A circle is a locus of points equidistant from a given point which is the center of the circle. The common distance from the center of a circle to its point is called radius.
Thus, the circle is entirely defined by its center (o) and radius R.
Any lining passing through the center and touched the two end on the circle, for e.g., the AC line on the circle is called the diameter of the circle.The diameter divides the circle into two halves such that they are equal in area. These two halves are referred as the semicircle.
Let us consider a point anywhere in the second half of the circle. Let it be point B
Let us join point A and point C to this point. It is one of the unique property of a semicircle that the triangle formed such that it always have a right angle triangle with AC as the base and AB and BC as respective chords.
Talking about the area of a semicircle
Since semicircle is half that of a circle, hence the area will be half that of a circle.
The area of a circle is the number of square units inside that circle.
Let us generate the following image. This polygon can be broken into n isosceles triangle(equal sides being radius)
The area of this triangle is given as ½(h*s)
Now for n number of polygons the area of a polygon is given as
The term n*s is equal to the perimeter of the polygon, as the polygon gets to look more and more like a circle , the value approaches the circle circumference, which is 2*3.14*r . So substituting 2*3.14*r for n*s.
Also, as the number of sides increases, the triangle gets narrower and so when s approaches zero, h and r have the same length. So substituting r for h:
Polygon area = h/2(2*3.14*r)
Rearranging this we get
Now the area of a semicircle is equal to half of that of a full circle
Therefore, the area of a semicircle is equal to