Polygon Apothem Worksheet - Calculate Perimeter and Apothem for Various Regular Polygons

To solve the problem of finding the perimeter and apothem of each polygon, we need to use the relationships between the area, perimeter, and apothem of a regular polygon. The key formulas are:

1. Area of a Regular Polygon:
\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]

2. Perimeter of a Regular Polygon:
\[
\text{Perimeter} = n \times s
\]
where \( n \) is the number of sides and \( s \) is the length of one side.

3. Apothem:
\[
\text{Apothem} = \frac{2 \times \text{Area}}{\text{Perimeter}}
\]

Let's solve each polygon step by step.

---

1) Pentagon

- Given: Area = 337.21 yd², Apothem = 13.86 yd
- Find: Perimeter

Using the area formula:
\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]
\[
337.21 = \frac{1}{2} \times \text{Perimeter} \times 13.86
\]
\[
337.21 = 6.93 \times \text{Perimeter}
\]
\[
\text{Perimeter} = \frac{337.21}{6.93} \approx 48.68 \text{ yd}
\]

- Perimeter: 48.68 yd
- Apothem: 13.86 yd

---

2) Hexagon

- Given: Area = 93.53 ft², Side length = 6.0 ft
- Find: Perimeter, Apothem

First, calculate the perimeter:
\[
\text{Perimeter} = n \times s = 6 \times 6.0 = 36.0 \text{ ft}
\]

Next, use the area formula to find the apothem:
\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]
\[
93.53 = \frac{1}{2} \times 36.0 \times \text{Apothem}
\]
\[
93.53 = 18.0 \times \text{Apothem}
\]
\[
\text{Apothem} = \frac{93.53}{18.0} \approx 5.19 \text{ ft}
\]

- Perimeter: 36.00 ft
- Apothem: 5.19 ft

---

3) Octagon

- Given: Area = 439.7 m², Apothem = 17.0 m
- Find: Perimeter

Using the area formula:
\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]
\[
439.7 = \frac{1}{2} \times \text{Perimeter} \times 17.0
\]
\[
439.7 = 8.5 \times \text{Perimeter}
\]
\[
\text{Perimeter} = \frac{439.7}{8.5} \approx 51.73 \text{ m}
\]

- Perimeter: 51.73 m
- Apothem: 17.00 m

---

4) Triangle

- Given: Area = 35.07 ft², Apothem = 6.9 ft
- Find: Perimeter

Using the area formula:
\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]
\[
35.07 = \frac{1}{2} \times \text{Perimeter} \times 6.9
\]
\[
35.07 = 3.45 \times \text{Perimeter}
\]
\[
\text{Perimeter} = \frac{35.07}{3.45} \approx 10.16 \text{ ft}
\]

- Perimeter: 10.16 ft
- Apothem: 6.90 ft

---

5) Octagon

- Given: Area = 1,044.73 in², Side length = 23 in
- Find: Perimeter, Apothem

First, calculate the perimeter:
\[
\text{Perimeter} = n \times s = 8 \times 23 = 184 \text{ in}
\]

Next, use the area formula to find the apothem:
\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]
\[
1,044.73 = \frac{1}{2} \times 184 \times \text{Apothem}
\]
\[
1,044.73 = 92 \times \text{Apothem}
\]
\[
\text{Apothem} = \frac{1,044.73}{92} \approx 11.36 \text{ in}
\]

- Perimeter: 184.00 in
- Apothem: 11.36 in

---

6) Octagon

- Given: Area = 309.01 yd², Apothem = 9.6 yd
- Find: Perimeter

Using the area formula:
\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]
\[
309.01 = \frac{1}{2} \times \text{Perimeter} \times 9.6
\]
\[
309.01 = 4.8 \times \text{Perimeter}
\]
\[
\text{Perimeter} = \frac{309.01}{4.8} \approx 64.38 \text{ yd}
\]

- Perimeter: 64.38 yd
- Apothem: 9.60 yd

---

7) Octagon

- Given: Area = 192.36 in², Side length = 7 in
- Find: Perimeter, Apothem

First, calculate the perimeter:
\[
\text{Perimeter} = n \times s = 8 \times 7 = 56 \text{ in}
\]

Next, use the area formula to find the apothem:
\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]
\[
192.36 = \frac{1}{2} \times 56 \times \text{Apothem}
\]
\[
192.36 = 28 \times \text{Apothem}
\]
\[
\text{Apothem} = \frac{192.36}{28} \approx 6.87 \text{ in}
\]

- Perimeter: 56.00 in
- Apothem: 6.87 in

---

8) Pentagon

- Given: Area = 688.15 yd², Side length = 20.0 yd
- Find: Perimeter, Apothem

First, calculate the perimeter:
\[
\text{Perimeter} = n \times s = 5 \times 20.0 = 100.0 \text{ yd}
\]

Next, use the area formula to find the apothem:
\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]
\[
688.15 = \frac{1}{2} \times 100.0 \times \text{Apothem}
\]
\[
688.15 = 50.0 \times \text{Apothem}
\]
\[
\text{Apothem} = \frac{688.15}{50.0} \approx 13.76 \text{ yd}
\]

- Perimeter: 100.00 yd
- Apothem: 13.76 yd

---

9) Square

- Given: Area = 144 ft²
- Find: Perimeter, Apothem

For a square, the side length \( s \) can be found using the area:
\[
\text{Area} = s^2
\]
\[
144 = s^2
\]
\[
s = \sqrt{144} = 12 \text{ ft}
\]

Calculate the perimeter:
\[
\text{Perimeter} = 4 \times s = 4 \times 12 = 48 \text{ ft}
\]

The apothem of a square is half the side length:
\[
\text{Apothem} = \frac{s}{2} = \frac{12}{2} = 6 \text{ ft}
\]

- Perimeter: 48.00 ft
- Apothem: 6.00 ft

---

Final Answers

\[
\boxed{
\begin{array}{ccc}
\text{Polygon} & \text{Perimeter} & \text{Apothem} \\
1) & 48.68 \text{ yd} & 13.86 \text{ yd} \\
2) & 36.00 \text{ ft} & 5.19 \text{ ft} \\
3) & 51.73 \text{ m} & 17.00 \text{ m} \\
4) & 10.16 \text{ ft} & 6.90 \text{ ft} \\
5) & 184.00 \text{ in} & 11.36 \text{ in} \\
6) & 64.38 \text{ yd} & 9.60 \text{ yd} \\
7) & 56.00 \text{ in} & 6.87 \text{ in} \\
8) & 100.00 \text{ yd} & 13.76 \text{ yd} \\
9) & 48.00 \text{ ft} & 6.00 \text{ ft} \\
\end{array}
}
\]