Calculating Polygon Angles and Sides Lengths

A polygon is any closed plane figure. It comes from the Latin poly meaning "many" and gōnia, meaning "angle." "Closed," in this context, means that the sides form a complete circuit. This definition does not exclude shapes such as an hourglass or a star where sides cross each other. When sides do not cross each other, we call them "simple polygons." For this article, we will only be using simple polygons.

Polygons must have at least three sides as a two-sided shape cannot be closed. The point where two sides meet is called a "vertex" (plural: "vertices"). When you draw a line segment drawn from one vertex to another, it is called a "diagonal."

A polygon is "convex" if any diagonal is drawn in the exterior of the polygon. The Red Cross logo is convex because the diagonal from one corner to the next is on the exterior. The "angles" of the polygon are all interior angles. Thus, the four angles where the cross pieces meet are 270° rather than 90°.

Naming Polygons

Naming polygons are generally based on the number of sides or number of angles. For example, an "equilateral" triangle has three equal sides, and an "equiangular" triangle has three equal angles. (Of course, they are the same.) Usually, however, names refer to the number of sides a polygon has. A hexagon has six sides, an octagon has eight, etc. The general name "n-gon" is a polygon with n sides. So, a hexagon is a 6-gon.

When a polygon's sides are the same length and angles are the same degree, we call it a "regular" polygon. A square is regular. All sides and all angles are equal. The Pentagon in Washington D.C. is a regular 5-gon.  A cutaway of most pencils is a regular hexagon, and stop signs are usually regular octagons. A rectangle has four equal (90°) angles, and a rhombus has four equal sides, but they are not "regular."

Angle Measurements

Sum of All Angles

The sum of the angles of any triangle will always equal 180° no matter how big it is.

You can divide larger polygons into triangles by drawing diagonals between vertices, quadrilaterals (4-sided polygons) into two triangles, pentagons (5-sided polygons) into three triangles, etc. You can divide any n-gon into n-2 triangles. Each triangle has a sum of 180°.

Thus, the sum of the angles of any polygon is:

S = (n – 2) * 180

For example, the sum of all eight angles of an octagon is:  S = (8 – 2) * 180  =  1080°.

This formula works whether or not the polygon is regular and even works if the polygon is convex. The Red Cross symbol is a convex 12-gon. It has four 270° angles where the cross pieces meet and eight 90° angles on the outside corners.

4(270) + 8(90) = 1800°.

Using our polygon formula, (12 – 2) * 180 = 1800°. Isn't that amazing?

Individual Angles

This formula can be used to find individual angles if the polygon is regular. For a regular octagon, such as a stop sign, the sum of all eight angles is 1080°, so each angle must be 1080/8 = 135°. Each angle in a regular hexagon is (6 – 2) * 180 / 6 = 120°.

For irregular polygons, if you know all angles except one, you can find the missing angle.

A gardener has walkways that form an almost pentagon—almost because one of the corners is covered by a fishpond. The shape has two right angles, and he measures the other two at 65° and 58°. The sum of the known angles is 303°.  The sum of all angles of a pentagon is 540°, so the angle under the fishpond is 540 – 303 = 237°.  This interior angle is greater than 180°.  Most likely, the gardener wants the exterior angle, which is 237 – 180 = 57°.

Side Lengths

There aren't many rules for finding the lengths of sides of polygons, but this usually isn't a problem. Side lengths are much easier to measure than angles, especially if you're working with a regular polygon. All sides are equal on regular polygons. If you measure one side, you'll know the length of the rest. In rectangles, opposite sides are equal.

Pythagorean Theorem

One way to calculate the sides of a right triangle is with the Pythagorean Theorem. A right triangle is a triangle that has a right angle (90°) made from two legs. The "hypotenuse" is the side across from the right angle. If you square (multiply a number by itself) the length of the two legs and then add the sums together, you will get the result of squaring the hypotenuse. If the leg lengths are represented by a and b, and the length of the hypotenuse is c, then the equation is a2 + b2 = c2

Here is a simple example.  An ordinary sheet of plywood is 4 ft wide by 8 ft long.  How long is the diagonal from one corner to the opposite corner?

42 + 82 = c2

16 + 64 = c2

80 = c2

√(80) = c ≈ 8.94

You would express this in carpenter math as: 0.94 ft = 0.94 x 12 in/ft = 11.28 inches. The length of the diagonal is about 8'-11¼".

Trigonometric Relations

Trigonometric relations are useful for right triangle ratios. They are based on an observation of similar triangles (same shape but not the same size). If two triangles have the same three angles, then the ratio of two sides of the first triangle will equal the ratio of the corresponding sides of the second triangle.

Here's how trig functions work. Consider a triangle with a right angle on one corner and a 31° angle on another corner. The third angle must be 180 – 90 – 31 = 59°.  All 31-59-90 triangles are similar, and the ratio of two sides of one will equal the ratio of the corresponding sides of all others.

Remember, the sides that form the right angle are called "legs" (usually designated a and b), and the side opposite the right angle is the "hypotenuse" (usually designated c).  More specifically, the side opposite the designated angle (in this case, 31°) is a, and the side adjacent to it is b, and the angle measures are α and β, respectively.

The ratio a/b or opposite/adjacent is given the name "tangent." You can determine the value of the tangent by measuring a and b and then dividing. But since it will be the same for every 31-59-90 triangle, you can find the value using trig tables or on scientific or online calculators.

The Pythagorean Theorem and Trig Functions only apply to right triangles, but very often, you can break down more complex polygons into several right triangles. You can then use known values to calculate the unknown ones.

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