Height of an Isosceles or Equilateral Triangle
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Every triangle has three sides. The bottom side of the triangle that rests on the "ground" is called the base. The two sides coming up from the ends of the base are referred to as the triangle's sides.
If the lengths of all the triangle's sides are the same, it is called an equilateral triangle.
If two of the sides are the same length it's called an isosceles triangle.
If all three sides are different lengths, it's called a scalene triangle.
Note: This help file focuses on only equilateral and isosceles triangles. For the help file for the height of a scalene triangle click the link in this sentence.
The height of a triangle is how high up the top point of the triangle reaches. There is not one single formula that is used to calculate an isosceles or equilateral triangle's height. Instead, there are a couple of small steps:
- Find the midpoint of the base. This is done by dividing the base by two:
After the midpoint is found, draw a line extending from it up to the top point of the triangle. This line represents the height of your triangle:
That will give you two right triangles. Using one of these triangles, you can use the Pythagorean Theorem to find the length of the line you drew through the center of the triangle to represent its height, thus finding the triangle's height.
2) Pythagorean Theorem
a² + b² = c²
The Pythagorean Theorem is used to find the length of a side of a right triangle. a stands for the first side, b stands for the second side, and c stands for the hypotenuse. The hypotenuse is the longest side of the triangle and is the only side not formed by a right angle.
When you know the sides of a right triangle that correspond to b and c, you can plug them into the Pythagorean Theorem to get the value of a, which, if you haven't noticed, will be the height of our original triangle.
If you're not sure how to calculate the value of b in one of the right triangles that make up our original triangle, go back to the first step where you found the base's midpoint.
The midpoint splits the triangle's base exactly in half. Since one of these halves makes up the base of the right triangle you're working with, the value you got for the midpoint is also the value you use for b in the Pythagorean Theorem.
Now that you have b and c, simply plug them into the Pythagorean Theorem and solve for a, the height of the triangle:
a² + b² = c²
Get a² by itself on one side of the equal sign by subtracting b² on both sides:
Example 1
The base of an isosceles triangle is 9 and its sides are both 7. Find the height of the triangle.
First, find the midpoint of the base:
9 / 2 = 4.5
Then plug that in for either a or b in the Pythagorean Theoremand plug in one of the sides of the original triangle for c in the Pythagorean Theorem:
a² + b² = c²
a² + (4.5)² = (7)²
Get a² by itself on one side of the equal sign:
What About Right Triangles?
Right triangles' heights are easier to find than any other triangle's height. It is simply the value of the right triangle's only straight vertical line (the one that's neither the base nor the hypotenuse.)
If you are not given the value of this line of your right triangle, simply use the Pythagorean Theorem to calculate it.
Helpful Definitions
Midpoint: the very center of any line from which both sides of the line are equal
Right triangle: a triangle that contains one angle that is ninety degrees
Pythagorean Theorem: the equation from which one can calculate one side of a right
triangle
Hypotenuse: The longest line of a right triangle that lies directly across from the right angle
Right angle: an angle of two perpendicular lines which always comes to be 90 degrees
Related websites:
http://mathcentral.uregina.ca/QQ/database/QQ.09.04/rao1.html
Works Cited
"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>.
"Quandaries and Queries." Math Central. Imperial Oil Limited. 13 June 2006.
<http://mathcentral.uregina.ca/QQ/database/QQ.09.02/dean1.html>.
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