Question

In the figure above, each circle is tangent to the other two circles and to the sides of the rectangle. If the diameter of each circle is 10, what is the area of the rectangle?

Answer 1

First, what does diameter of the circle mean?
It means that that distance of a straight line through the entire width of the circle.

So if that width is equal to 10, you can see that width of the rectangle is 2 circles.
Width of Rectangle = 2 x 10 = 20

Now the height of the rectangle involves a little more math, since the top circle is sort of overlapping into the bottom half of the rectangle. You can make a triangle like you did, and knowing the angle of the triangle will allow you to determine the height from a chord across the circle that you have actually drawn in. Therefore, you can find the height of the triangle (the vertical line drawn that splits the triangle into 2 parts).

This will be done by using properties of special triangles. This is a 30-60-90 triangle, so if the hypotenuse is equal to d/2 = 10/2 = 5, then we find the height.

In a 30-60-90 triangle the following is true:

So, our height of our triangle is the side that is labeled sqrt(3). This means, that is 5 is considered 2 times, then our 1 would be 2.5, and thus our sqrt(3) would be 2.5sqrt(3).

Now, this height has brought us to the center of the top circle. We can use the radius of 5 to reach the full height.

Rectangle Height = 10 + 2.5sqrt(3) + 5 = 15 + 2.5sqrt(3) = 19.33

So, the dimensions of the rectangle (width x height) are:
20 x 19.33