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
Read it, or press Listen and follow the words.
Try it first
In triangle ABC, ∠A = 71° and ∠B = 54°. Through A a line is drawn parallel to BC, and the two outer angles at A 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 17
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 17
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.
Problem 3 of 17
What do the interior angles of a 14-sided polygon add to, in degrees?
degrees
Problem 4 of 17
What is one exterior angle of a regular 18-sided polygon, in degrees?
degrees
Problem 5 of 17
What is one interior angle of a regular 18-sided polygon, in degrees?
degrees
Problem 6 of 17
A polygon's interior angles add to 1,980°. How many sides has it?
sides
Problem 7 of 17
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.
Problem 8 of 17
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°
Problem 9 of 17
Work it out properly. What is one interior angle of a regular 20-sided polygon, in degrees?
degrees
Problem 10 of 17
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
Problem 11 of 17
Work it out properly. A regular polygon has interior angles of 165°. How many sides has it?
sides
Problem 12 of 17
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°
Problem 13 of 17
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.
Problem 14 of 17
A regular polygon has interior angles of 175°. Which first move reaches the number of sides?
Problem 15 of 17
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.
Problem 16 of 17
Select every polygon whose interior angles add to 1,440°.
Problem 17 of 17
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.