Any polygon with n sides cuts into n − 2 triangles from one corner.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
So the interior sum
Each triangle contributes 180°, giving (n − 2) × 180° in total.
Exterior angles
They always sum to 360°, whatever n is — you have turned once round the shape.
Regular polygons
Every angle is equal, so divide the sum by n.
Cut the polygon into triangles
From one vertex of an n-sided polygon you can draw diagonals making n − 2 triangles. Each contributes 180°, so the interior angles total (n − 2) × 180°. The formula is a count of triangles, not something to memorise blind.
Regular polygons divide it evenly
In a regular polygon every interior angle is the total divided by n. A regular hexagon has (6 − 2) × 180 ÷ 6 = 120° at each vertex. Regularity is what licenses the division; an irregular hexagon has no single angle size.
Exterior angles always total 360°
Whatever the number of sides, the exterior angles sum to one full turn — imagine walking the perimeter and ending up facing your original direction. This makes many polygon problems far shorter through the exterior angle.
The formula needs convexity, carefully
The interior angle sum holds for concave polygons too, provided reflex angles are measured as reflex. The triangulation argument still works; it is the drawing that becomes awkward. Stating where a result applies is part of stating it.
Step 2: Try It Yourself
Tap and try it out.
Hexagon
Straight sides
6
Corners
6
A hexagon has 6 straight sides.
Turning it does not change what it is.
Step 3: Watch an Example
One step at a time.
Watch Amara Find One Angle of a Hexagon
A regular hexagon has six equal angles.
- Step 1
It cuts into 6 − 2 = 4 triangles.
Step 4: Your Turn
Practice makes it stick.
The Pentagon
Problem 1 of 2
Interior angle sum of a pentagon, in degrees?
Regular Pentagon
Problem 2 of 2
One angle of a regular pentagon, in degrees?
Sums and Single Angles
1 of 8
Interior sum of a quadrilateral, in degrees?
2 of 8
Interior sum of an octagon, in degrees?
3 of 8
One angle of a regular octagon, in degrees?
4 of 8
Exterior angle sum of any polygon, in degrees?
5 of 8
One exterior angle of a regular hexagon, in degrees?
6 of 8
Interior sum is 1440°. How many sides?
7 of 8
How many triangles does a decagon cut into?
8 of 8
Select every statement that is true for all polygons.
Step 5: Quick Check
Show what you know.
Question 1 of 1
Interior sum of a heptagon (7 sides), in degrees?
What You Learned
- A polygon with n sides cuts into n − 2 triangles, so its interior sum is (n − 2) × 180°.
- Exterior angles always sum to 360°, whatever the polygon.
- A regular polygon divides either sum evenly among its corners.