in the problem, it shows a regular pentagon, pointing upwards. There's an apothem from the center straight to the bottom, and it is perpendicular to the side. The length is 15. How do I find the perimeter and area with only that information? Please help. I really need it.
2007-03-19 03:47:33 · 5 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
Not only do you know that the length of the apothem is 15, but since it is a REGULAR pentagon, if we consider the trianlge formed by the center, call it A, and where the apothem intersects the side, call it B, and the adjacent vertex of the pentagon, call it C, then angle BAC is exactly one tenth of 360, or 36. Since, tan is opposite/adjacent and we know that the adjacent side of angle BAC is 15, side BC becomes to value 15 TAN 36, giving 30 TAN 36 as the side.
Therefore, with
p = ns ; perimeter = (number of sides)(length of side)
A = .5ap ; area = .5(apothem)(perimeter)
we have,
p = 5(30 TAN 36) = 150 TAN 36 = ~108.98
A = .5(15)(150 TAN 36) = 1125 TAN 36 = ~ 817.36
2007-03-19 04:21:05 · answer #1 · answered by boombabybob 3 · 0⤊ 0⤋
The main thing you need to do is figure out the angle made by joining one vertex to the center and then to an adjacent vertex. It's not hard; for a regular polygon with n sides, this angle will be 360/n, and in this case it's 360/5 = 72. The apothem creates a right triangle with one angle equal to half this, so 36 degrees, and the opposite leg is half of one side. The hypotenuse is a radius, and the apothem is your other leg.
You know that the apothem has length 15, and now you know that its adjacent angle is 36. You need the length of the other leg, x. By trigonometry, tan(36) = x/15 ==> x = 15*tan(36) = 10.9. Each side of the pentagon is 2x, so the perimeter is 5*2x = 10x = 10*10.9 = 109. With the lengths of the apothem and the side, you can also find the area. The pentagon is now divided into five congruent triangles where you know the base (side) and height (apothem). The total area of the pentagon is 5*bh/2 = 2.5bh = 2.5(2x)15 = 75x = 75*10.9 = 817.
You can also find the area of a regular polygon by simply multiplying half the perimeter by the apothem.
2007-03-19 10:51:19 · answer #2 · answered by DavidK93 7 · 1⤊ 0⤋
The apothem is the height of a triangle in the pentagon.
There are five of these identical triangles in the pentagon.
You should know the interior angle measure of a pentagonal angle.
Use this information to find the area of one triangle, and multiply by 5 to get the total area of the pentagon.
2007-03-19 10:52:39 · answer #3 · answered by MamaMia © 7 · 0⤊ 0⤋
Join centre to corner at either side of bottom side. Now you know what the angle in this corner is because it's half the angle in a regular pentagon. This gives you angle at the centre. Also it's in a right angled triangle so you can use tangent to find its base length which is half side length.
By the way, in all my years of doing maths (and that's a lot of years) I've never heard of the term apothem.
2007-03-19 10:55:19 · answer #4 · answered by mathsmanretired 7 · 0⤊ 0⤋
Area = 163.47210693, Perimeter = 48.73795444
2007-03-19 11:05:45 · answer #5 · answered by Surveyor 5 · 0⤊ 0⤋