Calculate Area of Regular Pentagon - 215 ft Perimeter

[maverick99]

If anyone could help me get started on this I would be really appreciative.

A concrete patio is to be built in the shape of a regular pentagon that has a perimenter of 215 feet. Find the area of the patio in square feet.

 

[maverick99]

ok i ended up getting an area of 750.35ft
Is this right?

 

[ceptimus]

If you consider that a 50 x 50 foot square has a perimeter of only 200 feet, but an area of 2,500 square feet, I think you will see that your answer must be a little off.

A regular pentagon is made of 5 equal triangles. The five apex angles at the centre of the pentagon make 360 degrees altogether, so each apex is 360 / 5 or 72 degrees. The 5 bases of the triangles (around the outside of the pentagon) total 215 feet, so each triangle base is 215 / 5 or 43 feet.

 

[maverick99]

Grrr, I'm not doing something right here. The attachment shows all of the details of the right triangle. After that, i did this equation to get the area
1/2 (21.5) (6.98) (10)=750.35

 

[ceptimus]

The 6.98 is wrong.

tan(36) = 21.5 / h

Total pentagon area = 21.5 * 21.5 * 5 / tan(36)

 

[maverick99]

Ohhhhh, ok i see.
Thx cept!

 

[maverick99]

I'm a little confused on your total pentagon area equation.
Wouldn't you do 1/2 *b*h*10??

 

[Diane_]

The general formula for the area of a regular polygon is

1/2 a*p

where p is the perimeter of the polygon and a is the "apothem". An apothem is a line drawn from the center of the polygon perpendicular to one of its sides. (Note: this only works for regular polygons - no others need apply.)

In your case, the perimeter is given as 215 feet. A little geometry gets you an apothem of 59.2 feet, so the area is 6362 square feet. I think ceptimus forgot the 1/2.

 

[ceptimus]

No I didn't forget the 1/2. The area of the right angled triangle is indeed 1/2 base * height, but two of these triangles are joined together to form each of the 5 isosceles triangles that make the pentagon.

So you can do 1/2 b * h * 10, or just b * h * 5, like I did. Diane's answer is wrong - she has the apothem twice as long as it really is.

 

[Earthground]

Very interesting answers all around but not quite correct. In general to find the area of a pentagon all one must do is to know the measurement of just one side and then square that number. Finally, multiply that number by 1.7. If you need to be more precise you should use 1.7205. In the problem of the pentagon with a diameter of 215 ft. I suppose we are going to assume that each side is of equal length. Therefore, 215' divided by 5 = 43'. 43' squared = 1,849' x 1.7205 = 3181.2045 sq. ft. If you do decide to use the 1.7, the answer is 3143.3 sq. ft. If you are going to tile the floor and use the second calculation you will have to go back to the store and buy about 38 more tiles, assuming you are using 12" x 12" tiles. Perhaps more if you mess up in those five triangular corners. Ha, Ha.

 

[Borek]

In general to find the area of a pentagon all one must do is to know the measurement of just one side and then square that number. Finally, multiply that number by 1.7. If you need to be more precise you should use 1.7205.

Are you an engineer?

This is bad case of thread necromancy.

Your answer is only approximate, while those earlier were ways of exact calculation of the area. Your answer requires knowledge about required constant, those listed earlier were universal, based just on properties of geometric figures.

 

[Earthground]

Question 1. yes

Wish I could contact the dead but even if I could they usually don't have much to say.
As far as the rest of it all I can say is...isn't that nice.