What Do the Angles in a Polygon Add Up To?
The interior angles of a pentagon
Angles in a polygon
A polygon is any 2D shape constructed from a finite number of straight lines. For example, a triangle is a three-sided polygon, and a square is a type of four-sided polygon, and so on.
All of these polygons have angles both on the inside and the outside, but what do these angles add up to? Do they always total the same thing? Is there a pattern to their totals? Let's start by looking at the interior angles of a triangle.
All of these polygons have angles both on the inside and the outside, but what do these angles add up to? Do they always total the same thing? Is there a pattern to their totals? Let's start by looking at the interior angles of a triangle.
The sum of the interior angles of a triangle
The interior angles of a polygon are, as the name suggests, the angles formed on the inside where the polygon's edges meet.
Before we start looking at the interior angles of many-sided polygons, we will start with three-sided polygons, otherwise known as triangles.
We can quickly work out the sum of the three interior angles of a triangle by considering a triangle with an extra straight line drawn parallel to the base of the triangle and touching the triangle's top corner, as in the diagram below.
To find the sum of the interior angles a + b + c, we note that the angle x, formed between the triangle and the parallel top line, is alternate to b, hence b = x. Likewise, c = y. We can also see that x, a, and y join together on the straight line, hence a + x + y = 180°.
Replacing x and y with b and c, we get that a + b + c = 180°.
We have just shown that the three interior angles of a triangle must always total 180°.
Before we start looking at the interior angles of many-sided polygons, we will start with three-sided polygons, otherwise known as triangles.
We can quickly work out the sum of the three interior angles of a triangle by considering a triangle with an extra straight line drawn parallel to the base of the triangle and touching the triangle's top corner, as in the diagram below.
To find the sum of the interior angles a + b + c, we note that the angle x, formed between the triangle and the parallel top line, is alternate to b, hence b = x. Likewise, c = y. We can also see that x, a, and y join together on the straight line, hence a + x + y = 180°.
Replacing x and y with b and c, we get that a + b + c = 180°.
We have just shown that the three interior angles of a triangle must always total 180°.
Proof of the interior angle sum of a triangle
The interior angles of a quadrilateral
A four-sided polygon is known as a quadrilateral. This group includes such familiar shapes as squares, rectangles, parallelograms, and more.
Now we know that a triangle's interior angles sum to 180°, we can quickly calculate the sum of the quadrilateral's interior angles.
Take a look at the diagram below. We have taken a quadrilateral and split it diagonally into two triangles. We can see from the diagram that the interior angles of the two triangles make up all four of the interior angles of the quadrilateral. Therefore, the quadrilateral angles must add up to twice that of the triangle angles.
180° × 2 = 360° and so the interior angles of any quadrilateral must always equal 360°.
Now we know that a triangle's interior angles sum to 180°, we can quickly calculate the sum of the quadrilateral's interior angles.
Take a look at the diagram below. We have taken a quadrilateral and split it diagonally into two triangles. We can see from the diagram that the interior angles of the two triangles make up all four of the interior angles of the quadrilateral. Therefore, the quadrilateral angles must add up to twice that of the triangle angles.
180° × 2 = 360° and so the interior angles of any quadrilateral must always equal 360°.
The interior angles of a quadrilateral
The interior angles of further polygons
We can use this same method for calculating the interior angles of polygons with more than four sides. Simply cut the polygon up into triangles by using diagonal lines, making sure that the interior angles of the triangles all match up with the interior angles of the polygon. Then multiply the number of triangles by 180° to get the interior angle sum of the polygon.
The diagram below shows how this can be used on a pentagon (a five-sided polygon).
The diagram below shows how this can be used on a pentagon (a five-sided polygon).
The interior angles of a pentagon
We can see in the pentagon diagram that a pentagon is split into three internal triangles, so it must have an interior angle sum of 3 × 180° = 540°.
It can be seen from our examples so far that the number of triangles that a polygon can be split into is two fewer than the total number of sides. For example, a four-sided quadrilateral is split into 4 − 2 = 2 triangles, and a five-sided pentagon is split into 5 − 2 = 3 triangles.
You can check quite quickly that this can be expanded to cover further polygons. Whatever the number of sides, we subtract two from this to give the number of triangles that we can split the polygon into. By multiplying this number of triangles by 180°, we then get the sum of the interior angles.
It can be seen from our examples so far that the number of triangles that a polygon can be split into is two fewer than the total number of sides. For example, a four-sided quadrilateral is split into 4 − 2 = 2 triangles, and a five-sided pentagon is split into 5 − 2 = 3 triangles.
You can check quite quickly that this can be expanded to cover further polygons. Whatever the number of sides, we subtract two from this to give the number of triangles that we can split the polygon into. By multiplying this number of triangles by 180°, we then get the sum of the interior angles.
Interior angle sums of various polygons
Polygon |
Number of sides |
Number of internal triangles |
Interior angle sum |
Triangle |
3 |
1 |
180 |
Quadrilateral |
4 |
2 |
360 |
Pentagon |
5 |
3 |
540 |
Hexagon |
6 |
4 |
720 |
Heptagon |
7 |
5 |
900 |
Octagon |
8 |
6 |
1080 |
Nonagon |
9 |
7 |
1260 |
Decagon |
10 |
8 |
1440 |
Icosagon |
20 |
18 |
3240 |
n-sided polygon |
n |
n-2 |
180(n-2) |
The general formula
As we can always split the polygon into two fewer triangles than there are number of sides, a polygon with n sides can be split into n − 2 triangles and hence has an interior angle sum of 180(n − 2)°.
Interior angle of an n-sided polygon = 180(n - 2)
What about exterior angles?
So far, we have been looking at the interior angles of a polygon, but what about exterior angles?
Exterior angles lie on the outside of the polygon and are created by extending a side and measuring the angle between the extended side and the next side. This can be seen in the diagram below.
As the exterior angles are created by the extended straight line, an exterior angle and its corresponding interior angle must always add together to make 180°.
Exterior angles lie on the outside of the polygon and are created by extending a side and measuring the angle between the extended side and the next side. This can be seen in the diagram below.
As the exterior angles are created by the extended straight line, an exterior angle and its corresponding interior angle must always add together to make 180°.
The exterior angles of a polygon
Calculating the sum of the exterior angles
To calculate the sum of the exterior angles, imagine you have a large polygon drawn on the floor and you are walking along a side. You reach the corner and then must rotate through the exterior angle in order to be facing along the next side. You walk along this second side, and again, when you reach the corner, you rotate through the next exterior angle in order to continue your journey.
This happens at each corner until you have returned to your starting point. At this point, you are now facing the same direction you started with, having turned through a full circle. A full circle is 360°, hence all of the exterior angles must sum to 360°.
This is the same regardless of how many sides the polygon has. The exterior angles of a triangle add up to 360°; the exterior angles of a decagon also add up to 360°.
This happens at each corner until you have returned to your starting point. At this point, you are now facing the same direction you started with, having turned through a full circle. A full circle is 360°, hence all of the exterior angles must sum to 360°.
This is the same regardless of how many sides the polygon has. The exterior angles of a triangle add up to 360°; the exterior angles of a decagon also add up to 360°.
Exterior angle sum of a polygon = 360°
Comments
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