Math worksheet for calculating area and perimeter of polygons.

To solve the problem, we need to identify each polygon, calculate its area and perimeter, and determine its type. Let's go through each polygon step by step.

---

1) Regular Hexagon

- Given: Side length \( s = 3.4 \, \text{ft} \), Apothem \( a = 1.7 \, \text{ft} \)
- Type: Regular Hexagon
- Perimeter:
\[
P = 6s = 6 \times 3.4 = 20.4 \, \text{ft}
\]
- Area:
\[
A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} = \frac{1}{2} \times 20.4 \times 1.7 = 17.34 \, \text{ft}^2
\]

---

2) Right Triangle

- Given: \( a = 8.2 \, \text{yds} \), \( b = 4.5 \, \text{yds} \), \( c = 9.4 \, \text{yds} \)
- Type: Right Triangle
- Perimeter:
\[
P = a + b + c = 8.2 + 4.5 + 9.4 = 22.1 \, \text{yds}
\]
- Area:
\[
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8.2 \times 4.5 = 18.45 \, \text{yds}^2
\]

---

3) Rectangle

- Given: \( a = 8.9 \, \text{ft} \), \( b = 4.4 \, \text{ft} \)
- Type: Rectangle
- Perimeter:
\[
P = 2(a + b) = 2(8.9 + 4.4) = 2 \times 13.3 = 26.6 \, \text{ft}
\]
- Area:
\[
A = a \times b = 8.9 \times 4.4 = 39.16 \, \text{ft}^2
\]

---

4) Regular Octagon

- Given: Side length \( s = 3.3 \, \text{yds} \), Apothem \( a = 1.65 \, \text{yds} \)
- Type: Regular Octagon
- Perimeter:
\[
P = 8s = 8 \times 3.3 = 26.4 \, \text{yds}
\]
- Area:
\[
A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} = \frac{1}{2} \times 26.4 \times 1.65 = 21.78 \, \text{yds}^2
\]

---

5) Parallelogram

- Given: Base \( a = 6.6 \, \text{cm} \), Height \( h = 5.98 \, \text{cm} \)
- Type: Parallelogram
- Perimeter: (Assume side lengths are not given, so we cannot calculate the perimeter without additional information.)
- Area:
\[
A = \text{base} \times \text{height} = 6.6 \times 5.98 = 39.588 \, \text{cm}^2
\]

---

6) Regular Octagon

- Given: Side length \( s = 7.6 \, \text{mm} \), Apothem \( a = 3.29 \, \text{mm} \)
- Type: Regular Octagon
- Perimeter:
\[
P = 8s = 8 \times 7.6 = 60.8 \, \text{mm}
\]
- Area:
\[
A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} = \frac{1}{2} \times 60.8 \times 3.29 = 100.456 \, \text{mm}^2
\]

---

7) Regular Pentagon

- Given: Side length \( s = 5.4 \, \text{mm} \), Apothem \( a = 2.57 \, \text{mm} \)
- Type: Regular Pentagon
- Perimeter:
\[
P = 5s = 5 \times 5.4 = 27 \, \text{mm}
\]
- Area:
\[
A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} = \frac{1}{2} \times 27 \times 2.57 = 34.815 \, \text{mm}^2
\]

---

8) Circle

- Given: Radius \( r = 1.25 \, \text{cm} \) (since \( s = 2.5 \, \text{cm} \) is the diameter, \( r = \frac{s}{2} = 1.25 \, \text{cm} \))
- Type: Circle
- Perimeter (Circumference):
\[
C = 2\pi r = 2\pi \times 1.25 = 2.5\pi \approx 7.85 \, \text{cm}
\]
- Area:
\[
A = \pi r^2 = \pi \times (1.25)^2 = \pi \times 1.5625 \approx 4.91 \, \text{cm}^2
\]

---

9) Right Triangle

- Given: \( a = 5.8 \, \text{inches} \), \( b = 8.66 \, \text{inches} \), \( c = 8.9 \, \text{inches} \), \( h = 5.4 \, \text{inches} \)
- Type: Right Triangle
- Perimeter:
\[
P = a + b + c = 5.8 + 8.66 + 8.9 = 23.36 \, \text{inches}
\]
- Area:
\[
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5.8 \times 8.66 = 25.354 \, \text{inches}^2
\]

---

Final Answers

1. Hexagon:
- Area: \( 17.34 \, \text{ft}^2 \)
- Perimeter: \( 20.4 \, \text{ft} \)
- Type: Regular Hexagon

2. Right Triangle:
- Area: \( 18.45 \, \text{yds}^2 \)
- Perimeter: \( 22.1 \, \text{yds} \)
- Type: Right Triangle

3. Rectangle:
- Area: \( 39.16 \, \text{ft}^2 \)
- Perimeter: \( 26.6 \, \text{ft} \)
- Type: Rectangle

4. Octagon:
- Area: \( 21.78 \, \text{yds}^2 \)
- Perimeter: \( 26.4 \, \text{yds} \)
- Type: Regular Octagon

5. Parallelogram:
- Area: \( 39.588 \, \text{cm}^2 \)
- Perimeter: Cannot be calculated without side lengths.
- Type: Parallelogram

6. Octagon:
- Area: \( 100.456 \, \text{mm}^2 \)
- Perimeter: \( 60.8 \, \text{mm} \)
- Type: Regular Octagon

7. Pentagon:
- Area: \( 34.815 \, \text{mm}^2 \)
- Perimeter: \( 27 \, \text{mm} \)
- Type: Regular Pentagon

8. Circle:
- Area: \( 4.91 \, \text{cm}^2 \)
- Perimeter: \( 7.85 \, \text{cm} \)
- Type: Circle

9. Right Triangle:
- Area: \( 25.354 \, \text{inches}^2 \)
- Perimeter: \( 23.36 \, \text{inches} \)
- Type: Right Triangle

\boxed{
\begin{array}{|c|c|c|c|}
\hline
\text{Polygon} & \text{Area} & \text{Perimeter} & \text{Type} \\
\hline
1) & 17.34 \, \text{ft}^2 & 20.4 \, \text{ft} & \text{Regular Hexagon} \\
2) & 18.45 \, \text{yds}^2 & 22.1 \, \text{yds} & \text{Right Triangle} \\
3) & 39.16 \, \text{ft}^2 & 26.6 \, \text{ft} & \text{Rectangle} \\
4) & 21.78 \, \text{yds}^2 & 26.4 \, \text{yds} & \text{Regular Octagon} \\
5) & 39.588 \, \text{cm}^2 & \text{Not calculable} & \text{Parallelogram} \\
6) & 100.456 \, \text{mm}^2 & 60.8 \, \text{mm} & \text{Regular Octagon} \\
7) & 34.815 \, \text{mm}^2 & 27 \, \text{mm} & \text{Regular Pentagon} \\
8) & 4.91 \, \text{cm}^2 & 7.85 \, \text{cm} & \text{Circle} \\
9) & 25.354 \, \text{inches}^2 & 23.36 \, \text{inches} & \text{Right Triangle} \\
\hline
\end{array}
}