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:
All comments are subject to approval before they are published, so they will not appear immediately. Comments should be civil, relevant, and substantive. Anonymous comments are not allowed and will be unceremoniously deleted. For more on my comments policy, please see this entry on my other blog.
AND A NEW RULE (per this post): comments critical of Trump's lying must include criticism of Biden's or Kamala's or some prominent leftie's lying on a one-for-one basis! Failure to be balanced means your comment will not be published.
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.