a shelving unit, a carpenter places three parallel shelves cut by two side battens, as shown in the figure. If the lengths of the segments on one…
To renew the design of a shelving unit, a carpenter places three parallel shelves cut by two side battens, as shown in the figure. If the lengths of the segments on one of the battens are $AB = 25$ in and $BC = 40$ in, and on the other batten the segment corresponding to $AB$ measures $A'B' = 30$ in, how long will the segment $B'C'$ be?
A parallel cut combines two ideas: lengths multiply by $k$, and areas multiply by $k^2$. The small triangle at the top is similar to the whole triangle, but the piece left over is a trapezoid, so you subtract.
A streetlight on a line of sight
Sometimes the "parallel cut" is a vertical pole standing between two vertical objects. The pole is parallel to both buildings, because all three are vertical. That makes similar triangles.
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A triangular plot of land with an area of $300 \text{ m}^2$ is to be divided into two plots by a fence parallel to one of its sides. The fence is placed so that it cuts…
A triangular plot of land with an area of $300 \text{ m}^2$ is to be divided into two plots by a fence parallel to one of its sides. The fence is placed so that it cuts the other two sides at a third of the distance from the common top vertex.
a) What is the scale factor between the small (top) triangle and the original plot of land?
b) Calculate the area of each of the two resulting plots.
c) If the base of the original plot of land measured $30 \text{ m}$, how long will the new fence be?
Two vertical buildings are separated by a street $20 \text{ m}$ wide. The taller building measures $45 \text{ m}$ and the shorter one $30 \text{ m}$. From the flat roof…
Two vertical buildings are separated by a street $20 \text{ m}$ wide. The taller building measures $45 \text{ m}$ and the shorter one $30 \text{ m}$. From the flat roof of the shorter building, the base of the taller building is observed. In that same line of sight, there is a streetlight which is $5 \text{ m}$ from the shorter building.
a) What height must the streetlight have so that its top is exactly in that line of sight?
b) If we throw a straight cable from the flat roof of the taller building to the flat roof of the shorter building, what will the length of the cable be?
c) At what height above the ground would that cable cross the streetlight (assuming the streetlight is tall enough)?
Draw the altitude $h$ from the right angle to the hypotenuse. It splits the triangle into two smaller right triangles. All three triangles (the big one and the two small ones) are similar to each other, because each has the same acute angles.
Here $a$ and $b$ are the legs, $c$ is the hypotenuse, and $m$ and $n$ are the pieces of the hypotenuse next to each leg: $m$ is next to $b$, and $n$ is next to $a$.
Some problems call these the Altitude Theorem and Euclid's Theorem. Whatever the name, they come from similar triangles.
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In the right triangle in the figure, the hypotenuse measures $25 \text{ cm}$. The height relative to the hypotenuse, $h$, divides it into two segments, $m$ and $n$. If…
In the right triangle in the figure, the hypotenuse measures $25 \text{ cm}$. The height relative to the hypotenuse, $h$, divides it into two segments, $m$ and $n$. If we know that the side $b$ measures $15 \text{ cm}$:
a) Calculate the length of the projection $m$ using Euclid's Theorem.
b) Calculate the value of the height $h$ using the Altitude Theorem.
c) Find the area and perimeter of the whole triangle.
A park has the shape of a regular hexagon with a side of $40 \text{ m}$. A circular fountain is to be built in its center that is similar to a small fountain of…
A park has the shape of a regular hexagon with a side of $40 \text{ m}$. A circular fountain is to be built in its center that is similar to a small fountain of $2 \text{ m}$ in diameter located at the entrance. If the scale factor between the radius of the inscribed circle in the hexagon (apothem) and the radius of the fountain must be $4$:
a) Calculate the apothem of the hexagon (you can use the Pythagorean theorem).
b) Calculate the diameter of the new central fountain.
c) What is the ratio between the area of the park and the area of the central fountain?