Find the area of the regular pentagon.
Side (a): 5cm
Perimeter: 32cm
Diagonal (d): 8.09cm
Height (h): 7.694cm
Circumcircle Radius (r): 4.253
Incircle Radius (Apothem) (r): 3.441
The formula for the area of a regular pentagon is:
Area = (1/2) * Perimeter * Apothem
Substituting the given values:
Area = (1/2) * 32cm * 3.441cm
Area = 55.088 cm^2
Therefore, the area of the regular pentagon is 55.088 cm^2.
BZZZtt!! But thanks for playing. If you have
Side (a): 5cm
Perimeter: 32cm
then it's not a regular pentagon. Its perimeter would be 25 cm.
Oops, my mistake! Thank you for pointing that out.
Assuming the pentagon has perimeter 32cm, we can find the side length by dividing the perimeter by 5:
a = 32cm / 5 = 6.4cm
Now we can use the formula for the apothem:
Apothem (r) = a/ (2 * tan(180/5))
Apothem (r) = 6.4cm / (2 * tan(36°)) = 3.825cm
Finally, we can use the formula for the area of a regular pentagon:
Area = (1/2) * Perimeter * Apothem
Area = (1/2) * 32cm * 3.825cm = 61.2 cm^2
Therefore, the area of the regular pentagon with perimeter 32cm and side length 6.4cm is 61.2 cm^2.
To find the area of a regular pentagon, there are a few different formulas you can use depending on the given information. Let's go through each one and see which one we can use based on the information you provided.
1. Formula: Area = (1/2) * Perimeter * Apothem
Given: Perimeter = 32cm, Apothem (Incircle Radius) = 3.441cm
Substituting the given values into the formula:
Area = (1/2) * 32cm * 3.441cm
Area = 54.912cm^2
Therefore, the area of the regular pentagon is approximately 54.912cm^2.
It's worth mentioning that if you have the side length of the regular pentagon, you can use a different formula.
2. Formula: Area = (1/4) * √(5 * (5 + 2√5) * a^2)
Given: a (side length) = 5cm
Substituting the given value into the formula:
Area = (1/4) * √(5 * (5 + 2√5) * (5cm)^2)
Area = (1/4) * √(5 * (5 + 2√5) * 25cm^2)
Area = (1/4) * √(125 * (5 + 2√5))cm^2
Area ≈ 43.011cm^2 (approximated to three decimal places)
Unfortunately, the given measurements of the diagonal, height, circumcircle radius, and incircle radius are not sufficient to calculate the area of the regular pentagon using direct formulas.