How to Calculate the Area of a Regular Hexagon - MywallpapersMobi

# Apothem

Not to be confused with apothegm .

Apothem of a hexagon

1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Regular_polygon_side_count_graph.svg/440px-Regular_polygon_side_count_graph.svg.png 2x” data-file-width=”512″ data-file-height=”768″ />

Graphs of side , s ; apothem , a and area , A of regular polygons of n sides and circumradius 1, with the base , b of a rectangle with the same area – the green line shows the case n = 6

The apothem (sometimes abbreviated as apo [1] ) 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 .

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.

For an equilateral triangle, 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.

## Contents

• 1 Properties of apothems
• 2 Finding the apothem
• 4 References

## Properties of apothems[ edit ]

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.

$\displaystyle A=\frac nsa2=\frac pa2.$

A
=

n
s
a

2

=

p
a

2

.

\displaystyle A=\frac nsa2=\frac pa2.

This formula can be derived by partitioning the n-sided polygon into n congruent isosceles 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.

This property can also be used to easily derive the formula for the area of a circle, because as the number of sides approaches infinity, the regular polygon’s area approaches the area of the inscribed circle of radius r = a.

$\displaystyle A=\frac pa2=\frac (2\pi r)r2=\pi r^2$

A
=

p
a

2

=

(
2
π
r
)
r

2

=
π

r

2

\displaystyle A=\frac pa2=\frac (2\pi r)r2=\pi r^2

## Finding the apothem[ edit ]

The apothem of a regular polygon can be found multiple ways.

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

$\displaystyle a=\frac s2\tan \left(\frac \pi n\right)=R\cos \left(\frac \pi n\right).$

a
=

s

2
tan

(

π
n

)

=
R
cos

(

π
n

)

.

\displaystyle a=\frac s2\tan \left(\frac \pi n\right)=R\cos \left(\frac \pi n\right).

The apothem can also be found by

$\displaystyle a=\frac s2\tan \!\left(\frac \pi (n-2)2n\right).$

a
=

s
2

tan

(

π
(
n

2
)

2
n

)

.

\displaystyle a=\frac s2\tan \!\left(\frac \pi (n-2)2n\right).

These formulae can still be used even if only the perimeter p and the number of sides n are known because

$\displaystyle s=\frac pn.$

s
=

p
n

.

\displaystyle s=\frac pn.

• Circumradius of a regular polygon
• Sagitta (geometry)
• Chord (trigonometry)
• Slant height

## References[ edit ]

1. ^ Shaneyfelt, Ted V. “德博士的 Notes About Circles, ज्य, & कोज्य: What in the world is a hacovercosine?” . Hilo, Hawaii: University of Hawaii . Archived from the original on 2015-09-19. Retrieved 2015-11-08.

 Look up apothem in Wiktionary, the free dictionary.
• Apothem of a regular polygon With interactive animation
• Apothem of pyramid or truncated pyramid
• Pegg, Jr., Ed . “Sagitta, Apothem, and Chord” . The Wolfram Demonstrations Project .

Categories :

• Polygons

### Personal tools

• Not logged in
• Talk
• Contributions
• Create account

• Article
• Talk

### Views

• Edit
• View history

### Search

• Main page
• Contents
• Featured content
• Current events
• Random article
• Donate to Wikipedia
• Wikipedia store

### Interaction

• Help
• Community portal
• Recent changes
• Contact page

### Tools

• Related changes
• Special pages
• Page information
• Wikidata item

### Print/export

• Create a book
• Printable version

### In other projects

• Wikimedia Commons

### Languages

• Asturianu
• Български
• Bosanski
• Català
• Deutsch
• Eesti
• Español
• Euskara
• Français
• Galego
• Հայերեն
• Italiano
• עברית
• Nederlands
• Oʻzbekcha/ўзбекча
• Polski
• Português
• Română
• Русский
• Slovenščina
• Српски / srpski
• Suomi
• தமிழ்
• Тоҷикӣ
• Українська
• 中文

• This page was last edited on 16 November 2018, at 12:57 (UTC).
• Disclaimers
• Contact Wikipedia
• Developers
• Mobile view

for Teachers
for Schools
for Enterprise

for Teachers
for Schools
for Enterprise

• Plans
• Plans
• Courses
Courses

Find Courses by Subject

• Science
• Math
• Psychology
• History
• English
• Social Science
• Humanities
• Spanish
• ACT & SAT Test Prep
• Teacher Certification
• Professional Development

By Education Level

• College
• High School
• Middle School

Explore over
4,100
video courses

Browse All Courses

• Credit
Credit

Credit Options

• Online College Credit
• High School & GED
• Certificates of Completion
• How it Works

Earn Transferable Credit & Get your Degree fast

• Degrees
Degrees

Find Degrees by Subject

• Agriculture
• Architecture
• Biological and Biomedical
Sciences
• Communications and Journalism
• Computer Sciences
• Culinary Arts and Personal
Services
• Education
• Engineering
• Legal
• Liberal Arts and Humanities
• Mechanic and Repair Technologies
• Medical and Health Professions
• Physical Sciences
• Psychology
• Transportation and Distribution
• Visual and Performing Arts

By Level

• High School Diploma
• Associates Degrees
• Bachelor Degrees
• Master Degrees
• Online Degrees

Find a degree that fits your goals

Search degrees

• Schools
Schools

Browse Schools by Degree Level

• High School Diplomas
• Certificate Programs
• Post Degree Certificates
Browse Schools

• Public Schools by State
• University Video Reviews

Career Counseling & Job Center

• Job Interviewing Tip Videos
• Job Networking Videos
• Résumé How To Videos
• Job Search Tips
• Career Videos
Career Research

• Researching Careers Videos
• Glossary of Careers
• Career Info by Degree
• Job Outlook by Region
• Degree & Career Research Articles

• Contact Support

# How to find the Apothem of a hexagon.

## Question:

How to find the Apothem of a hexagon.

