Angles in Polygons

Calculate the sum of interior angles for a pentagon.

Solution:

A pentagon has five sides, so using the formula, the sum of interior angles is $(5-2)\times 180=540°$

Calculate the sum of interior angles for a nonagon.

Solution:

A nonagon has nine sides, so using the formula, the sum of interior angles is $(9-2)\times 180=1260°$

Calculate the sum of interior angles for the shape below.

Angles in Polygons- 14 sided polygon, Jordan Madge- StudySmarter Originals

Solution:

The above shape has 14 sides and so the sum of interior angles is $(14-2)\times 180=2160°$

Calculate the sum of interior angles for a 24 sided shape.

Solution:

When id="2555885" role="math" $n=24$, the sum of interior angles is $(24-2)\times 180=3960°$

Calculate the size of the angle x in the image below.

Angles in Polygons- quadrilateral example, Jordan Madge- StudySmarter Originals

Solution:

This shape has five sides, so the sum of interior angles is $(5-2)\times 180=540°$

Each of the right angles in the shape is $90°$and so we can work out the missing angle by subtracting all of the given angles from 540. Thus, $x=540-90-90-90-130=140°$

Calculate each interior angle for a regular hexagon.

Solution:

Using table 1, we can see that the sum of interior angles for a hexagon is $720°$. Since this hexagon is regular, each of the angles is the same and thus we can work out each interior angle by dividing $720$ by $6$. Therefore, each interior angle is $120°$.

Below is part of a tiling pattern consisting of three regular pentagons. Calculate the angle labelled x.

Angles in Polygons- Pentagon Example, Jordan Madge- StudySmarter Originals

Solution:

The sum of interior angles for each regular hexagon is $720°$ (using the table of common interior angles).

Thus, each interior angle in each hexagon is $120°$.

Angles in Polygons- Pentagon Example, Jordan Madge- StudySmarter Originals

Recall that angles around a point sum to 360 degrees. Therefore, x can be found by subtracting the other known angles from 360. Thus, $x=360-108-108=144°$

In the image below, angles x and y are exterior angles. Calculate x and y.

Angles in Polygons- Pentagon with interior and exterior angles, Jordan Madge- StudySmarter Originals

Solution:

For exterior angle x, the interior angle is $109°$. Thus, since angles on a straight line add up to $180°$, $x=180-109=71°$. Angle $y$ is another exterior angle and since angles on a straight line add to 180, $y=180-81=99°$.

Each exterior angle of a regular polygon is 10. Work out the number of sides the polygon has.

Solution:

Since the sum of exterior angles is $360°$, and each exterior angle is $10°$, we can calculate the number of sides by $360÷10=36$. Thus, this polygon has 36 sides.

Each interior angle of a regular polygon is 165. Work out the number of sides the polygon has.

Solution:

If each interior angle is $165$, each exterior angle must be $180-165=15°$. Since the sum of the exterior angles is $360°$, there must be $360÷15=24$ sides.