Practical_Antenna_Handbook_0071639586

720 A p p e n d i x A : U s e f u l M a t h
Area of a Triangle
The area inside a triangle is given by
1
A = bh
(A.4.6)
2
where b is the base of a triangle of any shape and h is its height, obtained by creating a
right triangle for measurement purposes, as shown in Fig. A.4.4. Start by defining any
side of the original triangle as its base, then construct a right triangle inside or outside
the original one to reach the highest point of the triangle at the vertex opposite the base.
Of course, for a right triangle, the simplest orientation is to select the two sides a and b
as the base and height (or vice versa).
In an equilateral triangle, there is no way to tell one side from any other (because
they are all the same length by definition), and there is no way to distinguish one interior
angle from another. Therefore, each interior angle must be exactly 60 degrees. If we
draw a line from the junction of any pair of sides to the middle of the third side, we create
two right triangles inside the original equilateral triangle, as shown in Fig. A.4.5. If
the sides are of length d, for instance, then either right triangle has a hypotenuse of
length d and one of the other two sides is d/2. From that information, we can find the
height of an equilateral triangle by using what we know about right triangles:
2 2 2
2 d 4d d
h = d − ⎛ ⎝ ⎜ ⎞ −
2⎠
⎟ =
4
3
= d = d 0.75 = 0.866 d (A.4.7)
4
In other words, the height of an equilateral triangle is always 0.866 times the length of
any leg. Then the area inside the equilateral triangle is
1 1
2
A = bh = d(0.866 d) = 0.433d
(A.4.8)
2 2
Finally, if we cut a square in half by drawing a straight line between two opposing
corners, we create two isosceles triangles. But these are special because one of the inte-
h
a
c
A (area) = 1 bh
2
where h = c sin
Figure A.4.4 Finding the area inside a triangle.
b