Each circle has a radius of five centimeters.
Find the height of the figure.
Notice how we have a triangle of circles,
so it might be tempting to think that the
height of this figure is just going to be
three times the diameter or six times the radius
of five centimeters.
But this is not going to be the case because
if we look closely, notice how the top of this
first row of circles here
does not line up with the bottom
of the second row of circles, which is here.
And the top of the second row of circles here
doesn't line up with the bottom of the top circle,
which is here.
So the height is not going to be
three times the diameter or six times the radius.
So to help us get started with this,
let's first plot the center for each circle.
Notice how the centers of the circle form a triangle.
Let's go ahead and sketch that triangle.
And remember we know the radius of each circle
is equal to five,
so this length is five, this length would be 10,
and this length would be five.
So the length of this side of the triangle is equal to
20 centimeters.
And notice how this would be true
for each side of the triangle,
which means we have an equilateral triangle.
So all the sides have the same length,
and all the angles have the same measure,
and therefore the interior angles measure 60 degrees.
Notice the height of this triangle
would be this length here,
which we need in order to determine
the height of the entire figure.
So if we label this length x, notice how the length of x
plus the radius of this top circle,
which we know is five centimeters,
plus the radius of the bottom circle,
which we also know is five centimeters,
we would have the height of this figure.
So let's go ahead and write that down.
The height, h, of the figure,
is going to be equal to five plus x plus five,
or just x plus 10.
So if we can determine the length of x,
which is the height of this triangle,
we can determine the height of the entire figure.
And there are several ways to determine
the length of this side, x.
Notice how the height cuts this equilateral triangle
into two congruent right triangles.
We have a right triangle on the left,
and a right triangle on the right.
And notice how the length of this leg,
for the right triangle on the left,
would be five plus five or 10 centimeters.
Let's go and erase this 20.
And of course the same is true for the right triangle
on the right.
This would be 10 centimeters.
So because we have a right triangle,
and we know two of the lengths,
we could use the Pythagorean Theorem to find x.
So let's first find x using this method.
Using the Pythagorean Theorem, a squared plus b squared
equals c squared, remember a and b,
the lengths of the two legs,
and c is the length of the hypotenuse,
so let's let a be x
and b be 10.
So we'd have x squared plus 10 squared equals,
c would be 20, so 20 squared.
So we have x squared plus 100 equals
20 squared is 400.
Subtracting 100 on both sides, we'd have x squared equals
300.
And now we would take the principle square root
of both sides of the equation,
because we know the length, x, must be positive.
So we have x equals the square root of 300
is equal to the square root of 100 times three,
and the square root of 100 is equal to 10,
so now we know x equals 10 square root of three.
So now that we know that
x equals 10 square root of three centimeters,
we can determine the height of our figure,
but before we do this, let's discuss some additional ways
we can determine the value of x.
More specifically, let's find x using trigonometry.
Because we know the measure of this angle is 60 degrees,
and we're trying to determine the opposite side,
but we know the length of the hypotenuse
as well as the adjacent side, we could use a trig equation
using sine or tangent to determine the value of x.
Let's go ahead and show it using the sine function.
We could say the sine of 60 degrees
is equal to the ratio
of the opposite side to the hypotenuse,
which would be x divided by 20.
Or now to determine the sine of 60 degrees,
let's use a 30-60-90 reference triangle, shown here.
Or for any 30-60-90 right triangle, the sides will always
be multiples of one, two and the square root of three.
So notice how the sine of 60 degrees
would be equal to the ratio of the opposite side
of the hypotenuse, which would be square root of three
divided by two.
So we'd have the square root three divided by two
equals x divided by 20, and now here if multiply
both sides of the equation by 20, we can solve for x.
So notice on the right side,
20 divided by 20 simplifies to one so we just have x.
On the left side, notice how
we have a common factor of two here.
The two simplifies to one, the 20 simplifies to 10.
Once again we get 10 square root of three.
Another way to find the value of x
would be to recognize, again, any 30-60-90 right triangle
is going to be a multiple of this right triangle here,
meaning the sides are always multiples of one, two,
and square root of three.
Notice how because the hypotenuse is 20,
and here the hypotenuse is two,
if we multiply all three sides of this triangle by 10,
it would give us the length
of the three sides of this right triangle.
Notice two times 10 is 20, one times 10 is 10,
square root of three times 10, which would give us x,
would be 10 square root of three.
So whichever method we take,
the height of this triangle is
10 square root of three units.
Now to determine the height of our figure,
we would have h equals, again x is 10 square root of three
plus 10, the units here are centimeters,
so this would be the exact height of our figure.
Let's also get our decimal approximation.
So you would have h is approximately,
going to the calculator,
we would have 10 square root three,
right arrow, plus 10
equals.
The height would be approximately 27.32 centimeters.
I hope you found this helpful.