Lesson Solved problems on area of triangles
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<H2>Solved problems on area of triangles</H2> <H3>Problem 1</H3>Find the area of a triangle, if one of its sides is 5 cm long and the altitude drawn to this side has the measure of 6 cm. <B>Solution</B> The area of a triangle equals half the product of the measures of any of its sides and the altitude drawn to this side. In our case the area of the triangle equals {{{5*6/2}}} = {{{30/2}}} = 15 {{{cm^2}}}. <B>Answer</B>. The area of the triangle is 15 {{{cm^2}}}. <H3>Problem 2</H3>A triangle has two sides of 4 cm and 8 cm long. The altitude drawn to the side of 8 cm has the measure of 3 cm. Find the measure of the altitude drawn to the side which is 4 cm long. <B>Solution</B> Let <B>x</B> be the measure of the altitude drawn to the side which is 4 cm long. Since the area of a triangle equals half the product of the measure of any of its sides and the measure of the altitude drawn to this side, you can write the equation {{{4*x/2}}} = {{{8*3/2}}}. Solving this equation for <B>x</B>, you get {{{x}}} = {{{8*3/4}}} = {{{24/4}}} = 6 cm. <B>Answer</B>. The altitude drawn to the side which is 4 cm long has the measure of 6 cm. <H3>Problem 3</H3>Find the area of a triangle circumscribed about the circle of the radius of 5 cm, if the perimeter of the triangle is of 60 cm. <B>Solution</B> You can use the formula expressing the area of the triangle via its perimeter and the radius of the inscribed circle {{{S}}} = {{{P*r/2}}}, where {{{P}}} is the perimeter of the triangle and {{{r}}} is the radius of the inscribed circle (see the lesson <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Formulas-for-area-of-a-triangle.lesson>Formulas for area of a triangle</A> under the topic <B>Area and Surface Area</B> of the section <B>Geometry</B> in this site). In our case {{{S}}} = {{{60*5/2}}} = {{{150}}} {{{cm^2}}}. <B>Answer</B>. 150 {{{cm^2}}}. <H3>Problem 4</H3>Find the area of a triangle, if its sides are of 13 cm, 14 cm and 15 cm long. <B>Solution</B> You can use the Heron's formula (see the lesson <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Formulas-for-area-of-a-triangle.lesson>Formulas for area of a triangle</A> under the topic <B>Area and Surface Area</B> of the section <B>Geometry</B> in this site) {{{S}}} = {{{sqrt(p*(p-a)*(p-b)*(p-c))}}}, where {{{a}}}, {{{b}}} and {{{c}}} are the measures of the triangle sides and {{{p}}} is its semi-perimeter {{{p}}} = {{{(a+b+c)/2}}}. In our case {{{p}}} = {{{(13 + 14 + 15)/2}}} = 21 cm and {{{S}}} = {{{sqrt(21*(21-13)*(21-14)*(21-15))}}} = {{{sqrt(21*8*7*6)}}} = 84 {{{cm^2}}}. <B>Answer</B>. 84 {{{cm^2}}}. <H3>Problem 5</H3>Find the shortest altitude of a triangle, if its sides are of 5 cm, 5 cm and 8 cm long. <B>Solution</B> First, let us find the area of the triangle. You can use the Heron's formula: {{{S}}} = {{{sqrt(p*(p-a)*(p-b)*(p-c))}}} = {{{sqrt(9*(9-5)*(9-5)*(9-8))}}} = {{{sqrt(9*4*4*1)}}} = 12 {{{cm^2}}}. Now, let <B>x</B> be the measure of the altitude drawn to the side which is 8 cm long. Since the area of a triangle equals half the product of the measure of any of its sides and the measure of the altitude drawn to this side, you can write the equation {{{8*x/2}}} = {{{12}}}. Solving this equation for <B>x</B>, you get {{{x}}} = {{{12*2/8}}} = {{{24/8}}} = 3 cm. It is obvious that this altitude is the shortest one. <B>Answer</B>. The shortest altitude is 3 cm long. 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/-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> - <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-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>.
