Area of an equilateral triangle
The area of an equilateral triangle can be found by using the pythagorean formula:
Start with any equilateral triangle
Label the sides.
Draw the perpendicular bisector of the equilateral triangle as shown below. Note how the perpendicular bisector
breaks down side a into its half or a/2
Now apply the pythagorean theorem to get the height (h) or the length of the line you see in red
a
^{2} = (a/2)
^{2} + h
^{2}
a
^{2} = a
^{2}/4 + h
^{2}
a
^{2} − a
^{2}/4 = h
^{2}
4a
^{2}/4 − a
^{2}/4 = h
^{2}
3a
^{2}/4 = h
^{2}
h = √(3a
^{2}/4)
h = (√(3)×a)/2
Area = (base × h)/2
base × h = (a × √(3)×a)/2 = (a
^{2}× √(3))/2
Dividing by 2 is the same as multiplying the denominator by 2. Therefore, the formula is
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