Connect the circles' centers to form an equilateral triangle (which therefore has three 60-degree interior angles). Note that the answer to the question means calculating the triangle's area, then subtracting the areas of the three circle sectors inside the triangle.
Since the circles all have a radius of 2, we know the equilateral triangle has sides of length 4.
Since the triangle's interior angles are each 60 degrees, we know that each fraction of a circle is 1/6 the area of a whole circle, and there are three such fractions of a circle inside the triangle.
The formula for the area of an equilateral triangle is
[(s^2)√3]/4
So for this triangle, the area is
(16√3)/4, or 4√3.
The area of any circle is πr^2, so the area of one of these circles is 4π. A sixth of that is 4π/6, or 2π/3. There are three "slices of pie" inside the triangle, so multiply that by 3, and you get
2π
So the answer you're looking for requires subtracting 2π from 4√3:
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Connect the circles' centers to form an equilateral triangle (which therefore has three 60-degree interior angles). Note that the answer to the question means calculating the triangle's area, then subtracting the areas of the three circle sectors inside the triangle.
ReplyDeleteSince the circles all have a radius of 2, we know the equilateral triangle has sides of length 4.
Since the triangle's interior angles are each 60 degrees, we know that each fraction of a circle is 1/6 the area of a whole circle, and there are three such fractions of a circle inside the triangle.
The formula for the area of an equilateral triangle is
[(s^2)√3]/4
So for this triangle, the area is
(16√3)/4, or 4√3.
The area of any circle is πr^2, so the area of one of these circles is 4π. A sixth of that is 4π/6, or 2π/3. There are three "slices of pie" inside the triangle, so multiply that by 3, and you get
2π
So the answer you're looking for requires subtracting 2π from 4√3:
4√3 - 2π
In decimal form, that's
0.645 square units.
QED.