# Area of "My Space" Project

## What is this?

The link above will send you to a Google Drawing of the neighborhood Bayberry in Little River, South Carolina. On this drawing, you can see two triangles. I used these two triangles to find all of the different angles and length of the sides. The numbers in black are the ones given to me on Google Maps and Geogebra. The numbers in red were found by using the Law of Sines and the Law of Cosines.

## Law of Cosines Triangle ABC

In Triangle ABC, we are given the lengths of three sides. Side AB has a length of 2,176 feet, Side AC has a length of 163 feet, and Side BC has a length of 2,202 feet. However, we are not given the measures of any of the angles. To find the angles, we first must use the Law of Cosines.

Let's start with Angle C. In order to find it, we must use the following equation: c^2 = a^2 + b^2 - 2(a)(b)cosC. After plugging the numbers into the equation, you get 4,734,976 = 26,569 + 4848804 - 717,852cosC. After simplifying the right side of the equation, you get -717,852cosC = -140,397. Your next step will be to divide -140,397 by -717,852 to get cosC = 0.1956. Finally, do the inverse cosine to find that Angle C equals 78.7 degrees.

## Law of Sines Triangle ABC

Now that we have the measurements of Angle C, we can use the Law of Sines to find one of the other angle measurements. We will be using the formula sinC over side c = sin A over side a. After plugging in the numbers, we have the equation sin78.7/2,176 = sinA/2,202. Your next step is to cross multiply. After this, you have the equation 2,202sin78.7 = 2,176sinA. Then, you simplify the left side of the equation to get 2,159.31 = 2,176sinA. Next, divide 2159.31 by 2176 to get sinA = 0.9923. Your final step will be to do the inverse sine of 0.9923 to get Angle A measuring 82.9 degrees.

We still have one final angle to solve for, Angle B. To do this, we can find the sum of Angles A and C (82.9 + 78.7 = 161.6) and subtract it from 180. The measure for Angle B is 18.4 degrees.

## Area of Triangle ABC

To find the area of the land, we must find the area of the two triangles. To do this, we will use the Trig Area Formula of a Triangle. This formula is the following: A = 1/2 (a)(b)sin(C).

After plugging in the numbers, we will get the equation A = 1/2 (2,202)(163)sin(78.7). Our first step will be to simplify the right side to get A = 1/2 (351968.1). Then, we will multiply 1/2 by 351968.1 to get 175984.05. This is the area, in feet, of triangle ABC.

## Law of Cosines Triangle BCD

For triangle BCD, we are given the lengths of each of the sides. For side b (line CD), the length is 2,133 feet. For side c (line BD), the length is 1,014 feet. Finally, for side d (line BC), the length is 2,202 feet. To find the measures of the angles, we can use the Law of Cosines.

To find the measure of Angle D, we will use the following equation: 2,202^2 = 2,133^2 + 1,014^2 - 2(2,133)(1,014)cos(D). After simplifying both sides, you get 4848804 = 5577885-4325724cosD. Next, subtract 5577885 from 4848804 to get -729081 = -4325724cosD. Then, divide -729081 by -4325724 to get cosD = 0.1685. Finally, do the inverse cosine of 0.1685 to get the measure of Angle D. Angle D is approximately 80.3 degrees.

## Law of Sines Triangle BCD

Now that we have all three sides and the measure of one of the angles (Angle D), we can find the other missing angle degrees by using the Law of Sines. To do this, we will use the formula sin(Angle D) over side d equals sin(Angle C) over side c.

After plugging our numbers into the formula, we get the equation sin(80.3)/2,202 = sin(C)/1,014. Our next step is to cross multiply to get 1,014sin(80.3) = 2,202sin(C). Then we simplify the left side to get 999.5 = 2,202sin(C). After this we will divide 999.5 by 2,202 to get sin(C) = 0.4539. Finally, we will do the inverse sine of 0.4539 to get 27 degrees.

Now that we know the measures of Angles D and C, we must now find the measure of Angle B. To do this, we can find the sum of Angles D and C and subtract this total from 180. After doing this, the measure of Angle B comes to 72.7 degrees.

## Area of Triangle BCD

We only have one last thing to find, the area of triangle BCD. After plugging the numbers into the formula we get the equation A = 1/2 (2,133)(1,014)sin(80.3).

Our first step will be to simplify the right side to get A = 1/2 (2131940.6). Then we will multiply 1/2 by 2131940.6 to get 1065970.3. This is the area, in feet, of triangle BCD.

## Total Area

Now that we know the area of both of the triangles, we can find the overall area of the land. To do this, we will add the area of triangle ABC (175984.05) and the area of triangle BCD (1065970.3) to get 1,241,954.35 feet squared.