proove that the area of a circle with radiur r = area of a rectangle with breadth r and length circumference/2

Answer 1

area of circle eqn = pi*r^2
area of rectangle = r*circumference/2

these must equal the same

circumference = pi*diametre = pi*2*r

cancel out the 2 form the area of the rectangle and -


pi*r^2 = pi*r*r

=> pi*r^2 = Pi*r^2

Answer 2

Is this the sort of thing you mean?

http://www.mathguide.com/bbs/Messages2002/2665.html

The formula for the area of a circle can be derived from the formula for the circumference and the formula for the area of a triangle, as follows. Imagine a regular hexagon (six-sided figure) divided into equal triangles, with their apices at the centre of the hexagon. The area of the hexagon may be found by the formula for triangle area by adding up the lengths of all the triangle bases (on the exterior of the hexagon), multiplying by the height of the triangles (distance from the middle of the base to the center) and dividing by two. This is an approximation of the area of a circle. Then imagine the same exercise with an octagon (eight-sided figure), and the approximation is a little closer to the area of a circle. As a regular polygon with more and more sides is divided into triangles and the area calculated from this, the area becomes closer and closer to the area of a circle. In the limit, the sum of the bases approaches the circumference 2πr, and the triangles' height approaches the radius r. Multiplying the circumference and radius and dividing by 2, we get the area, π r².

Answer 3

length of the rectangle=circumference/2
circumference of a circle=2*(pi)*r
Thus, length of the rectangle=2*(pi)*r/2=(pi)*r
Thus, Area of the rectangle=l*b=[(pi)*r]*r=(pi)r^2
Also, Area of the circle=(pi)r^2 [Formula]
Therefore, Area of the circle= Area of the rectangle

Answer 4

It's obvious.

area of the circle = (pi)r^2

area of the rectangle = breath * length = r(2(pi)r/2) = r((pi)r) = (pi)r^2

area of the circle = area of the rectangle. #

Answer 5

Given:
radius of circle 'R' = r
breadth of rectangle 'b' = r
length of rectangle 'l' = (circumference of circle)/2
= 2(pi)(r)/2 -->formula
=(pi)(r)
To Prove:
area of circle=area of rectangle

Solution:
Area of circle=(pi)(r^2)
Area of rectangle='l' * 'b'
=(pi)(r)*(r)
=(pi)(r^2) -->from given
=area of circle

hence proven.

Answer 6

no... YOU prove it!