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Inscribed Circle

This topic has 3 expert replies and 3 member replies

Inscribed Circle

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Given a circle inscribed inside an equilateral triangle with side 4. Find the area of the circle.
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Radius of the circle inscribed inside an equilateral triangle = (1/3)*(Length of the median of the triangle)

Again length of the median of triangle = sqrt(4^2 - 2^2) = sqrt(12)

Hence radius of the circle = (1/3)*sqrt(12)
And area of the circle = (12/9)*pi = (4/3)*pi

Also we can find the radius by using the formula,


Where, r = radius of the incircle and P is the perimeter ans s is the semiperimeter of the triangle.

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Given overlapping shapes, look for what the shapes have in common.
Since we're being asked for the area of the circle, the triangle likely will help us to determine the radius of the circle.
Since each angle of the equilateral triangle is 60 degrees, look for a 30-60-90 triangle:



The sides of a 30-60-90 triangle are proportioned x: x√3: 2x.
In the triangle above, x√3 = 2.
Thus, radius = x = 2/√3.
Area of the circle = ∏(2/√3)² = (4/3)∏.

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draw perpendiculars from center of the circle to the sides.

thus in the triangle base = 2, leg = radius(r) and angle = 60/2=30 (angle bisector)

thus tan 30 = 1/3^(1/2) = r/2

thus r= 2/3^(1/2).

hence A = pi* 4/3.

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In any triangle, centroid divides the median in the ratio 2:1.
In an equilateral triangle, median is altitude(h=√3/2 times side)
radius is therefore, 1/3*h=4/3 times ∏

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GMATGuruNY wrote:
Given overlapping shapes, look for what the shapes have in common.
Since we're being asked for the area of the circle, the triangle likely will help us to determine the radius of the circle.
Since each angle of the equilateral triangle is 60 degrees, look for a 30-60-90 triangle:



The sides of a 30-60-90 triangle are proportioned x: x√3: 2x.
In the triangle above, x√3 = 2.
Thus, radius = x = 2/√3.
Area of the circle = ∏(2/√3)² = (4/3)∏.
I received a PM requesting that I explain how the 30-60-90 triangle shown above can be derived.
An equilateral triangle can be split into 6 congruent 30-60-90 triangles:


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