Height of a Scalene Triangle
Where:
A = the area of the triangle
b = the base of the triangle
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The process of finding the height of scalene triangles (triangles whose sides are all different lengths,) is a little more difficult than that of an isosceles or equilateral triangle. This is because you must first find the area of a scalene triangle before you can find its height.
Why is it Harder Than it is for Isosceles and Equilateral Triangles?
All you need to do to find the height of isosceles and equilateral triangles is to draw a line from the top of the triangle down to the base. This line will meet the base at its midpoint, thus splitting the base of the triangle in half. This creates two right triangles and with one of these you can use the Pythagorean Theorem to find the height of the triangle.
If you draw a line starting from the top point of a scalene triangle to its base, though, where you end will not be the midpoint of the base and may not even be inside of the triangle. Therefore, you won't know the value of either side of the right triangle that your line created and you won't be able to calculate its height.
Ways of Finding Area
Sine
One way of finding the area of a scalene triangle is to use the trigonometric function Sine. The formula is:
Sine is a trigonometric function that uses an angle in a triangle to find the value of one or more of the triangl's sides. This means that in our case, you must know the angle of C to use the equation to find the area of triangle ABC.
For more information on Sine and its calculator, click this links in this sentence.
Heron/Hero's Formula
If you aren't given any angles of the triangle and you know what its three sides are, you can use Heron's Formula to find its area.
Heron's Formula is a seemingly complicated equation that uses the value of a triangle's three sides to find its area. Why is it seemingly complicated? Well, at first sight it looks much harder to work with than it actually is. Don't let it intimidate you too much.
For a much deeper explanation of this formula and its calculator, click the links provided in this sentence.
Once you have the area of the triangle, you can go on to calculate its height. The formula for that is:
When this equation is solved for h, it becomes the equation our calculator uses (the equation for height that you see above.)
Example 1
The sides of a scalene triangle are 4 in, 5 in, and 6 in , the base being 4in. Use Heron's Formula to find its area and then find its height.
Heron's Formula:
Plug 7.5 as s and the values for a, b, and c into the overall equation:
Since our calculator rounds our answer to four decimals, we'll round that answer to four decimals to make it 9.9216 in².
Now that we know the area of our triangle is 9.9216, let's plug that into our equation:
Given angle W is 36º, side j is 6 in, and side a is 9 in, find the area of the triangle AWJ. Then find its height.
With our equation:
Now, we find the Sine of 60 degrees. This is most easily done by using our Sine calculator.
According to the calculator:
Sin 60 = 0.86602540378444
Now, we plug into our equation:
Now, divide:
46.7654 / 9 = 5.1961555...
The height of triangle AWJ is 5.1962.
Related Websites:
http://mathcentral.uregina.ca/QQ/database/QQ.09.04/rao1.html
http://www.ajdesigner.com/phptriangle/scalene_triangle_area_height.php
http://home.alltel.net/okrebs/page93.html
Works Cited
Bottomley, Henry. "Area of a Triangle." Henry Bottomley. October 2001. Henry
Bottomley. 30 June 2006. <http://www.btinternet.com/~se16/hgb/triangle.htm>.
"AJ Triangle Formulas Calculator." AJ Design Software. 2005. AJ Online Design and
Engineering Software. 13 June 2006.
<http://www.ajdesigner.com/phptriangle/scalene_triangle_area_height.php>.
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