How to Calculate the Apothem of a Hexagon

Step-by-Step Guide

1. 1

   Step 1: Divide the hexagon into six congruent

   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. , 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. , The formula is a2+b2=c2{\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.
   
   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 of about
   1.732{\displaystyle
   1.732} inches (3{\displaystyle {\sqrt {3}}}), the Pythagorean Theorem would state that 12+32=22{\displaystyle 1^{2}+{\sqrt {3}}^{2}=2^{2}}, which is true when you complete the calculations: 1+3=4{\displaystyle 1+3=4}. , Substitute for b{\displaystyle b}.
   
   For example, if the length of the base is 4 cm, your formula will look like this: a2+42=c2{\displaystyle a^{2}+4^{2}=c^{2}}. , 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.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: a2+42=82{\displaystyle a^{2}+4^{2}=8^{2}}. , Remember that squaring a number means to multiply it by itself.
   
   For example, squaring the known values, your formula will look like this: a2+16=64{\displaystyle a^{2}+16=64}. , To do this, subtract the squared value of b{\displaystyle b} from both sides of the equation.
   
   For example:a2+16−16=64−16{\displaystyle a^{2}+16-16=64-16}a2=48{\displaystyle a^{2}=48} , 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.
2. 2

   Step 2: equilateral triangles.To do this

3. 3

   Step 3: draw a line connecting each vertex

4. 4

   Step 4: or point

5. 5

   Step 5: with the vertex opposite.

6. 6

   Step 6: Choose one triangle and label the length of its base.

7. 7

   Step 7: Create two right triangles.

8. 8

   Step 8: Set up the formula for the Pythagorean Theorem.

9. 9

   Step 9: Plug the length of the right triangle’s base into the formula.

10. 10

    Step 10: Plug the length of the hypotenuse into the formula.

11. 11

    Step 11: Square the known values in the formula.

12. 12

    Step 12: Isolate the unknown variable.

13. 13

    Step 13: Solve for a{\displaystyle a}.

Detailed Guide

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