area math problem

Watch this video and see whether you can solve the math problem. You'll need to pause the video immediately after you hear the problem described. The guy doesn't give a decimal answer, but I solved it and got both the exact answer and the approximate decimal answer. Highlight the area between the brackets to see my answer. He ended up solving the problem the same way I did. Or vice versa.

ANSWER:

[The problem says to draw or imagine three mutually tangent unit circles. A unit circle has a radius of 1.  You must then calculate the area "inside" the three circles. 

You immediately know that, if you connect the circles' centers, this creates an equilateral triangle with a side length of 2. The formula for the area of an equilateral triangle is (s²√3)/4. So (2²√3)/4 = √3. 

To find the inside area, you need to subtract the area of the three "wedges" or "pie slices" from the area of the equilateral triangle. Each pie slice has a 60º angle, so each pie slice is exactly a sixth of the unit circle's area. 

A unit circle has an area of π, so a single wedge is π/6. There are three such wedges, so the area of all three wedges is 3(π/6), i.e., π/2. 

Subtract that from the triangle's area, and you have the area you're looking for: √3 - (π/2). That's the exact answer. The decimal approximation is 1.7321 - 1.5708 = 0.1613. QED.]