altitude of a equilateral

Let ABC be the triangle.

Let AD be the median from A to BC. In an equilateral triangle, median is same as the height.

So, Angle of ADC = 90 degrees.

So, ADC is a right angled triangle with AC as the hypotenuse.

If 'a' is the length of a side,

AC = a units
DC = (a/2) units. (as AD is the median to BC).

Using Pythagorus theorem, one can deduce that AD = height of the equi triangle = side * sqrt(3)/2

Hope this helps. Thanks.

If you know the ratio of the sides of a 30-60-90 triangle you can quickly get to the altitude of an equilateral. Try drawing a 30-60-90. Your short leg will be x, the hypotenuse 2x and the other leg x times the square root of three. Then rotate that triangle around the longer leg (the x root3 leg). You will have a 60-60-60 triangle (equilateral). The sides will be 2x and the altitude x root 3.

Similarly, the 45-45-90 triangle when rotated around the hypotenuse gives you a square. So the diagonal of a square is always the side times the square root of 2.

Two great short cuts when dealing with mixed figures!