Eighth Grade Mathematics: Interior and Exterior Angles of Convex Polygons
Beyond the three-sided triangle lies the broader family of convex polygons: four-sided quadrilaterals, five-sided pentagons, six-sided hexagons, and polygons with dozens of sides. Although the total interior angle sum changes as more sides are added, the geometric progression follows an orderly algebraic rule based on partitioning shapes into non-overlapping triangles. In this chapter, you will derive the formula for the interior angle sum of an n-sided polygon, investigate the remarkable constant sum of exterior angles, and solve angle problems for regular polygons.
Deriving the Interior Angle Sum Formula: (n - 2) * 180
To find the sum of the interior angles of any convex polygon with n sides, choose one single vertex and draw all possible diagonals from that vertex to all non-adjacent vertices.
Partitioning Convex Polygons into Triangles from One Vertex:
Triangle (n = 3) Quadrilateral (n = 4) Pentagon (n = 5)
* *-------* *
/ \ / \ / / / \ / \ / / / \ / \ / * *
*-------* *-------* \ /
*-*
1 Triangle 2 Triangles 3 Triangles
Sum = 1 * 180 = 180 Sum = 2 * 180 = 360 Sum = 3 * 180 = 540
Notice the consistent mathematical pattern:
- A triangle (n = 3) contains 1 triangle: 1 = 3 - 2
- A quadrilateral (n = 4) contains 2 triangles: 2 = 4 - 2
- A pentagon (n = 5) contains 3 triangles: 3 = 5 - 2
- A hexagon (n = 6) contains 4 triangles: 4 = 6 - 2
Every convex polygon with n sides can be partitioned into exactly (n - 2) non-overlapping triangles sharing the same vertices. Because the angles of these triangles combine to form all the interior angles of the polygon, and each triangle contains 180 degrees, we arrive at the universal formula:
Interior Angle Sum Formula for ANY n-sided Convex Polygon:
S = (n - 2) * 180 degrees
Interior Angle Sum Reference:
Polygon Name Sides (n) Triangles (n - 2) Total Interior Angle Sum
Triangle 3 1 1 * 180 = 180 degrees
Quadrilateral 4 2 2 * 180 = 360 degrees
Pentagon 5 3 3 * 180 = 540 degrees
Hexagon 6 4 4 * 180 = 720 degrees
Heptagon 7 5 5 * 180 = 900 degrees
Octagon 8 6 6 * 180 = 1080 degrees
Decagon 10 8 8 * 180 = 1440 degrees
Regular Polygons: Equal Sides and Equal Angles
A regular polygon is both equilateral (all side lengths are congruent) and equiangular (all interior angle measures are congruent). Because all n interior angles are equal, you find the measure of a single interior angle by dividing the total sum S by the number of sides n:
Individual Interior Angle of a Regular n-sided Polygon:
(n - 2) * 180
Angle = -------------
n
Examples of Regular Polygon Interior Angles:
Regular Triangle (Equilateral): (3 - 2)*180 / 3 = 180 / 3 = 60 degrees
Regular Quadrilateral (Square): (4 - 2)*180 / 4 = 360 / 4 = 90 degrees
Regular Pentagon: (5 - 2)*180 / 5 = 540 / 5 = 108 degrees
Regular Hexagon: (6 - 2)*180 / 6 = 720 / 6 = 120 degrees
Regular Octagon (Stop sign): (8 - 2)*180 / 8 = 1080 / 8 = 135 degrees
The Exterior Angle Sum Constant: Always 360 Degrees
While the interior angle sum grows larger with every added side, the sum of the exterior angles (one exterior angle drawn per vertex) behaves in an extraordinary way: it NEVER changes.
The Exterior Angle Sum Theorem:
For ANY convex polygon, regardless of the number of sides n,
the sum of the exterior angle measures (one at each vertex) is ALWAYS 360 degrees!
To visualize why this is true, imagine walking along the perimeter of any polygon. As you turn each corner, the angle you turn through is an exterior angle. When you complete your walk around the polygon and return to your starting position facing the original direction, your body has completed exactly one full 360-degree rotation.
For a regular n-sided polygon, because all exterior angles are congruent, each individual exterior angle measures:
Individual Exterior Angle of a Regular Polygon = 360 / n degrees
Notice that the interior angle and exterior angle at each vertex form a linear pair: 108 + 72 = 180 degrees for a pentagon; 120 + 60 = 180 degrees for a hexagon; 135 + 45 = 180 degrees for an octagon.
Chapter Exercises
Exercise 1: Calculate the total sum of the interior angles of a convex polygon with 12 sides (a dodecagon). If the dodecagon is a regular polygon, determine the measure of each individual interior angle and each individual exterior angle.
Exercise 2: An unknown regular polygon has an individual interior angle that measures 144 degrees. Determine how many sides this regular polygon has, showing two distinct methods of solution (one using the interior angle formula and one using the exterior angle theorem).
Exercise 3: Five of the interior angles of a convex hexagon measure 110 degrees, 125 degrees, 130 degrees, 105 degrees, and 140 degrees. Calculate the measure of the sixth interior angle.
Solutions and Explanations
Solution 1: To find the total interior angle sum of a 12-sided polygon, use the formula S = (n - 2) * 180. Substituting n = 12 yields S = (12 - 2) * 180 = 10 * 180 = 1800 degrees. If the dodecagon is regular, all 12 interior angles are congruent, so divide the total sum by 12: 1800 / 12 = 150 degrees. To find the measure of each individual exterior angle, use the Exterior Angle Sum Theorem, which states that the exterior angles sum to 360 degrees: 360 / 12 = 30 degrees. Checking the linear pair: 150 + 30 = 180 degrees. The total sum is 1800 degrees, each interior angle measures 150 degrees, and each exterior angle measures 30 degrees.
Solution 2: Method 1 (Using the Exterior Angle): At each vertex of a regular polygon, the interior angle and exterior angle form a linear pair and sum to 180 degrees. If the interior angle measures 144 degrees, the exterior angle measures 180 - 144 = 36 degrees. Because the sum of all exterior angles is always 360 degrees, the number of sides n is found by dividing 360 by the measure of one exterior angle: n = 360 / 36 = 10 sides. Method 2 (Using the Interior Angle Formula): Set the individual interior angle formula equal to 144: [(n - 2) * 180] / n = 144. Multiply both sides by n: 180*(n - 2) = 144n. Distribute the 180: 180n - 360 = 144n. Subtract 144n from both sides: 36n - 360 = 0. Add 360 to both sides: 36n = 360. Divide by 36: n = 10 sides. Both methods confirm that the regular polygon has 10 sides (a decagon).
Solution 3: A hexagon has 6 sides (n = 6). First, calculate the total sum of the interior angles of a hexagon using S = (n - 2) * 180: S = (6 - 2) * 180 = 4 * 180 = 720 degrees. Next, sum the five given interior angles: 110 + 125 + 130 + 105 + 140 = 610 degrees. Let x represent the measure of the unknown sixth interior angle. Set up the equation: 610 + x = 720. Subtract 610 from both sides: x = 720 - 610 = 110 degrees. The sixth interior angle of the hexagon measures 110 degrees.