Finding the Angle Sum of a Pentagon

What is the sum of the interior angles of a pentagon?

Identify the number of sides: A pentagon has 5 sides, so $n = 5$ = $n = 5$

Apply the formula: Sum $= (n - 2) \times 180°$ = Sum $= (5 - 2) \times 180°$

Calculate the number of triangles: $5 - 2 = 3$ triangles = $3$ triangles

Multiply by 180°: $3 \times 180° = 540°$ = $540°$

Answer: The sum of interior angles in a pentagon is $540°$

Finding a Missing Angle in a Quadrilateral

A quadrilateral has angles of $85°$, $110°$, and $95°$. Find the fourth angle.

Find the total angle sum: Quadrilateral: $n = 4$, Sum $= (4 - 2) \times 180° = 360°$ = Total $= 360°$

Add the known angles: $85° + 110° + 95° = 290°$ = Known angles $= 290°$

Subtract from total: $360° - 290° = 70°$ = Missing angle $= 70°$

Answer: The fourth angle is $70°$

Finding Each Angle in a Regular Hexagon

What is the measure of each interior angle in a regular hexagon?

Find the total angle sum: Hexagon: $n = 6$, Sum $= (6 - 2) \times 180° = 720°$ = Total $= 720°$

Understand 'regular': In a regular polygon, all angles are equal = 6 equal angles

Divide by the number of angles: $720° \div 6 = 120°$ = Each angle $= 120°$

Answer: Each interior angle in a regular hexagon is $120°$

Mistake: Using $n \times 180°$ instead of $(n - 2) \times 180°$

Why: Students forget to subtract 2 from the number of sides. The formula counts triangles, not sides.

Correct: Always subtract 2 first: $(n - 2) \times 180°$. A pentagon has 5 sides but only 3 triangles.

Mistake: Confusing interior angles with exterior angles

Why: The exterior angle sum is always $360°$ for any convex polygon, regardless of the number of sides.

Correct: Interior angles are inside the polygon. The formula $(n - 2) \times 180°$ is for interior angles only.

Mistake: Forgetting that triangles sum to $180°$

Why: The whole formula is based on dividing the polygon into triangles.

Correct: Remember: triangle = $180°$, that's why we multiply $(n - 2)$ by $180°$.

Architecture and Design

Architects use the angle sum formula when designing polygonal rooms, windows, or floor plans.

A pentagonal window has 5 sides. The angle sum is $(5 - 2) \times 180° = 540°$. If 4 corners are right angles (90° each), the fifth angle must be $540° - 360° = 180°$... which is impossible! This tells the architect the design needs adjustment.

Sports Field Design

Sports facilities sometimes use polygonal shapes for efficiency or aesthetics.

A hexagonal practice area has interior angles summing to $720°$. If it's regular, each angle is $720° \div 6 = 120°$.

Q: Does this formula work for all polygons?

A: Yes, for all simple (non-self-intersecting) polygons, whether regular or irregular, convex or concave.

Q: Why do we subtract 2 in the formula?

A: When you draw diagonals from one vertex, you create $(n - 2)$ triangles. Try it with a quadrilateral: 2 triangles. Pentagon: 3 triangles. The pattern is always $n - 2$.

Q: What about exterior angles?

A: The sum of exterior angles is always $360°$ for any convex polygon, regardless of the number of sides. This is a separate formula!