Lesson 3: Angles in Polygons

Property

• A polygon is a closed figure made up of line segments called sides.
• A triangle is a polygon with three sides.
• A quadrilateral is a polygon with four sides.
• A pentagon is a polygon with five sides.
• A hexagon is a polygon with six sides.
• An $n$-gon is a polygon with $n$ sides.

Examples

Property

The sum of the angles of a triangle is $180^\circ$.
If you cut out the angles and put all the vertices together, so that the angles are adjacent and not overlapping, you will get a straight angle.

Examples

• A triangle has angles measuring $50^\circ$ and $70^\circ$. The third angle is $180^\circ - (50^\circ + 70^\circ) = 180^\circ - 120^\circ = 60^\circ$.
• A right triangle has one angle of $90^\circ$. If another angle is $35^\circ$, the third angle must be $180^\circ - 90^\circ - 35^\circ = 55^\circ$.
• An isosceles triangle has two equal angles. If the unique angle is $40^\circ$, the other two angles together are $180^\circ - 40^\circ = 140^\circ$. Each equal angle is $140^\circ / 2 = 70^\circ$.

Explanation

No matter what a triangle looks like—tall, short, wide, or skinny—if you add its three corner angles together, you will always get exactly $180^\circ$. It is a fundamental rule for all triangles!

Property

The sum of the interior angles of a quadrilateral is $360^\circ$. For a polygon with $n$ sides, the sum of the interior angles is given by the formula:

This can be reasoned by considering that the sum of the exterior angles of any convex polygon is $360^\circ$. With $n$ vertices, the sum of all interior and exterior angles is $n \times 180^\circ$. So, the sum of interior angles is $(n \times 180^\circ) - 360^\circ = (n-2) \times 180^\circ$.

Examples

• For a pentagon ($n=5$), the sum of the interior angles is $(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$.

• A regular hexagon has 6 equal sides and angles. The sum of its angles is $(6-2) \times 180^\circ = 720^\circ$. Each individual angle is $720^\circ \div 6 = 120^\circ$.