I don't know what I should call this

Geometry Level 3

Use only Euclidean geometry to solve this problem.

There exists a right isosceles triangle with an incircle. What percentage of the area of the triangle does the area of the circle take up?

Answer as a percent.


The answer is 53.9.

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Chew-Seong Cheong
Feb 24, 2015

A right isosceles triangle is a 4 5 45^\circ - 9 0 90^\circ - 4 5 45^\circ triangle (see figure above). Let the equal sides be of length a a units, therefore, the hypothenuse b = 2 a b=\sqrt{2}a . Let the radius for the incircle be r r .

The area of A B C \triangle ABC , A = a 2 2 = ( a + a + b ) r 2 A=\dfrac {a^2}{2} = \dfrac {(a+a+b) r}{2}

a 2 = ( 2 a + 2 a ) r r = a ( 2 + 2 ) \Rightarrow a^2 = (2a+\sqrt{2}a)r \quad \Rightarrow r = \dfrac {a}{(2+\sqrt{2})}

Now the area of incircle A = π r 2 A_{\circ} = \pi r^2

Therefore, the ratio,

x = A A = π r 2 a 2 2 = 2 π ( 2 + 2 ) 2 = 2 π 6 + 4 2 x = \dfrac {A_{\circ}}{A} = \dfrac {\pi r^2}{\frac{a^2}{2}} = \dfrac {2\pi}{(2+\sqrt{2})^2} = \dfrac {2\pi}{6+4\sqrt{2}}

= π 3 + 2 2 = 0.539012084 = 53.9 % \quad = \dfrac {\pi}{3+2\sqrt{2}} = 0.539012084 = \boxed{53.9} \%

Curtis Clement
Feb 24, 2015

Without loss of generality: let the shorter sides of the triangle be 1 and the hypotenuse 2 \sqrt{2} such that: a r e a o f t r i a n g l e = A T = 1 2 \large \ area \ of \ triangle = \ A_T = \frac{1}{2} . Now the radius of the incircle can be given by: r = 2 A T a + b + c = 1 2 + 2 = 2 2 2 r = \frac{2A_T}{a+b+c} = \frac{1}{2+\sqrt{2}} = \frac{2-\sqrt{2}}{2} Let I A \ I_{A} be the area of the incircle such that I A = π × ( 2 2 2 ) 2 0.2695 = x \Rightarrow\ I_A = \pi \times (\frac{2-\sqrt{2}}{2})^2 \approx 0.2695 = x p e r c e n t a g e o f a r e a = x 1 / 2 × 100 53.9 \therefore\ percentage \ of \ area = \frac{x}{1/2}\times\ 100 \approx 53.9

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...