Area of a Rectangle: Subtraction or addition to a larger shape

As we know that EFGD is a rectangle, we also know that DE is equal to 2 and DG is equal to 4

In a rectangle, each pair of opposite sides are equal and parallel, therefore:

$ED=FG=2$

$DG=EF=4$

Now we calculate the area of the orange rectangle EFGD by multiplying the length by the width:

$2\times4=8$

Now we calculate the total area of the white rectangle ABCD:

$AD=AE+ED=2+2=4$

$DC=DG+GC=4+5=9$

The area of the entire rectangle ABCD is:

$4\times9=36$

Now to find the area of the white part that is not covered by the area of the orange rectangle, we will subtract the area of the rectangle EFGD from the rectangle ABCD:

$36-8=28$

Let's calculate the side DC:

$3+5=8$

Now we can calculate the area of square ABCD:

$4\times8=32$

Let's calculate the area of triangle ADE:

$\frac{4\times3}{2}=\frac{12}{2}=6$

Let's calculate the area of triangle FCE:

$\frac{5\times1.5}{2}=\frac{7.5}{2}=3.75$

Now let's calculate the area of AEFB by subtracting the other areas we found:

$AEFB=32-6-3.75=22.25$

The area of the entire shape equals the area of the rectangle plus the area of each of the semicircles.

Let's label each semicircle with a number:

Therefore, we can claim that:

The area of the entire shape equals the area of the rectangle plus 2A1+2A3

Let's calculate the area of semicircle A1:

$\frac{1}{2}\pi r^2$

$\frac{1}{2}\pi4^2=8\pi$

Let's calculate the area of semicircle A3:

$\frac{1}{2}\pi r^2$

$\frac{1}{2}\pi2^2=2\pi$

The area of the rectangle equals:

$4\times8=32$

Now we can calculate the total area of the shape:

$32+2\times8\pi+2\times2\pi=32+16\pi+4\pi=32+20\pi$

Let's first calculate the area of triangle ABC:

$\frac{6\times7}{2}=\frac{42}{2}=21$

Since the shape before us is a rectangle, we can claim that:

AC=GD=7

Now let's calculate the area of triangle HGD:

$\frac{7\times3}{2}=\frac{21}{2}=10.5$

Let's draw an imaginary line between B and H to get square BEFH where each side equals 3 cm.

Let's calculate the area of BEFH:

$3\times3=9$

Let's calculate the area of rectangle ACDG:

$7\times12=84$

Now we can calculate the area of the brown shape by subtracting the other areas we found: