Circle Areas #2: The Basics

Geometry Level 3

In the diagram above, Circle O is circumscribed about Triangle ABC, where AB= $4$ , AC= $4$ , and BC= $6$ . The area of Circle O can be expressed as $\frac { a\pi }{ b }$ , where $a,b$ are constants, are positive integers, and share no common factors greater than 1. What is $a+b$ ?

Note: Not to scale

This is part of the set Circles , made by Chris H.

The answer is 71.

1 solution

Finn Hulse
Jun 7, 2014

Let $R$ be the radius of the circumscribed circle. First we need to find the area of the triangle, denoted $A$ . Using Heron's, this is

$\sqrt{7(1)(3)(3)}$

Or $3\sqrt{7}$ . The formula we need now is

$R=\dfrac{abc}{4A}$

where $a$ , $b$ , and $c$ are the sides of the triangle. Plugging these in, $R=\frac{8}{\sqrt{7}}$ . Thus the area, or $\ piR^2$ is $\frac{64\pi}{7}$ . The desired answer is $64+7=\boxed{71}$ .

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Guiseppi Butel - 7 years ago

Circle Areas #2: The Basics

The answer is 71.

1 solution

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