Anonymous
Anonymous asked in Science & MathematicsMathematics · 2 months ago

# Find colored area?

Find area of the colored part

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• 2 months ago

The center of an inscribed circle (or "in-circle") of a triangle is where the three angle bisectors meet.  In an equilateral triangle, that is also where the three medians of the triangle meet, and that turns out to be the easy way to find the sizes of those circles.

A median is the segment from one vertex of a triangle to the midpoint of the opposite side.  In an equilateral triangle, that bisects the angle at that vertex; and is also perpendicular to the opposite side, making the median an altitude of the triangle.

The three medians of any triangle meet a single point (the "centroid") and that point divides each median in a 2:1 ratio, where the distance from the centroid to a vertex is twice the distance to the opposite side.

Any one median divides that triangle into two congruent right triangles with sides 18, 9 and h; where h is the altitude of the equilateral triangle and 18 is the hypotenuse of the right triangle.  Pythagoras say  h = √(18² - 9²) = 9√(2² - 1²) = 9√3.  The centroid (and also the center of the big circle) is 1/3 of that, making the radius of the big circle R = h/3 = 3√3.  The area of the big circle is πR² = 27π square units.

And the area of the triangle is 18h/2 = 9h = 81√3

Each small circle has a radius r = R/3.  To see why, look at the point where a small circle meets the large one. Draw the common tangent line to both circles at that point, and symmetry tells you that the common tangent cuts off an equilateral triangle from the large one, and that the small circle is centered in that smaller triangle. The common tangency point is a distance 2R = 2h/3 from the bottom of the big triangle so the height of the small triangle is h - 2h/3 = h/3.  That's one third the height of the large triangle, which makes the radius of the in-circle of the small triangle 1/3 of the larger circle's radius, or R/3.

1/3 of the radius means the small circle has 1/9 the area, or (27π)/9 = 3²π square units.

FINALLY, the shaded area is the triangle area, minus the four circles:

A = 81√3 - 27π - 3*(3π) = 81√3 - 18π

• 2 months ago

The area of the triangle is Area = 140.29612 units^2

The area of the larger circle

The area of the three smaller circles