Any polygon with n sides cuts into n − 2 triangles from one corner. Each triangle contributes 180°, so the interior angles add to (n − 2) × 180°.
Step 1: Let's Learn
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Try it first
In triangle ABC, ∠A = 71° and ∠B = 54°. Through A a line is drawn parallel to BC, putting three angles side by side on a straight line at A: a copy of ∠B, then ∠A itself, then a copy of ∠C — the two copies are alternate interior angles across the parallel lines. What is ∠C, in degrees?
degrees
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Why n − 2
From one corner, draw diagonals to every other corner except its two neighbors. That is n − 3 diagonals, making n − 2 triangles. A pentagon: 2 diagonals, 3 triangles, 540°.
Exterior angles
Walk round the shape, turning at each corner by the exterior angle. One full circuit is one full turn, so the exterior angles always sum to 360°, whatever n is.
Interior and exterior at one corner
They lie on a straight line, so they add to 180°. For a regular hexagon: interior 120°, exterior 60°, and six of 60° is 360°.
Regular polygons
Every angle is equal, so divide either sum by n. Each exterior angle is 360 ÷ n; each interior is 180 minus that. A regular octagon: exterior 45°, interior 135°.
Finding n from an angle
A regular polygon with interior angles of 150° has exterior angles of 30°. 360 ÷ 30 = 12 sides. The exterior angle is usually the quicker route.
Step 2: Try It Yourself
Tap and try it out.
A regular polygon has interior angles of 150°. How many sides?
- 1exterior = 180 − 150 = 30°interior and exterior lie on a straight line
Step 0 of 3
Hexagon
Straight sides
6
Corners
6
A hexagon has 6 straight sides.
Turning it does not change what it is.
Step 3: In Real Life
A gazebo
An octagonal gazebo’s interior angles sum to 1,080 degrees, 135 each. A carpenter cuts every corner joint at 135 degrees, straight from the formula.
Step 4: Watch an Example
One step at a time.
Watch Priya Find the Fifth Angle of a Pentagon
A five-sided garden bed has corners of 110°, 95°, 130° and 120°. The fifth corner was never measured. Priya finds it without going outside.
- Step 1
A pentagon has five sides, so from one corner it cuts into 5 − 2 = 3 triangles, and its interior angles add to 3 × 180 = 540°. Nothing here has to be regular: the cut into triangles works on any pentagon, however uneven its corners look.
Watch Dev Find the Angles of a Quadrilateral
A quadrilateral has angles of 3x, 4x, 5x and 6x. Dev writes the first two lines; you supply the rest.
- Step 1
A quadrilateral has four sides, so it cuts into 4 − 2 = 2 triangles from one corner, and its four angles add to 2 × 180 = 360°.
Two Roads to One Angle of a Regular Decagon
A regular decagon has ten equal sides and ten equal angles. There are two honest routes to one of those angles, and the difference between them is worth having.
- Step 1
The first road goes through the inside. Ten sides cut into 10 − 2 = 8 triangles, so the angles add to 8 × 180 = 1,440°, and ten equal angles share that out: 1,440 ÷ 10 = 144°.
Step 5: Your Turn
Practice makes it stick.
The Tile Order
Problem 1 of 2
A glass shop cuts regular polygon tiles and charges $3 for every side it has to cut. A customer orders one tile whose corners are each 140°. What does that tile cost, in dollars?
dollars
The Trophy Base
Problem 2 of 2
A trophy base is a polygon whose interior angles add to 1,620°. The engraver charges $6 for each corner. What does engraving every corner cost, in dollars?
dollars
Sums, Single Angles, and Sides
On the ones that name a regular polygon, two routes are open: through the triangles inside, or round the outside. Either is fine.
1 of 5
What do the interior angles of a 14-sided polygon add to, in degrees?
degrees
2 of 5
What is one exterior angle of a regular 18-sided polygon, in degrees?
degrees
3 of 5
What is one interior angle of a regular 18-sided polygon, in degrees?
degrees
4 of 5
A polygon's interior angles add to 1,980°. How many sides has it?
sides
5 of 5
A regular polygon has interior angles of 156°. How many sides has it?
sides
Whose Step?
One line in each of these is a slip. Find it, then work the problem out properly.
1 of 6
Maya worked out one interior angle of a regular 20-sided polygon and reached 180°. One line is a slip. Which?
Maya's working
- 1n = 20
- 2interior sum = 20 × 180 = 3600°
- Answer3600 ÷ 20 = 180°
2 of 6
Work it out properly. What is one interior angle of a regular 20-sided polygon, in degrees?
degrees
3 of 6
A regular polygon has interior angles of 165°. Dev worked out the number of sides and reached 2.2. One line is a slip. Which?
Dev's working
- 1interior angle = 165°
- 2n = 360 ÷ 165
- Answern ≈ 2.2
4 of 6
Work it out properly. A regular polygon has interior angles of 165°. How many sides has it?
sides
5 of 6
Leo worked out one exterior angle of a regular pentagon and reached 252°. One line is a slip. Which?
Leo's working
- 1interior sum = (5 − 2) × 180 = 540°
- 2one interior angle = 540 ÷ 5 = 108°
- Answerone exterior angle = 360 − 108 = 252°
6 of 6
Work it out properly. What is one exterior angle of a regular pentagon, in degrees?
degrees
Pick the Move
Choose the first move only. Nothing here needs finishing.
1 of 2
A regular polygon has interior angles of 175°. Which first move reaches the number of sides?
2 of 2
A polygon's interior angles add to 2,700°. Which first move reaches the number of sides?
Same Answer?
Two descriptions can name the same total, or the same polygon. Decide without working every one of them out.
1 of 2
Select every polygon whose interior angles add to 1,440°.
2 of 2
A regular polygon has interior angles of 150°. Select every description that names the same polygon.
Step 6: Quick Check
Show what you know.
Question 1 of 1
Interior sum of a heptagon (7 sides), in degrees?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
Three regular polygons meet at one point on a floor, and their three interior angles fill the 360° around that point exactly, with no gap and no overlap. One is a square and one is a regular hexagon. How many sides has the third?
sides
Stretch 2 of 2
The five interior angles of a pentagon are five consecutive whole numbers of degrees. What is the largest of them, in degrees?
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°: one full turn round the shape.
- At each corner, interior plus exterior is 180°.
- A regular polygon divides either sum evenly, and the exterior angle finds n fastest.
For the grown-up
The misconception this lesson targets is that a polygon has one triangle for every side, so the interior angles add to n × 180°. It survives because it is nearly right and is never contradicted by a picture: an octagon does look like eight triangles round a middle point, and it is, but that construction adds a whole 360° of angle at the centre that no corner of the octagon owns. Cutting from one corner instead gives n − 2 triangles and adds nothing. Its cousin is the belief that the exterior angles add to more when a shape has more sides; they never do, because one walk round any polygon is exactly one full turn, which is why the outside route stays small while the inside route grows. A third slip worth watching for is dividing an interior total by n when nobody said the polygon was regular: only equal angles can be shared out equally.