# Area of pentagon

1. Nov 28, 2004

### 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.

2. Nov 28, 2004

### maverick99

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

3. Nov 28, 2004

### 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.

4. Nov 28, 2004

### maverick99

Grrr, I'm not doin 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

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Last edited: Nov 28, 2004
5. Nov 28, 2004

### ceptimus

The 6.98 is wrong.

tan(36) = 21.5 / h

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

6. Nov 28, 2004

### maverick99

Ohhhhh, ok i see.
Thx cept!

7. Nov 28, 2004

### maverick99

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

8. Nov 29, 2004

### 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.

9. Nov 29, 2004

### 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.

Last edited: Nov 29, 2004
10. Nov 29, 2009

### 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.

11. Nov 29, 2009

### Staff: Mentor

Are you an engineer?