The area of a square can be found as follows:

\(\displaystyle Area=x^2\)

Where:

\(\displaystyle x=Side\ Length\)

So we can write:

\(\displaystyle \frac{Area\ 2}{Area\ 1}=\frac{b^2}{a^2}\)

The area of a square is found by multiplying one side by itself.

\(\displaystyle A=s\times s=s^2\)

We are given the side length, allowing us to solve.

\(\displaystyle s=\sqrt{3}\)

\(\displaystyle A=(\sqrt{3})^2=3\)

To find the area of a trapezoid, multiply one half (or 0.5, since we are working with decimals) by the sum of the lengths of its bases (the parallel sides) by its height (the perpendicular distance between the bases). This quantity is

\(\displaystyle A = 0.5 \cdot (7.2 + 13.4) \cdot 10.6 =0.5 \cdot 20.6 \cdot 10.6 = 109.18\textrm{ m}^{2}\)

The area of a trapezoid can be determined using the equation \(\displaystyle A=\frac{1}{2}(b_1+b_2)h\).

\(\displaystyle A=\frac{1}{2}(6+8)(4)\)

\(\displaystyle A=\frac{1}{2}(14)(4)\)

\(\displaystyle A=(7)(4)=28\)

To find the area of a trapezoid, multiply the sum of the bases (the parallel sides) by the height (the perpendicular distance between the bases), and then divide by 2.

\(\displaystyle A = \frac{1}{2} \cdot (7 + 15) \cdot 9 = \frac{1}{2} \cdot 22 \cdot 9 = 99 \textrm{ m}^2\)

The area of a triangle is found by multiplying the base by the height and dividing by two:

\(\displaystyle \small \frac{b\cdot h}{2}\)

In this problem we are given the base, which is \(\displaystyle \small 43\), and the area, which is \(\displaystyle \small 2042.5\).  First we write an equation using \(\displaystyle \small h\) as our variable.

\(\displaystyle \small \frac{43\cdot h}{2}=2042.5\)

To solve this equation, first multply both sides by \(\displaystyle \small 2\), becuase multiplication is the opposite of division and therefore allows us to eliminate the \(\displaystyle \small 2\).

The left-hand side simplifies to:

\(\displaystyle \small \frac{43\cdot h}{2}\cdot 2=43\cdot h\)

The right-hand side simplifies to:

\(\displaystyle \small 2042.5\cdot 2=4085\)

So our equation is now:

\(\displaystyle \small \small 43\cdot h=4085\)

Next we divide both sides by \(\displaystyle \small \small 43\), because division is the opposite of multiplication, so it allows us to isolate the variable by eliminating \(\displaystyle \small \small 43\).

\(\displaystyle \small \frac{43\cdot h}{43}= h\)

\(\displaystyle \small \frac{4085}{43}=95\)

\(\displaystyle \small h=95\)

So the height of the triangle is \(\displaystyle \small 95\).

The area of a triangle is one half the product of its base and its height - in the above diagram, that means

\(\displaystyle A = \frac{1}{2}xy\).

Substitute \(\displaystyle A = 36, x = 4.5\), and solve for \(\displaystyle y\).

\(\displaystyle \frac{1}{2} \cdot 4.5 \cdot y = 36\)

\(\displaystyle 2.25 y = 36\)

\(\displaystyle 2.25 y \div 2.25= 36\div 2.25\)

\(\displaystyle y = 16 \textrm{ in}\)

From this shape we are able to see that we have a square and a triangle, so lets split it into the two shapes to solve the problem. We know we have a square based on the 90 degree angles placed in the four corners of our quadrilateral. 

Since we know the first part of our shape is a square, to find the area of the square we just need to take the length and multiply it by the width. Squares have equilateral sides so we just take 5 times 5, which gives us 25 inches squared.

We now know the area of the square portion of our shape. Next we need to find the area of our right triangle. Since we know that the shape below the triangle is square, we are able to know the base of the triangle as being 5 inches, because that base is a part of the square's side. 

To find the area of the triangle we must take the base, which in this case is 5 inches, and multipy it by the height, then divide by 2. The height is 3 inches, so 5 times 3 is 15. Then, 15 divided by 2 is 7.5. 

We now know both the area of the square and the triangle portions of our shape. The square is 25 inches squared and the triangle is 7.5 inches squared. All that is remaining is to added the areas to find the total area. Doing this gives us 32.5 inches squared. 

Area of a triangle can be determined using the equation:

\(\displaystyle \small A=\frac{1}{2}bh\)

\(\displaystyle \small A=\frac{1}{2}(14)(5)=35\)

In order to find the area of a triangle, we multiply the base by the height, and then divide by 2.

\(\displaystyle \small \frac{b\cdot h}{2}\)

In this problem we are given the base and the area, which allows us to write an equation using \(\displaystyle \small h\) as our variable.

\(\displaystyle \small \frac{8\cdot h}{2}=40\)

Multiply both sides by two, which allows us to eliminate the two from the left side of our fraction.

The left-hand side simplifies to:

\(\displaystyle \small \frac{8\cdot h}{2}\cdot 2=8\cdot h\)

The right-hand side simplifies to:

\(\displaystyle \small 40\cdot 2=80\)

Now our equation can be rewritten as:

\(\displaystyle \small 8\cdot h=80\)

Next we divide by 8 on both sides to isolate the variable:

\(\displaystyle \small \frac{8\cdot h}{8}=h\)

\(\displaystyle \small \frac{80}{8}=10\)

\(\displaystyle \small h=10\)

Therefore, the height of the triangle is \(\displaystyle \small 10 \: feet\).