Lesson Proof of the formula for the area of a triangle via the radius of the inscribed circle

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<H2>Proof of the formula for the area of a triangle via the radius of the inscribed circle</H2>

In this lesson you will learn the proof of the formula for the area of a triangle via the radius of the inscribed circle:

{{{A}}} = {{{r*s}}},

where &nbsp;{{{r}}}&nbsp; is the radius of the inscribed circle in the triangle, &nbsp;{{{s}}}&nbsp; is the triangle semiperimeter &nbsp;{{{s}}} = {{{(a + b + c)/2}}}, &nbsp;and &nbsp;{{{a}}}, &nbsp;{{{b}}}&nbsp; and &nbsp;{{{c}}}&nbsp; are the measures of the triangle sides.

<TABLE>
 <TR>
 <TD>
Let &nbsp;{{{DELTA}}}ABC&nbsp; be a triangle with the side measures &nbsp;{{{a}}}, &nbsp;{{{b}}}&nbsp; and &nbsp;{{{c}}}&nbsp; (Figure 1).&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

As you know, &nbsp;the angle bisectors of any triangle are concurrent &nbsp;(see the lesson

<A HREF=>Angle bisectors of a triangle are concurrent</A>&nbsp; under the topic &nbsp;Triangles&nbsp; of the

section &nbsp;Geometry&nbsp; in this site). &nbsp;Their common intersection point is equidistant

from the sides of the triangle and is the center of the inscribed circle.

In &nbsp;Figure 1&nbsp; the angle bisectors are shown in green; &nbsp;P&nbsp; is their intersection

point, &nbsp;and the inscribed circle is shown in red.

The angle bisectors &nbsp;AP, &nbsp;BP&nbsp; and &nbsp;CP&nbsp; divide the triangle &nbsp;{{{DELTA}}}ABC&nbsp; in three

smaller triangles &nbsp;{{{DELTA}}}APB, &nbsp;{{{DELTA}}}APC&nbsp; and &nbsp;{{{DELTA}}}BPC&nbsp; (Figure 2). &nbsp;Therefore, &nbsp;the

area of the triangle &nbsp;{{{DELTA}}}ABC&nbsp; is the sum of the areas of the triangles &nbsp;{{{DELTA}}}APB,

{{{DELTA}}}APC&nbsp; and &nbsp;{{{DELTA}}}BPC:

</TD>
 <TD>
{{{drawing( 240, 240, 0, 6, 0, 6, 
 line( 1.0, 1.0, 5.0, 1.0), 
 line( 1.0, 1.0, 4.0, 5.0),
 line( 4.0, 5.0, 5.0, 1.0),

locate(1.0, 1.0, A),
            locate(5.0, 1.0, B),
            locate(4.0, 5.4, C),

circle(3.1, 1.0, 0.06, 0.06),
            locate(2.9, 1.0, c),

circle(4.6, 2.8, 0.06, 0.06),
            locate(4.6, 3.2, a),

circle(2.4,  2.9,  0.06, 0.06),
            locate(2.05, 3.05, b),

green(line(4.0, 5.0, 3.1, 1.0)),
      green(line(4.6, 2.8, 1.0, 1.0)),
      green(line(2.4, 2.9, 5.0, 1.0)),

circle(3.35, 2.21, 0.06, 0.06),
            locate(3.20, 2.7, P),

red(circle(3.4, 2.22, 1.2, 1.2))
)}}}
&nbsp;&nbsp;&nbsp;&nbsp;Figure 1. To the formulation&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;of the statement
 </TD>
 <TD>
{{{drawing( 240, 240, 0, 6, 0, 6, 
 line( 1.0, 1.0, 5.0, 1.0), 
 line( 1.0, 1.0, 4.0, 5.0),
 line( 4.0, 5.0, 5.0, 1.0),

locate(1.0, 1.0, A),
            locate(5.0, 1.0, B),
            locate(4.0, 5.4, C),

locate(2.9, 1.0, a),

locate(4.6, 3.2, c),

locate(2.05, 3.05, b),

green(line(1.0, 1.0, 3.4, 2.22)),
      green(line(4.0, 5.0, 3.4, 2.22)),
      green(line(5.0, 1.0, 3.4, 2.22)),

circle(3.35, 2.21, 0.06, 0.06),
            locate(3.20, 2.7, P),

red(circle(3.4, 2.22, 1.2, 1.2)),

red(line (3.38, 2.22, 3.38, 1.0)),
            locate(3.45, 1.9, r),

red(line (3.35, 2.22, 2.46, 2.95)),
            locate(2.7, 2.67, r),

red(line (3.35, 2.20, 4.5, 2.6)),
 locate(4.1, 2.5, r)
)}}}
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 2. To the proof 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;of the statement
 </TD>
 </TR>
</TABLE>

{{{A}}} = {{{A[ABC]}}} = {{{A[APB]}}} + {{{A[APC]}}} + {{{A[BPC]}}}.

