How To Calculate The Apothem Of A Hexagon: 15 Steps - WikiHow

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Terms of Use wikiHow is where trusted research and expert knowledge come together. Learn why people trust wikiHow How to Calculate the Apothem of a Hexagon PDF download Download Article Explore this Article methods 1 Using the Pythagorean Theorem (Given Side Length or Radius) 2 Using Trigonometry (Given Side Length or Radius) Other Sections Questions & Answers Tips and Warnings Related Articles References Article Summary Co-authored by David Jia

Last Updated: March 20, 2025 Fact Checked

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This article was co-authored by David Jia. David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. There are 7 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 266,300 times.

A hexagon is a six-sided polygon. When a hexagon is regular it has six equal side lengths and an apothem. An apothem is a line segment from the center of a polygon to the middle point of any one side. You usually need to know the length of the apothem when calculating the area of a hexagon.[1] As long as you know the side length of the hexagon, you can calculate the length of the apothem.

Steps

Method 1 Method 1 of 2:

Using the Pythagorean Theorem (Given Side Length or Radius)

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  1. Step 1 Divide the hexagon into six congruent, equilateral triangles. 1 Divide the hexagon into six congruent, equilateral triangles. To do this, draw a line connecting each vertex, or point, with the vertex opposite.[2]
  2. Step 2 Choose one triangle and label the length of its base. 2 Choose one triangle and label the length of its base. This is equal to the side length of the hexagon.
    • For example, you might have a hexagon with a side length of 8 cm. The base of each equilateral triangle, then, is also 8 cm.
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  3. Step 3 Create two right triangles. 3 Create two right triangles. To do this, draw a line from the top vertex of the equilateral triangle perpendicular to its base. This line will cut the base of the triangle in half (and thus is the apothem of the hexagon). Label the length of the base of one of the right triangles.
    • For example, if the base of the equilateral triangle is 8 cm, when you divide the triangle into two right triangles, each right triangle now has a base of 4 cm.
  4. Step 4 Set up the formula for the Pythagorean Theorem. 4 Set up the formula for the Pythagorean Theorem. The formula is a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}} , where c {\displaystyle c} equals the length of the hypotenuse (the side opposite the right angle), and a {\displaystyle a} and b {\displaystyle b} equal the lengths of the other two sides of the triangle.[3]
    • For example, if a right triangle had a hypotenuse of 2 {\displaystyle 2} inches, one leg of 1 {\displaystyle 1} inch, and another leg of about 1.732 {\displaystyle 1.732} inches ( 3 {\displaystyle {\sqrt {3}}} ), the Pythagorean Theorem would state that 1 2 + 3 2 = 2 2 {\displaystyle 1^{2}+{\sqrt {3}}^{2}=2^{2}} , which is true when you complete the calculations: 1 + 3 = 4 {\displaystyle 1+3=4} .
  5. Step 5 Plug the length of the right triangle’s base into the formula. 5 Plug the length of the right triangle’s base into the formula. Substitute for b {\displaystyle b} .
    • For example, if the length of the base is 4 cm, your formula will look like this: a 2 + 4 2 = c 2 {\displaystyle a^{2}+4^{2}=c^{2}} .
  6. Step 6 Plug the length of the hypotenuse into the formula. 6 Plug the length of the hypotenuse into the formula. You know the length of the hypotenuse because you know the side length of the hexagon. The side length of a regular hexagon is equal to the radius of the hexagon. The radius is a line that connects the central point of a polygon with one of its vertices.[4] You’ll note that the hypotenuse of your right triangle is also a radius of the hexagon, thus, the side length of the hexagon is equal to the length of the hypotenuse.
    • For example, if the side length of the hexagon is 8 cm, then the length of the right triangle’s hypotenuse is also 8 cm. So your formula will look like this: a 2 + 4 2 = 8 2 {\displaystyle a^{2}+4^{2}=8^{2}} .
  7. Step 7 Square the known values in the formula. 7 Square the known values in the formula. Remember that squaring a number means to multiply it by itself.
    • For example, squaring the known values, your formula will look like this: a 2 + 16 = 64 {\displaystyle a^{2}+16=64} .
  8. Step 8 Isolate the unknown variable. 8 Isolate the unknown variable. To do this, subtract the squared value of b {\displaystyle b} from both sides of the equation.
    • For example: a 2 + 16 − 16 = 64 − 16 {\displaystyle a^{2}+16-16=64-16} a 2 = 48 {\displaystyle a^{2}=48}
  9. Step 9 Solve for a {\displaystyle a} . 9 Solve for a {\displaystyle a} . To do this, find the square root of each side of the equation. This will give you the length of the missing side of the triangle, which is equal to the length of the hexagon’s apothem.
    • For example, using a calculator, you can calculate 48 = 6.93 {\displaystyle {\sqrt {48}}=6.93} . Thus, the missing length of the right triangle, and the length of the hexagon’s apothem, equals 6.93 cm.
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Method 2 Method 2 of 2:

Using Trigonometry (Given Side Length or Radius)