## Apothem of a Hexagon:

A regular hexagon is a six-sided polygon with all of its sides having equal length. The apothem of a regular hexagon is the line segment that runs from the center of the hexagon to the midpoint of one of its sides. We can find the apothem of a hexagon using a couple of different formulas.

We have two formulas we can use to find the apothem of a regular hexagon with side length s, and they are as follows:

• Apothem = (√(3) / 2)…

Become a Study.com member to unlock this answer!
Create your account

Try it risk-free

Apothem: Definition & Formula

from High School Geometry: Homework Help Resource

Chapter 3
/ Lesson 9

58K

#### Related to this Question

• Related Lessons

• Related Courses

What is the expression for the determinant of the…

In mathematics, what is the difference between…

Solve: (x + 2)^3 = -252

How do you find a missing length when given an…

what is the root of f(x) = e^x – x^2

Simplify (2 / (x^2+x)) – (4 / x).

Simplify (1 / (x^2 – x)) + (1 / (x^2 + x))

Simplify (2 / (x + 2)) – (x / (x + 2)^2)

Constructing Similar Polygons

Measuring the Area of Regular Polygons: Formula & Examples

Oblique Prism: Definition & Volume

Surface Area of a Pentagonal Prism

Hexagonal Prism: Properties, Formula & Examples

Arc Measure: Definition & Formula

How to Find the Surface Area of a Right Prism

How to Find the Altitude of a Trapezoid

How to Find the Measure of an Inscribed Angle

Lateral Area: Definition, Formula & Examples

Volume, Faces & Vertices of an Octagonal Pyramid

How to Find the Height of a Parallelogram

Glide Reflection in Geometry: Definition & Example

Geometric Probability: Definition, Formula & Examples

Volume of a Frustum of Pyramids & Cones

Midsegment: Theorem & Formula

Chord Theorems of Circles in Geometry

How to Find the Volume of a Right Prism

Slant Height: Definition & Formula

Surface Area of Composite Figures

High School Geometry: Help and Review

High School Geometry: Tutoring Solution

ELM: CSU Math Study Guide

High School Precalculus: Help and Review

High School Trigonometry: Help and Review

High School Algebra II: Tutoring Solution

McDougal Littell Algebra 1: Online Textbook Help

CUNY Assessment Test in Math: Practice & Study Guide

High School Trigonometry: Homework Help Resource

High School Trigonometry: Tutoring Solution

Holt McDougal Algebra I: Online Textbook Help

Precalculus: Help and Review

Precalculus: Homework Help Resource

Holt Geometry: Online Textbook Help

Intro to Calculus

AP Calculus AB & BC: Homework Help Resource

Praxis Mathematics – Content Knowledge (5161): Practice & Study Guide

SAT Prep: Practice & Study Guide

ACT Prep: Help and Review

High School Algebra II: Help and Review

#### Become a member and unlock all Study Answers

Try it risk-free for 30 days!

Try it risk-free

#### Explore our homework questions and answer library

Browse
by subject
• Math
• Social Sciences
• Science
• Humanities
• History
• Art and Design
• Tech and Engineering
• Health and Medicine

Support

###### You are joining:

 30 day money back guarantee Starting Original Price /yr / Just Just /day DiscountFor months – %– / Price after trial Starting Price starting today / Just Just /day
Cancel before and your credit card will not be charged.

Study.com video lessons have helped over 30 million
students.

#### Students Love Study.com

“I learned more in 10 minutes than 1 month of chemistry classes”

– Ashlee P.

#### Earn College Credit

“I aced the CLEP exam and earned 3 college credits!

– Clair S.

Study.com video lessons have helped over half a million teachers engage their students.

#### Teachers Love Study.com

“The videos have changed the way I teach! The videos on Study.com accomplish in
5 minutes what would take me an entire class.”

– Chris F.

#### Did you know…

Students in online learning conditions performed better than those receiving face-to-face
instruction.

U.S. Department of Education

Study.com video lessons have helped over 500,000
teachers engage their students.

Just a few seconds while we find the right plan for you

###### You are joining:

 30 day money back guarantee Starting Original Price /yr / Just Just /day DiscountFor months – %– / Price after trial Starting Price starting today / Just Just /day
Cancel before and your credit card will not be charged.

Study.com video lessons have helped over 30 million
students.

#### Students Love Study.com

“I learned more in 10 minutes than 1 month of chemistry classes”

– Ashlee P.

#### Earn College Credit

“I aced the CLEP exam and earned 3 college credits!

– Clair S.

Study.com video lessons have helped over half a million teachers engage their students.

#### Teachers Love Study.com

“The videos have changed the way I teach! The videos on Study.com accomplish in
5 minutes what would take me an entire class.”

– Chris F.

#### Did you know…

Students in online learning conditions performed better than those receiving face-to-face
instruction.

U.S. Department of Education

Study.com video lessons have helped over 500,000
teachers engage their students.

Secure Server

tell me more

403. Forbidden.