Degree measure of multiple polygons

Question

I made this design on the Desmos calculator, and I was wondering what the quickest way was to find the degree measure of each individual angle.

What I know so far:

• The measures of each of the angles of the pentagon are 108
• The measures of each of the angles of the rectangle are 90
• The measure of the sector angle is 107.53
• The measure of the smaller angle of the rhombus is 35.53
• The measure of the larger angle of the rhombus is 144.47
• The measure of the largest angle of the triangle is 162
• The measure of the angle in the triangle opposite the smallest side is 5.8
• The measure of the last angle in the triangle is 14.6
• The measure the bottom left angle of the trapezoid is 57.4
• The measure of the top left angle of the trapezoid is 84.2
• The measure of the bottom right angle of the trapezoid is 104.1
• The measure of the top right angle of the trapezoid is 114.3

What I still need:

• Nothing left, this is now finished!

Thanks.

Answer 1

If you know the coordinates of every point in your shape, you can find any angle using a combination of Pythagoras' theorem and the Cosine rule.

Let us say that you have the coordinates of 3 points: $A(a_1, a_2)$, $B(b_1, b_2)$, $C(c_1, c_2)$.

To find $\angle ACB$ you would do the following:

First, find the lengths of the 3 sides of $\triangle ABC$ using Pythagoras: $AB$, $BC$, $AC$.

$$AB=\sqrt{(a_1-b_1)^2+(a_2-b_2)^2}$$ $$BC=\sqrt{(b_1-c_1)^2+(b_2-c_2)^2}$$ $$AC=\sqrt{(a_1-c_1)^2+(a_2-c_2)^2}$$

Next you can use the Cosine rule to find the angle:

$$\cos(\angle ACB)={{AB}^2+{BC}^2-{AC}^2\over 2{AB}.{BC}}$$

Using this, the final formula you would obtain is:

$$\cos(\angle ACB)={{(a_1-b_1)^2+(a_2-b_2)^2}+{(b_1-c_1)^2+(b_2-c_2)^2}-{(a_1-c_1)^2-(a_2-c_2)^2}\over 2{\sqrt{(a_1-b_1)^2+(a_2-b_2)^2}}.{\sqrt{(b_1-c_1)^2+(b_2-c_2)^2}}}$$

$$\cos(\angle ACB)={b_1^2+b_2^2-a_1b_1-a_2b_2-b_1c_1-b_2c_2+a_1c_1+a_2c_2\over {\sqrt{(a_1-b_1)^2+(a_2-b_2)^2}}.{\sqrt{(b_1-c_1)^2+(b_2-c_2)^2}}}$$

although I would advise doing this in 4 steps as shown above rather than like this to avoid making mistakes.

For example, with the smallest angle of the rhombus, $\angle BED$, you know that the 3 points which create the angle are: $B(9.2, 5.6)$, $E(2, -6.2)$, $D(10, -6.2)$.

Now substitute $B$ for $A$, $E$ for $B$ and $D$ for $C$ in the above formula to find the angle.