A Nice Geometry Problem – Can You Find the Radius ?

Added: 2026-05-07

[00:00:00]Hello everyone. You're welcome. Today we have a very beautiful geometry math problem. Here we have given a triangle with three sides length is given.

[00:00:09]And there is a circle inside a triangle.

[00:00:12]Our target is to find out the radius of this circle. And to find out the radius of circle, first first we will try to find out the area of this triangle.

[00:00:21]Looking to this one triangle, it has three different lengths. So here to find out the area of this triangle, one of the easiest method is the Hero's formula. Here we will use the Hero's formula and we will find out the area of this triangle.

[00:00:35]So by Hero's formula, we can write the area of this triangle as area of this triangle that is square root of S times S minus A times S minus B times S minus C.

[00:00:56]Where S is the sum of all sides of this triangle ABC divided by two.

[00:01:05]So for that, let us suppose this is side A, this is side B and this is side C.

[00:01:13]So we substitute these values here and we will find out the value of S. So this will become S 15 B is 20 and C is 25 divided by two.

[00:01:25]And here 25 plus 15 is simply 40. 40 plus 20 is 60.

[00:01:29]So this is 60 divided by two. And 60 divided by two is simply 30.

[00:01:34]And we will substitute S is equal to 30 and ABC is equal to these two values in this one formula.

[00:01:40]So the area will become So finally, our area of this triangle that will become square root of Here S is 30. So this is 30 times this will become 30 minus 15 times 30 minus 20 times 30 minus 25.

[00:02:06]So let's simplify this right hand side.

[00:02:09]So this becomes square root of This is 30 times this will become 15 times this is 10 times five.

[00:02:19]Next we can write this as this is simply 30 times 15 times here we can write this 10 as two times five times this one five.

[00:02:30]Now let us rearrange these numbers in square forms. This become this is square root of This is 30 times 15 times two is simply 30 times and this is five times five which is five square.

[00:02:44]And we can also write this number in square form as this will become square root of 30 square times five square.

[00:02:53]The 30 square is simply this is 30 and this is five by simplifying the square root.

[00:03:00]So this will become the final area of this triangle that will become 150 square units. So let us suppose this is our equation number first.

[00:03:13]Looking to the figure here, we will try to find out the radius of this circle.

[00:03:17]Now to find out the radius of this circle here, we will join the center of the circle with these point of tangencies and also the vertices of this triangle. So this figure will become Now by joining the center with the vertices of this triangle and also the point of tangencies here we have three triangles. This one triangle, this one triangle and this one triangle inside this one bigger triangle.

[00:03:41]Now let us suppose this is the radius small r of this one triangle. This is the radius or height of this one triangle. And this is the radius and height of this one triangle.

[00:03:53]Now actually here the area of this bigger triangle is actually the sum of the areas of these three triangles.

[00:04:00]So from here, from this figure we can write the area of the bigger triangle is simply that is 150 square units. And the area of this triangle will become here it has base 20 and its height is r.

[00:04:15]It has base 15 and height is r. It has base 25 and height is r.

[00:04:22]>> [clears throat] >> So therefore the area of this first triangle will become that is half times base time base is 20 times its height.

[00:04:30]That is r. Plus the area of this triangle will become half times its base is 15 times its height is r.

[00:04:38]Plus the area of the third triangle. So that is one over two times the base is 25 times its height is small r.

[00:04:49]Now let us simplify this one equation for the value of r. So this is simply 150 and here the denominator is same. So this is two. This is 20 r plus this is 15 r plus this is 25 r.

[00:05:06]Now let but let us first simplify this right hand side. So this is here 15 plus 25 is 40. Then 40 plus 20 is simply 60.

[00:05:14]This is 60 r divided by two.

[00:05:18]And this is two time minus two. Two times 30 is 60.

[00:05:22]So this will become 150 is equal to 30 r.

[00:05:28]And dividing both sides by 30 this gives him here 30 and 30 will be cancelled.

[00:05:34]Zero and zero will be cancelled. Three time minus three. Three time five is 15.

[00:05:38]So [snorts] the final value of r will become five.

[00:05:41]So here the value of r is five. So finally the radius of this circle inside the triangle is simply five units. So here this radius is just five units. And that is our final answer.