Sum of exterior angles of a hexagon

Q: How do you find the exterior angle of a hexagon?

d=180(n−1)n, d” represents interior angle and n is the number of sides in the polygon. So, the value of d or interior angle is 120 degrees. Therefore, the exterior angle of a regular hexagon will be = 180 – 120 = 60 degrees.

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Table of Contents Show

Interior angles in a regular hexagon
Formula to find the angles in a hexagon
How to find a missing angle in a hexagon?
Solved examples of interior angles of a hexagon
How do you find the exterior angle of a hexagon?
Is the sum of exterior angles always 360?
How do you find the sum of exterior angles?
What is the sum of exterior and interior angle of a hexagon?

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Hint: Hexagon is a polygon having six sides. Sum of its interior angles is ${720^ \circ }$. But the sum of exterior angles of any polygons is ${360^ \circ }$ irrespective of the number of sides in the polygon.

Complete step-by-step answer:

A hexagon is a polygon consisting of six sides. The total of the internal angles of a hexagon is ${720^ \circ }$.
If the hexagon is a regular hexagon, all the six sides of it are equal and all the six internal angles are also equal. And the measure of each angle is ${120^ \circ }$.
While the sum of interior angles of a polygon varies with the number of sides in a polygon, the sum of exterior angles remains the same for all polygons and it is ${360^ \circ }$.
In our case, if the hexagon is a regular hexagon, all the six exterior angles will be equal and each of them will measure ${60^ \circ }$. But if it is not regular, their values will differ. But the sum will always be the same which is ${360^ \circ }$.
Therefore, in the figure:
$ \Rightarrow a + b + c + d + e + f = {360^ \circ }$

(A) is the correct option.

Note: Sum of the interior angles of a polygon is determined by the formula $\left( {n - 2} \right) \times {180^ \circ }$ where $n$ is the number of sides in the polygon. And the sum of exterior angles is ${360^ \circ }$ irrespective of the number of sides in the polygon.

Hexagons have a sum of interior angles of 720°. A regular hexagon has all its angles with the same measure, so each interior angle measures 120°. On the other hand, irregular hexagons have angles with different measures, but they always have a total sum equal to 720°.

Here, we will learn about the interior angles of hexagons in more detail. We will learn the formula used to find the sum of these angles and we will look at some examples.

GEOMETRY

Relevant for…

Learning about the interior angles of a hexagon with examples.

See angles

Contents

Interior angles in a regular hexagon
Formula to find the angles in a hexagon
How to find a missing angle in a hexagon?
Solved examples of interior angles of a hexagon
See also

GEOMETRY

Relevant for…

Learning about the interior angles of a hexagon with examples.

See angles

Interior angles in a regular hexagon

A regular hexagon has the main characteristic that all its sides have the same length, all the sides of the hexagon are congruent. This also means that all interior angles have the same measure.

The sum of the interior angles of any hexagon is always equal to 720°. We can find the measure of each interior angle of a regular hexagon by dividing 720° by 6. Therefore, we have:

720°÷6 = 120°

Each interior angle in a regular hexagon measures 120°.

In the following diagram, we have a regular hexagon, which has sides of the same length and angles of the same measure. We can verify that by adding the six 120° angles, we get a total of 720°.

Formula to find the angles in a hexagon

We can find the sum of the interior angles of any polygon by applying the following formula:

$latex (n-2)\times 180$°

In this formula, n is equal to the number of sides of the polygon. In this case, we use $latex n = 6$ for a hexagon. Using this value, we have $latex (4) \times 180 = 720$°. This shows that the sum of the interior angles of a hexagon is equal to 720°.

How to find a missing angle in a hexagon?

A missing angle of an irregular hexagon can be found by adding the measures of all known angles and subtracting the obtained value from 720°.

For example, in the following hexagon, we have the angles 110°, 140°, 100°, 120°, 150°.

We add all the known angles:

110°+140°+100°+120°+150° = 620°

Now, we subtract the obtained value from 720°:

720°-620° = 100°

The missing angle of the hexagon has a measure of 100°.

In the following example, we have to find more than one missing angle. In this case, the angles that are different have different symbols. For example, angles that have a double line are equal.

We know that the angles that have a double line are equal, therefore we have a=130°.

Similarly, angles b and c are equal since they have triple lines. We can find the measure of these angles by following a process similar to the previous example. We start by adding the known angles:

100°+130°+130°+120° = 480°

Now, we have to subtract this value from 480° to find the missing angles:

720°-480° = 240°

The missing angles have the same measure, so we divide the obtained value by 2 to obtain the value of each angle. Therefore, we have b=120° and c=120°.

Solved examples of interior angles of a hexagon

EXAMPLE 1

Determine the missing angle in the hexagon below.

Solution: We can look at the angles 130°, 90°, 115°, 140°, and 150°. Therefore, we start by adding them:

130°+90°+115°+140°+150° = 625°

Now, we subtract the obtained value from 720° to find the missing angle:

720°-615° = 95°

The missing angle measures 95°.

EXAMPLE 2

Find the missing angles in the hexagon below.

Solution: The angles shared by the three lines are equal. Therefore, we have a=120°.

Angles b and c also have the same measure since they are both represented by a double line. We can determine its measure by starting by adding the angles we know:

115°+120°+120°+105°=460°

Now, we subtract the obtained value from 720° and we have:

720°-460° = 260°

The result represents the sum of the two missing angles. Since both angles are equal, we have to divide by 2 to get the value of one angle. Therefore, we have b = 130° and c = 130°.

How do you find the exterior angle of a hexagon?

d=180(n−1)n, “d” represents interior angle and n is the number of sides in the polygon. So, the value of d or interior angle is 120 degrees. Therefore, the exterior angle of a regular hexagon will be = 180 – 120 = 60 degrees.

Is the sum of exterior angles always 360?

Exterior angles of a polygon are formed when by one of its side and extending the other side. The sum of all the exterior angles in a polygon is equal to 360 degrees.

How do you find the sum of exterior angles?

Polygon Exterior Angle Sum Theorem.

If a polygon is convex, then the sum of the measures of the exterior angles, one at each vertex, is 360° ..

Consider the sum of the measures of the exterior angles for an n -gon..

N=180n−180(n−2).

Distribute 180 ..

N=180n−180n+360 =360..

What is the sum of exterior and interior angle of a hexagon?

There are six angles, so 720 ÷ 6 = 120°. Each interior angle of a regular hexagon has a measure of 120°. Because the sum of these angles will always be 360°, then each exterior angle would be 60° (360° ÷ 6 = 60°).