The Maraquan Circle?

Question

The Maraquan Circle is between Mystery Island, Krawk Island, and Maraqua. It should be noted that Mystery Island and Krawk Island are 240 kilometres apart, Krawk Island and Maraqua are 330 kilometres apart, and Mystery Island and Maraqua are 270 kilometres apart. The Maraquan Circle, according to legend, is defined as the largest possible circle that fits entirely within the triangle formed by those three islands.

What is the area of the Maraquan Circle, in square kilometres?

Answer 1

This is a neat problem. There are different issues to resolve.
1. Find the area of the triangle bounded by a, b, and c
2. Use that result to find the height of the triangle (using c as the base)
3. using h (height) as the diameter of the largest circle that fits within the triangle, find the area using a=Pi * r squared

one formula to find the area of a triangle is:
Area = sqrt(s*(s-a)*(s-b)*(s-c)) where s=(a+b+c)/2 is the semi-perimeter.
abcs
240270330420

ss-as-bs-cArea
4201801509031947
find the height of the only side that is within the bounded sides of the triangle
A=1/2 (b*h)
h=2A/B
h 2 A b
193.6 2 31947 330
The radius of a circle is 1/2 diameter (this diameter is the same as h)
Area of a circle is
A=Pi * R2
r= 96.81 (1/2 h)
r2= 9372.00

Pi R2
3.1416 9372.00

area= 29443 Sq Km

You need to space out the columns above... I cannot get the format correct

Answer 2

Using the law of cosines we have:

c^2 = a^2 + b^2 -2abcos(C)
so cos(C) =(c^2-a^2-b^2)/(-2ab)

where a=240km, b=330km, c=270km and C is the angle between the route from Krawk to Mystery and the route from Krawk to Maraqua.

So, substituting we have

C = 53.78 degrees

Using the law of sines we have:

Sin(C)/c = 1/d where d is the diameter of the circle

so substituting we have:

d = 334.68 km

so radius, r = 167.34 km

Area = (pi)r^2 = 87973.57 square km