Area of a Rectangle: Using additional geometric shapes

We can calculate the area of rectangle ABCD:

$20\times30=600$

Let's divide the deltoid along its length and width and add the following points:

Now we can calculate the area of deltoid PMNK:

$PMNK=\frac{PN\times MK}{2}=\frac{20\times30}{2}=\frac{600}{2}=300$

The area of the rectangle is equal to length multiplied by width.

Let's set up the data in the formula:

$27=3x\times x$

$27=3x^2$

$\frac{27}{3}=\frac{3x^2}{3}$

$9=x^2$

$x=\sqrt{9}=3$

Let's calculate side BG using the Pythagorean theorem:

$BG^2+GC^2=BC^2$

We'll substitute the known data:

$BG^2+3^2=5^2$

$BG^2+9=25$

$BG^2=16$

$BG=\sqrt{16}=4$

Now we can calculate the area of rectangle ABGE since we have the length and width:

$5\times4=20$

Let's first calculate the sides of the rectangle:

$AEDF=AE\times ED$

Let's input the known data:

$35=7\times ED$

Let's divide the two legs by 7:

$ED=5$

Since AEDF is a rectangle, we can claim that:

ED=FD=7

Let's calculate side CD:

$2+7=9$

Let's calculate the area of parallelogram ABCD:

$ABCD=CD\times ED$

Let's input the known data:

$ABCD=9\times5=45$

The area of rectangle ABCD equals length multiplied by width.

Let's input the known data into the formula in order to find side AB:

$36=3\times AB$

Let's divide both sides by 3:

$AB=12$

Since we are given that AE equals a quarter of AB, we can substitute the known data and calculate side AE:

$AE=\frac{1}{4}AB=\frac{1}{4}\times12=3$

The area of rectangle ABCD equals length multiplied by width.

Let's use the known data in the formula to find side AB:

$70=7\times AB$

Let's divide both sides by 7:

$AB=10$

Since we are given that EB equals one-fifth of AB, we can use the known data and calculate side EB:

$EB=\frac{1}{5}AB=\frac{1}{5}\times10=2$

The area of rectangle ABCD equals length times width.

Let's write down the known data in a formula to find side CD:

$45=5\times CD$

Let's divide both sides by 5:

$CD=9$

We know that in a rectangle, each pair of opposite sides are equal, therefore:

$AB=CD=9$

Since we are given that ED equals one-third of CD, we can substitute the known data and calculate side ED:

Remember that AB equals CD

$ED=\frac{1}{3}AB=\frac{1}{3}\times9=3$

Let's look at triangle ADK to calculate side AD:

$AD^2+DK^2=AK^2$

Let's input the given data:

$AD^2+4^2=5^2$

$AD^2+16=25$

We'll move 16 to the other side and change the appropriate sign:

$AD^2=25-16$

$AD^2=9$

We'll take the square root and get:

$AD=3$

Since AD is a side of rectangle ABCD, we can calculate side AB as follows:

$S=AB\times AD$

Let's input the given data:

$24=3\times AB$

We'll divide both sides by 3:

$AB=8$

Let's draw the center of the circle and we can divide the diameter of the circle into two equal radii

Now let's calculate the length of the radii as follows:

$7\times2r=28x$

$14r=28x$

We'll divide both sides by 14:

$r=\frac{28}{14}x$

$r=2x$

Let's calculate the circumference of the circle:

$P=2\pi\times r=2\pi\times2x=4\pi x$

Let's calculate the area of triangle ABC:

$8x=\frac{(x+4)x}{2}$

Multiply by 2:

$16x=(x+4)x$

Divide by x:

$16=x+4$

Let's move 4 to the left side and change the sign accordingly:

$16-4=x$

$12=x$

Now let's calculate the area of the rectangle, multiply the length and width where BC equals 12 and AB equals 16:

$16\times12=192$

Since we are given the area of the rectangle, let's find the length of the missing side:

$7\times a=28$

We'll divide both sides by 7 and get:

$a=4$

Since in a hexagon all sides are equal to each other, each side is equal to 4.

Now let's calculate the area of the hexagon:

$\frac{6\times a^2\times\sqrt{3}}{4}$

$\frac{6\times4^2\times\sqrt{3}}{4}$

Let's simplify the exponent in the denominator of the fraction and we'll get:

$6\times4\times\sqrt{3}=24\times\sqrt{3}=41.56$