calculation of area and perimeter of cleared areas (without using integral calculation)

Question

How can I find the area and perimeter of [these cases], can you help me?

Thanks

Answer 1

Let [.] denote areas of various shapes below.

Case 1.

One of the little grey areas is

$$I_1 = [ABCD] - [CDS] - 2[DAS]$$

where the areas of the triangle CDS and the circle sector DAS are given by

$$[CDS] = \frac{\sqrt 3}{4}L^2, \>\>\> [DAS] = \frac{\pi}{12}L^2$$

Thus,

$$ I_1 = \left(1- \frac{\sqrt 3}{4} - \frac{\pi}{6}\right)L^2 $$

and its perimeter is $\left( 1+\frac{\pi}{3}\right)L$.

Case 2.

One of the four areas in the second case is

$$ I_2 = [ABCD] - [ADB] - 2I_1 = \left(\frac{\sqrt 3}{2}-1 + \frac{\pi}{12}\right)L^2 $$

and its perimeter is $\frac{\pi}{2}L$.

Case 3.

The area in the case of the middle area,

$$ I_3 = (1-4I_1-4I_2)L^2=\left(1-\sqrt 3 + \frac{\pi}{3}\right)L^2$$

with perimeter $\frac{2\pi}{3}L$.

Answer 2

Call the area in the first picture A, the one in the second picture 4B, the one on the third picture 4C.

Then using equivalence of areas we get that: \ $A+4B+4C=L^{2} \\ A+3B+2C=\frac{\pi L^{2}}{4} \\ A+2B=\frac{(\pi - 2)L^{2}}{2}$ \

Solve the equations for A, B, C.

Answer 3

BIG HINT

An area of $A+2B+C$ is formed by the union of two sectors of angle $60^o$ of a circle of radius $L$. These sectors overlap in an equilateral triangle of side $L$. This gives us the extra equation we need $$A+2B+C=\frac{1}{3} \pi L^2-\frac{\sqrt(3)}{4}L^2.$$