Polygon Properties: Step-by-Step Guide
A polygon is a closed shape made of straight sides. Polygons have properties you can figure out, like how many sides they have and what their angles add up to. The inside angles of a polygon with n sides always add up to (n - 2) x 180 degrees. The outside angles always add up to 360 degrees. In a regular polygon, all sides and angles are equal, so you can find each angle.
Math Racer skill guide · Elementary · Grades K-5
How to solve it, step by step
- Count the sides. Count the number of sides. Call that number n. A polygon has the same number of corners as sides.\[n = \text{number of sides}\]
- Find the sum of the inside angles. Use the rule: sum = (n - 2) x 180 degrees.\[S_{\text{int}} = (n - 2) \cdot 180^\circ\]
- Find each inside angle (regular polygons). If all the angles are equal, divide the sum by n.\[\text{each interior angle} = \frac{(n-2)\cdot 180^\circ}{n}\]
- Find each outside angle (regular polygons). The outside angles always add up to 360 degrees. If all the angles are equal, divide 360 by n.\[\text{each exterior angle} = \frac{360^\circ}{n}\]
Worked example
Find the sum of interior angles, the measure of each interior angle, and the measure of each exterior angle of a regular hexagon.
Step 1: A hexagon has 6 sides, so n = 6.
Step 2: Sum of inside angles: (6 - 2) x 180 = 4 x 180 = 720 degrees.
Step 3: Each inside angle: 720 / 6 = 120 degrees.
Step 4: Each outside angle: 360 / 6 = 60 degrees.
Step 5: Check: an inside angle and an outside angle make a straight line, so 120 + 60 should be 180. It is.
So the regular hexagon has an inside angle sum of 720 degrees, each inside angle is 120 degrees, and each outside angle is 60 degrees.
Tips and common mistakes
- Why (n - 2) x 180? Any polygon can be cut into (n - 2) triangles, and each triangle has 180 degrees.
- The outside angles of any polygon always add up to 360 degrees, no matter how many sides it has.
- At each corner, the inside angle and the outside angle add up to 180 degrees.
- Only use 'sum divided by n' if all the angles are equal, which means the polygon is regular.
- Good ones to remember: each angle of a square is 90, a regular pentagon is 108, a regular hexagon is 120, and a regular octagon is 135.
- If you know one outside angle of a regular polygon, divide 360 by it to find the number of sides.
- Always check that the polygon is regular before you divide to find a single angle.