Question

5a - b) what are the area and perimeter of each of these shapes? 5c) what are the area and perimeter of this shape? if this equiangular shape was changed to a concave polygon with the same side lengths, would its perimeter increase, decrease, or stay the same? why? would its area increase, decrease, or stay the same? why? this a+ question is not organized by success criteria.

Explanation:

Step1: Analyze the first - composite shape

The first shape is composed of a square of side - length 2 in and two semi - circles (which is equivalent to one full circle). The radius of the circle $r = 1$ in.
The perimeter of the square part is $3\times2=6$ in (we don't count the side where the circle is attached). The circumference of the circle is $C = 2\pi r=2\pi(1)=2\pi$ in. So the perimeter $P_1=6 + 2\pi\approx6+2\times3.14 = 6 + 6.28=12.28$ in. The area of the square is $A_{square}=2\times2 = 4$ in², and the area of the circle is $A_{circle}=\pi r^{2}=\pi(1)^{2}=\pi\approx3.14$ in². So the area $A_1=4+\pi\approx4 + 3.14=7.14$ in².

Step2: Analyze the circle

For a circle with diameter $d = 12$ cm, radius $r = 6$ cm. The perimeter (circumference) is $C=2\pi r=2\pi\times6 = 12\pi\approx12\times3.14 = 37.68$ cm. The area is $A=\pi r^{2}=\pi\times6^{2}=36\pi\approx36\times3.14 = 113.04$ cm².

Step3: Analyze the triangle

Using the distance formula $d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}$. Let the vertices be $A(-3,-1)$, $B(2,-1)$, and $C(2,3)$.
The length of $AB=\sqrt{(2+3)^{2}+(-1 + 1)^{2}}=5$. The length of $BC=\sqrt{(2 - 2)^{2}+(3 + 1)^{2}}=4$. The length of $AC=\sqrt{(2 + 3)^{2}+(3 + 1)^{2}}=\sqrt{25 + 16}=\sqrt{41}\approx6.4$. The perimeter $P_3=5 + 4+\sqrt{41}\approx5 + 4+6.4 = 15.4$. The area of the right - triangle with base $b = 5$ and height $h = 4$ is $A=\frac{1}{2}\times5\times4 = 10$ square units.

For the equiangular polygon changed to a concave polygon with the same side - lengths:
The perimeter stays the same because the lengths of the sides do not change. Mathematically, perimeter is the sum of the lengths of the sides of a polygon, and if the side - lengths are unchanged, $P=\sum_{i = 1}^{n}s_i$ remains the same.
The area decreases. In a concave polygon, there is an "indentation" which reduces the amount of space enclosed by the polygon compared to the equiangular (convex) polygon with the same side - lengths.

Answer:

First shape: Perimeter $\approx12.28$ in, Area $\approx7.14$ in²
Circle: Perimeter $\approx37.68$ cm, Area $\approx113.04$ cm²
Triangle: Perimeter $\approx15.4$ units, Area $ = 10$ square units
For the polygon transformation: Perimeter stays the same because side - lengths are unchanged. Area decreases because of the "indentation" in the concave polygon.