The area of each of the three smaller triangle is half the product of the measure of the corresponding side of the original triangle and the radius of the inscribed circle, &nbsp;because the radius drawn to the tangent point is perpendicular to that side and, &nbsp;hence, &nbsp;is the altitude in the smaller triangle. &nbsp;In other words,

{{{A}}} = {{{A[ABC]}}} = {{{ar/2}}} + {{{br/2}}} + {{{cr/2}}} = {{{r*(a + b + c)/2}}} = {{{rs}}}.

It is exactly what has to be proved.

<H3>Example 1</H3>Find the radius of the inscribed circle in the triangle with the side measures of &nbsp;4 cm, &nbsp;13 cm&nbsp; and &nbsp;15 cm.

Solution

The semiperimeter of the triangle is {{{s}}} = {{{(4 +13 + 15)/2}}} = {{{32/2}}} = {{{16}}}.

The area of the triangle, &nbsp;according to the Heron's formula &nbsp;(see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/-Proof-of-the-Heron%27s-formula-for-the-area-of-a-triangle.lesson>Proof of the Heron's formula for the area of a triangle</A>&nbsp; and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/One-more-proof-of-the-Heron%27s-formula-for-the-area-of-a-triangle.lesson>One more proof of the Heron's formula for the area of a triangle</A>

in this site), &nbsp;is

{{{A}}} = {{{sqrt(16*(16-4)*(16-13)*(16-15))}}} = {{{sqrt(16*12*3*1)}}} = {{{sqrt(16*4*9)}}} = {{{8*3}}} = {{{24}}} {{{cm^2}}}.

Now,  &nbsp;using the formula {{{A}}} = {{{rs}}} proved above,  &nbsp;you can calculate the radius of the inscribed circle.  &nbsp;It is

{{{r}}} = {{{A/s}}} = {{{24/16}}} = 1.5 cm.

Answer. &nbsp;The radius of the inscribed circle is &nbsp;1.5 cm.

My other lessons on the topic &nbsp;Area&nbsp; in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/What-is-area.lesson>WHAT IS area?</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Formulas-for-area-of-a-triangle.lesson>Formulas for area of a triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Proof-of-the-formula-for-the-radius-of-the-circumscribed-circle.lesson>Proof of the formula for the radius of the circumscribed circle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-a-parallelogram.lesson>Area of a parallelogram</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-a-trapezoid.lesson>Area of a trapezoid</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-a-quadrilateral.lesson>Area of a quadrilateral</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-a-sircumscribed-quadrilateral.lesson>Area of a quadrilateral circumscribed about a circle</A> &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-a-quadrilateral-inscribed-in-a-circle.lesson>Area of a quadrilateral inscribed in a circle</A>

under the topic &nbsp;Area and surface area&nbsp; of the section &nbsp;Geometry, &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-triangles.lesson>Solved problems on area of triangles</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-right-angled-triangles.lesson>Solved problems on area of right-angled triangles</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-regular-triangles.lesson>Solved problems on area of regular triangles</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-the-radius-of-inscribed-circles-and-semicircles.lesson>Solved problems on the radius of inscribed circles and semicircles</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-the-radius-of-a-circumscribed-circle.lesson>Solved problems on the radius of a circumscribed circle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-Math-circle-level-problem-on-area-of-a-triangle.lesson>A Math circle level problem on area of a triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-parallelograms.lesson>Solved problems on area of parallelograms</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-rhombis-rectangles-and-squares.lesson>Solved problems on area of rhombis, rectangles and squares</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-trapezoids.lesson>Solved problems on area of trapezoids</A> &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-quadrilaterals.lesson>Solved problems on area of quadrilaterals</A>

under the topic &nbsp;Geometry&nbsp; of the section &nbsp;Word problems.

For navigation over the lessons on &nbsp;Area of Triangles&nbsp; use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Surface-area/REVIEW-OF-LESSONS-ON-AREA-OF-TRIANGLES.lesson>OVERVIEW of lessons on area of triangles</A>.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.