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  1. Step 1 Set up the formula for finding the apothem of a regular polygon. 1 Set up the formula for finding the apothem of a regular polygon. The formula is apothem = s 2 tan ⁡ ( 180 n ) {\displaystyle {\text{apothem}}={\frac {s}{2\tan({\frac {180}{n}})}}} , where s {\displaystyle s} equals the side length of the polygon and n {\displaystyle n} equals the number of sides the polygon has.[5]
  2. Step 2 Plug the side length into the formula. 2 Plug the side length into the formula. Remember to substitute for the variable s {\displaystyle s} .
    • For example, for a hexagon with a side length of 8 cm, the formula will look like this: 8 2 tan ⁡ ( 180 n ) {\displaystyle {\frac {8}{2\tan({\frac {180}{n}})}}} .
  3. Step 3 Plug the number of sides into the formula. 3 Plug the number of sides into the formula. A hexagon has 6 sides. Remember to substitute for the variable n {\displaystyle n} .
    • For example: 8 2 tan ⁡ ( 180 6 ) {\displaystyle {\frac {8}{2\tan({\frac {180}{6}})}}} .
  4. Step 4 Complete the calculation in parentheses. 4 Complete the calculation in parentheses. You are finding the degrees you will use to calculate the tangent.[6]
    • For example, 180 6 = 30 {\displaystyle {\frac {180}{6}}=30} , so the formula now look like this: 8 2 tan ⁡ ( 30 ) {\displaystyle {\frac {8}{2\tan(30)}}} .
  5. Step 5 Find the tangent. 5 Find the tangent. To do this, use a calculator or a trigonometry table.[7]
    • For example, the tangent of 30 is about .577, so the formula will now look like this: 8 2 ( .577 ) {\displaystyle {\frac {8}{2(.577)}}} .
  6. Step 6 Multiply the tangent by 2, then divide the side length by this number. 6 Multiply the tangent by 2, then divide the side length by this number. This will give you the length of the apothem of your hexagon.[8]
    • For example: apothem = 8 2 ( .577 ) {\displaystyle {\text{apothem}}={\frac {8}{2(.577)}}} apothem = 8 1.154 {\displaystyle {\text{apothem}}={\frac {8}{1.154}}} apothem = 6.93 {\displaystyle {\text{apothem}}=6.93} So, the apothem of a regular hexagon with 8-cm sides is about 6.93 cm.
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Calculator, Practice Problems, and Answers

Sample Hexagon Apothem Calculator Sample Calculating the Apothem of a Hexagon Practice Problems Sample Calculating the Apothem of a Hexagon Practice Answers

Community Q&A

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  • Question Is there a formula to calculate the apothem if you only know how long one side of the hexagon is, and its perimeter? Community Answer Community Answer Since all you need to know to calculate the apothem is the length of one side of the hexagon, the length of the perimeter is irrelevant. Use the length of the side with any of the methods in the article. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 9 Helpful 11
  • Question Is there away to calculate perimeter of a hexagon by only the area? Donagan Donagan Top Answerer Yes. When you know the area, set it equal to [(3√3) s²] / 2, where s is the length of a side. Solve the equation for s, and multiply that length by 6. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 6 Helpful 9
  • Question With Method two it metions it can also be used with the radius. However in the explanation it only shows side length. So would I use the same formula but use the radius instead of the side length? Pangdi Xiong Pangdi Xiong Community Answer The radius of a hexagon is the same as the side length, because the circle is inscribed in the hexagon. So just use the same formula and plug it in. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 6 Helpful 7
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Tips

  • The word “apothem” can refer to either the actual line segment or to the length of that line segment. Thanks Helpful 0 Not Helpful 0
  • Remember that this technique only works for regular hexagons. Irregular hexagons do not have apothems. Thanks Helpful 0 Not Helpful 0
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References

  1. ↑ https://www.mathsisfun.com/geometry/regular-polygons.html
  2. ↑ https://www.omnicalculator.com/math/hexagon
  3. ↑ https://en.neurochispas.com/geometry/apothem-of-a-hexagon-formulas-and-examples/
  4. ↑ https://www.mathsisfun.com/definitions/radius-polygon-.html
  5. ↑ https://www.mathopenref.com/apothem.html
  6. ↑ https://www.varsitytutors.com/high_school_math-help/how-to-find-the-area-of-a-hexagon
  7. ↑ https://www.mathopenref.com/apothem.html
  8. ↑ https://www.cuemath.com/geometry/apothem/

About This Article

David Jia Co-authored by: David Jia Math Tutor This article was co-authored by David Jia. David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. This article has been viewed 266,300 times. 42 votes - 57% Co-authors: 13 Updated: March 20, 2025 Views: 266,300 Categories: Geometry Article SummaryX

To calculate the apothem of a hexagon, start by dividing the hexagon into 6 triangles. Then, divide one of the triangles in half to create 2 right triangles. Next, plug the length of one of the right triangle's base and hypotenuse into the Pythagorean Theorem. Finally, solve for a to get the apothem of the hexagon. To learn how to calculate the apothem of a hexagon using trigonometry, scroll down! Did this summary help you?YesNo

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