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# how do i find the apothem of a regular polygon

From Wikipedia

Apothem

The apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. The word "apothem" can also refer to the length of that line segment. Regular polygons are the only polygons that have apothems. Because of this, all the apothems in a polygon will be congruent and have the same length.

For a regular pyramid, which is a pyramid whose base is a regular polygon, the apothem is the slant height of a lateral face; that is, the shortest distance from apex to base on a given face. For a truncated regular pyramid (a regular pyramid with some of its peak removed by a plane parallel to the base), the apothem is the height of a trapezoidal lateral face. [http://www.bymath.com/studyguide/geo/sec/geo15.htm]

For a triangle (necessarily equilateral), the apothem is equivalent to the line segment from the midpoint of a side to any of the triangle's centers, since an equilateral triangle's centers coincide as a consequence of the definition.

## Properties of apothems

The apothem a can be used to find the area of any regular n-sided polygon of side length s according to the following formula, which also states that the area is equal to the apothem multiplied by half the perimeter since ns = p.

A = \frac{nsa}{2} = \frac{pa}{2}.

This formula can be derived by partitioning the n-sided polygon into ncongruentisosceles triangles, and then noting that the apothem is the height of each triangle, and that the area of a triangle equals half the base times the height.

An apothem of a regular polygon will always be a radius of the inscribed circle. It is also the minimum distance between any side of the polygon and its center.

## Finding the apothem

The apothem of a regular polygon can be found multiple ways, of which two are described here.

The apothem a of a regular n-sided polygon with side length s, or circumradiusR, can be found using the following formula:

a=\frac{s}{2\tan(\pi/n)}=R\cos(180^\circ/n).

The apothem can also be found by

a=\frac{1}{2}s\tan\!\left(\frac{90^\circ(n-2)}{n}\right).

Both formulas can still be used even if only the perimeter p and the number of sides n are known because s = \frac{p}{n}.

Question:I need to know how to find the apothem of a regular octagon with side length of 's'.

Answers:From the center of the octagon, draw lines connecting to each vertex. Now, focus on one of the eight isosceles triangles. The angle at the center is 360/8 = 45 degrees. Dropping an altitude from the center to the midpoint of the triangle's base (this is the apothem "a"), the angle at the center is split in half to 45/2 degrees and the base is split into two segments having length "s/2". This gives a right triangle! Using the definition of the tangent: tan(45/2) = opp/adj = (s/2) / a ==> a = s / (2 tan(45/2)). I hope this helps!

Question:Find the number of sides of each regular polygon. 1. A regular polygon in which A=90.9 in and P=35 in. 2. A regular polygon in which A=84.9 cm and the side lengths are 14 cm. 3. A regular polygon in which A=324.9 m and the apothem is 10m. I need help figuring out these. If possible, provide how you got the answer. You don't need to answer all three, but do as many as you can. Thanks.

Answers:A regular polygon's area is equivalent to (pa)/2, where p represents the perimeter and a the apothem, so just plug these numbers in. Furthermore, the apothem is equivalent to the square root of the circumradius and half of a side length. You should be able to apply these to find the answer. Good luck! =]

Question:The only 2 information given is that the pentagon is a regular polygon and that the side lengths are 5. Can someone not only give me the answer, but also explain it? Thank you

Answers:1) Draw a straight horizontal line. It doesn't have to be too long. This line will represent one of the sides of your polygon. In your case, it's one of the 5 sides of the regular polygon. 2) Now from the middle of that line, draw a line straight up, so that the two lines are perpendicular. This second line drawn is the "apothem." This is the segment's length your trying to find. Let's call it "b". 3) Finally draw one more line...connecting the top of the apothem (which is the center of your polygon) to either end of the first line you drew (it doesn't matter which end - left or right). We don't know this length either. Let's call it "c". Now that you have a triangle picture, let's call 1/2 of the first line you drew "a", and its length = 2.5, since it is half the length of each side of the polygon. Using the fact that this pentagon is "regular" (convex and equilateral), you can find the angle that is formed between sides "a" and "c". To do this, first determine what each interior angle in this regular pentagon equals. You've probably already learned that a 5-sided convex polygon's interior angles add up to 3 x 180 degrees = 540 degrees. So, since the pentagon is regular: 540 / 5 = 108 degrees. That's how large each interior angles is. But, the angle we're interested in (the one between "c" and "a") is exactly half of 108 degrees or 54 degrees. Ok, so here's your picture: |\ | \  c |__\  a In this picture "b" is the vertical line on the left. Remember, side "b" is the "apothem". You know the length of "a" and the angle between "a" and "c". Finally, using trigonometry, specifically tangent, you can find the length of "b". tan 54 = b / 2.5 I'm getting b = 3.44 (rounded to two decimal places) Hope this helps!