Area of a Rectangle: Using additional geometric shapes

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:

$5\times4=20$

We can calculate the area of rectangle ABCD as follows:

$20\times30=600$

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

Finally, we can calculate the area of deltoid PMNK as follows:

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

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 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 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 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:

$16\times12=192$

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 draw a line in the middle of the drawing that divides the house into 2

Meaning it divides the triangle and the rectangular part.

The 2 lines represent the heights in both shapes.

If we connect the height of the roof with the height of the rectangular part, we get the total height

Let's put the known data in the formula:

$\frac{1}{2}h_{\text{square}}+h_{square}=12x+9$

$\frac{3}{2}h_{\text{square}}=12x+9$

We'll multiply by two thirds and get:

$h_{\text{square}}=\frac{2(12x+9)}{3}=2(4x+3)$

$h_{\text{square}}=8x+6$

If the height of the triangle equals half the height of the rectangular part, we can calculate it using the following formula:

$h_{\text{triangle}}=\frac{1}{2}(8x+6)=4x+3$

Now we can calculate the area of the triangular part:

$\frac{(x+2y)\times(4x+3)}{2}=\frac{4x^2+3x+8xy+6y}{2}=2x^2+1.5x+4xy+3y$

Now we can calculate the rectangular part:

$(x+2y)\times(8x+6)=8x^2+6x+16xy+12y$

Now let's combine the triangular area with the rectangular area to express the total area of the shape:

$S=2x^2+1.5x+4xy+3y+8x^2+6x+16xy+12y$

$S=10x^2+20xy+7.5x+15y$

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

$7\times a=28$

We'll now divide both sides by 7 to get:

$a=4$

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

Now let's calculate the area of the hexagon:

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

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

Finally, we simplify the exponent in the denominator of the fraction to get:

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