Answer:
Length of one side (s) ≈ 51.96 m
Perimeter (P) ≈ 155.88 m
Area (A) ≈ 1169.13 m²
Step-by-step explanation:
To calculate the length of one side, perimeter, and area of a regular polygon with 3 sides (a triangle) and an apothem of 15 m we can use the following formulas:
Length of one side (s) of a regular polygon:
[tex]\large\boxed{\boxed{ \sf s =2a \cdot \tan\left(\dfrac{\pi}{n}\right) }}[/tex]
Where:
- a is the length of the apothem
- n is the number of sides
Perimeter (P) of a regular polygon:
[tex]\large\boxed{\boxed{ \sf P = n \times s}} [/tex]
Where:
- n is the number of sides
- s is the length of one side
Area (A) of a regular polygon:
[tex]\large\boxed{\boxed{\sf A = \dfrac{1}{2} \times a \times P }}[/tex]
Where:
- a is the apothem
- P is the perimeter
Given:
- Number of sides (n) = 3
- Apothem (a) = 15 m
Let's calculate:
Length of one side (s):
[tex] \sf s = 2\cdot 15 \cdot \tan(\dfrac{\pi}{3}) [/tex]
[tex] \sf s =30\cdot \tan(60^\circ) [/tex]
[tex] \sf s = 30\cdot 1.7320508075688 [/tex]
[tex] \sf s = 51.961524227066 [/tex]
[tex] \sf s \approx 51.96 \, \textsf{m (in nearest hundredth)} [/tex]
Perimeter (P):
[tex] \sf P = 3 \times 51.961524227066 [/tex]
[tex] \sf P \approx 155.88457268119 [/tex]
[tex] \sf P \approx 155.88 \, \textsf{m (in nearest hundredth)} [/tex]
Area (A):
[tex] \sf A = \dfrac{1}{2} \times 15 \times 155.88 [/tex]
[tex] \sf A = 1169.1342951089 [/tex]
[tex] \sf A \approx 1169.13 \, \textsf{m$^2$ (in nearest hundredth)}[/tex]
In summary:
Length of one side (s) ≈ 51.96 m
Perimeter (P) ≈ 155.88 m
Area (A) ≈ 1169.13 m²
