A hexagon is a six-sided polygon with six interior angles. The sum of the angles within this polygon is 720 degrees, with each individual interior angle at 120 degrees. This shape can be found in honeycombs and in nuts used to tighten mechanical components. In order to calculate the side length of a hexagon, you need at least one length value of the legs that form triangles within the hexagon. Because all sides of a hexagon are equal in length, you need only to find the length of one side of a hexagon in order to know the lengths of all of the sides.
Draw a Hexagon
Draw a hexagon on a sheet of paper. Use your ruler to ensure that all sides are equal in length.
Label each angle within the hexagon at 120 degrees. The sum of the interior angles of a hexagon is 720 degrees.
Draw a line from the top-left axis to three opposite axes, to form four triangles within the hexagon.
Label each of the smaller angles in the left-most triangle as 30 degrees. Because the left-most triangle is an isosceles triangle, its two sides are equal in length, meaning that its two smaller angles are equal in degree. Because the large angle is 120 degrees, the two remaining angles must be equal and total 60 degrees, meaning that each angle must be 30 degrees.
Label the smallest angle within the second triangle from the left at 30 degrees. The top four angles, creating the four triangles from the original axis point, should all be equal at 30 degrees.
Label the bottom left angle in the second triangle from the left as 90 degrees. Because its complementary angle is 30 degrees, this angle must be 90 degrees, as each interior hexagon angle is 120 degrees.
Label the third angle within the triangle, second from the left, at 60 degrees. Because a triangle equals 180 degrees and the other two angles are 30 and 90 degrees, the final must be 60 degrees. You now have a 30-60-90 right triangle.
Observe that within a 30-60-90 right triangle, the length of the hexagon side, which is opposite the 30 degree angle, is equal to one-half the length of the hypotenuse, or the side opposite the 90 degree angle. So if the hypotenuse is 8 inches long, the hexagon side length is 4 inches.
Observe also that the hexagon side length, or the side opposite the 30 degree angle, is equal to the quotient of the side length opposite the 60 degree angle divided by the square root of 3. That is, if the length of the side opposite the 60 degree angle is 17.5 centimeters, then the hexagon side length is that number divided by the square root of 3, or about 10 centimeters.
Calculating Side Length
Draw your hexagon so that one of its sides is parallel to the top of the paper. This will make it easier to visualize the angles. Draw your hexagon with equal sides. This will make it easier to visualize the angle degree and side length values. Use a pencil in case you make a mistake.
Plug any values that you have into the hexagon. You need at least one value in order to calculate the length of a hexagon side. The values could be the length of the line that completes any of the triangles within the hexagon.
Divide your value by the square root of 3 if your given value is the length of the line that completes the left-most or the right-most isosceles triangle in the hexagon. The quotient is the length of the hexagon side. If the value is 7, then the length of one side of the hexagon is 8 divided by the square root of 3, which is approximately 4.074.
Divide your value by 2 if your given value is the length of the center line that creates the middle two triangles within the hexagon. The quotient is the length of the hexagon side. If this value is 8, then the length of one side of the hexagon is 8 divided by 2, which is 4.
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