Lesson Proof of the Heron's formula for the area of a triangle
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<H2>Proof of the Heron's formula for the area of a triangle</H2> In this lesson you will learn the proof of the Heron's formula for the area of a triangle: {{{S}}} = {{{sqrt(s(s-a)(s-b)(s-c))}}} = {{{sqrt(((a+b+c)/2)*((a+b-c)/2)*((a-b+c)/2)*((b+c-a)/2))}}}, where {{{a}}}, {{{b}}} and {{{c}}} are the measures of a triangle sides, {{{S}}} is the triangle area and {{{s}}} is the triangle semiperimeter: {{{s}}} = {{{(a + b + c)/2}}}. <TABLE> <TR> <TD> Let {{{DELTA}}}<B>ABC</B> be a triangle with the side measures {{{a}}}, {{{b}}} and {{{c}}} (<B>Figure 1</B>). The area of the triangle {{{DELTA}}}<B>ABC</B> equals half product of the measure of its side {{{a}}} = |<B>BC</B>| and the measure of the altitude {{{h}}} = |<B>AD</B>| drawn to this side: {{{S}}} = {{{a*h/2}}}.   (1) By the <A HREF=http://www.algebra.com/algebra/homework/Pythagorean-theorem/The-Pythagorean-Theorem.lesson>Pythagorean theorem</A> you have {{{c^2}}} = {{{h^2}}} + {{{d^2}}} and {{{b^2}}} = {{{h^2}}} + {{{(a-d)^2}}} </TD> <TD> {{{drawing( 320, 200, -0.5, 7.5, -0.5, 3.5, line( 0.0, 0.0, 7.0, 0.0), line( 0.0, 0.0, 4.0, 3.0), line( 4.0, 3.0, 7.0, 0.0), locate( -0.2, 0.0, B), locate( 6.9, 0.0, C), locate( 3.9, 3.4, A), line( 4.0, 3.0, 4.0, 0.0), locate( 3.9, 0.0, D), locate( 3.1, 0.0, a), locate( 5.7, 1.7, b), locate( 1.6, 1.7, c), locate( 4.1, 1.7, h), locate( 2.3, 0.4, d), locate( 4.7, 0.4, a-d) )}}} <B>Figure 1</B>. To the proof of the Heron's formula </TD> </TR> </TABLE> in accordance with the <B>Figure 1</B>. Distract the second equality from the first one. You will get {{{c^2}}} - {{{b^2}}} = {{{2ad}}} - {{{a^2}}}. Thus you can express {{{d}}} via the triangle side measures {{{d}}} = {{{(a^2 + c^2 - b^2)/2a}}}. Next, you can express {{{h}}} via the triangle side measures {{{h^2}}} = {{{c^2}}} - {{{d^2}}} = {{{c^2}}} - {{{((a^2 + c^2 - b^2)/2a)^2}}} = {{{((2ac)/(2a))^2}}} - {{{((a^2 + c^2 - b^2)/2a)^2}}} = {{{((2ac)^2 - (a^2 + c^2 - b^2)^2)/(4a^2)}}} = {{{((2ac - a^2 -c^2 + b^2)*(2ac + a^2 + c^2 - b^2))/(4a^2)}}} = {{{(b^2 - (a-c)^2)*((a+c)^2 -b^2)/(4a^2)}}} = {{{(b-a+c)*(b+a-c)*(a+c-b)*(a+c+b)/(4a^2)}}} So, h = {{{(1/2a)*sqrt((a+b+c)*(a+b-c)*(a+c-b)*(b+c-a))}}}. Substitute it into the formula (1). You will get for the area of the triangle {{{S}}} = {{{sqrt(((a+b+c)/2)*((a+b-c)/2)*((a-b+c)/2)*((b+c-a)/2))}}} = {{{sqrt(s(s-a)(s-b)(s-c))}}}, exactly as the Heron's formula states. <H3>Example 1</H3>Find the area of a triangle with the side measures of 4 cm, 13 cm and 15 cm. <B>Solution</B> The semiperimeter of the triangle is {{{s}}} = {{{(4 +13 + 15)/2}}} = {{{32/2}}} = {{{16}}}. The area of the triangle, according to the Heron's formula, is {{{S}}} = {{{sqrt(16*(16-4)*(16-13)*(16-15))}}} = {{{sqrt(16*12*3*1)}}} = {{{sqrt(16*4*9)}}} = {{{8*3}}} = {{{24}}} {{{cm^2}}}. <B>Answer</B>. The area of the triangle is of 24 {{{cm^2}}}. <H3>Example 2</H3>Find the measure of the altitude of a triangle with the side measures of 4 cm, 13 cm and 15 cm drawn to the shortest side. <B>Solution</B> We just have found the area of this triangle in the solution for the <B>Example 1</B>. It is of {{{24}}} {{{cm^2}}}. From the formula for the area of a triangle {{{S}}} = {{{ah/2}}} you have for the altitude {{{h}}} = {{{2S/a}}}. Substituting here {{{S}}} = {{{24}}} {{{cm^2}}} and {{{a}}} = {{{4}}} {{{cm}}} you get {{{h}}} = {{{2*24/4}}} = {{{48/4}}} = {{{12}}} {{{cm}}}. <B>Answer</B>. The altitude of the triangle drawn to its shortest side is of 12 {{{cm}}}. My other lessons on the topic <B>Area</B> in this site are - <A HREF=http://www.algebra.com/algebra/homework/Surface-area/What-is-area.lesson>WHAT IS area?</A> - <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Formulas-for-area-of-a-triangle.lesson>Formulas for area of a triangle</A> - <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> - <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Proof-of-the-formula-for-the-area-of-a-triangle-via-the-radius-of-the-inscribed-circle.lesson>Proof of the formula for the area of a triangle via the radius of the inscribed circle</A> - <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> - <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-a-parallelogram.lesson>Area of a parallelogram</A> - <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-a-trapezoid.lesson>Area of a trapezoid</A> - <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-a-quadrilateral.lesson>Area of a quadrilateral</A> - <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> and - <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 <B>Area and surface area</B> of the section <B>Geometry</B>, and - <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-triangles.lesson>Solved problems on area of triangles</A> - <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> - <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> - <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> - <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> - <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> - <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-parallelograms.lesson>Solved problems on area of parallelograms</A> - <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> - <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-trapezoids.lesson>Solved problems on area of trapezoids</A> and - <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 <B>Geometry</B> of the section <B>Word problems</B>. For navigation over the lessons on <B>Area of Triangles</B> use this file/link <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 <A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.
