Finding the Circumradius

Geometry Level 2

What is the circumradius of a triangle which has sides of length 3, 4, and 5 units?

2.5 units 3 units 2.75 units 2 units

Krishna Keshav by Krishna Keshav , 24, India

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2 solutions

Here a , b , c a,b,c are sides of △ \triangle , r r radius of circle inscribed and s s semi-perimeter of triangle.

r × s = Area of △ r×s=\text{Area of }\triangle
r × s = s ( s − a ) ( s − b ) ( s − c ) r×s=\sqrt{s(s-a)(s-b)(s-c)}
r = ( s − a ) ( s − b ) ( s − c ) r=\sqrt{(s-a)(s-b)(s-c)}
r = ( 3 ) × ( 2 ) × ( 1 ) r=\sqrt{(3)×(2)×(1)}
r = 6 = 2.5 r=\sqrt{6}=\boxed{2.5}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

We could also just note that, since a 3 / 4 / 5 3/4/5 triangle is right-angled, the hypotenuse corresponds to a diameter of the circumcircle, (by Thales' theorem ), and thus the circumradius is one-half the length of the hypotenuse, namely 5 / 2 = 2.5 5/2 = 2.5 units.

Brian Charlesworth - 5 years, 5 months ago

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That's also nice Sir .

A Former Brilliant Member - 5 years, 5 months ago

This was correct one.. I was not able to type that solution

Krishna Keshav - 5 years, 4 months ago
Harshendu Mahto
Jan 17, 2016

It is clear that 3,4,5 are pythagorean triplets.

Therefore the triangle must be a right angled triangle.

We know that in a circle angle in a semi circle is a right angle.

So AC must be the diameter. but AC = 5

and, diameter/2 = radius

Therefore ,

radius = 5/2 = 2.5

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Good solution Harshendu . I also tried the same.

abhigyan adarsh - 5 years, 4 months ago

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