Sum and Difference of Angles Practice Problems with Solutions

Understanding Sum of Angles in a Polygon

Complete explanation with examples

In any polygon, you can calculate the sum of its internal angles using the following formula:

Sum of the internal angles of a polygon: $=180\times\left(n-2\right)$
while
$n=$ The number of edges or sides of the polygon

Steps to find the sum of the internal angles of a polygon:

Count how many sides it has.
Place it in the formula and we will obtain the sum of the internal angles of the polygon.

Important

In the formula, there are parentheses that require us to first perform the operations of subtraction (first we will subtract $2$ from the number of edges and only then multiply by $180º$ .

First of all, observe how many sides the given polygon has and write it as $=n$ .
Then, note the correct n in the formula and discover the sum of the internal angles.

When it comes to a regular polygon (whose sides are all equal to each other) its angles will also be equal and we can calculate the size of each one of them.
For example, when it comes to a four-sided polygon (like a rectangle, rhombus, trapezoid, kite or diamond), the sum of its angles will be $360º$ degrees.
However, when it comes to a polygon of $7$ sides, the sum of its angles will be $900º$ degrees.

The sum of the external angles of a polygon will always be $360º$ degrees.

Detailed explanation

Practice Sum of Angles in a Polygon

Test your knowledge with 28 quizzes

Find the measure of the angle $\alpha$

Examples with solutions for Sum of Angles in a Polygon

Step-by-step solutions included

Exercise #1

Indicates which angle is greater

Step-by-Step Solution

Answer B is correct because the more closed the angle is, the more acute it is (less than 90 degrees), meaning it's smaller.

The more open the angle is, the more obtuse it is (greater than 90 degrees), meaning it's larger.

Answer:

Video Solution

Exercise #2

Indicates which angle is greater

Step-by-Step Solution

In drawing A, we can see that the angle is an obtuse angle, meaning it is larger than 90 degrees:

While in drawing B, the angle is a right angle, meaning it equals 90 degrees:

Therefore, the larger angle appears in drawing A.

Answer:

Video Solution

Exercise #3

Which angle is greatest?

Step-by-Step Solution

In drawing A, we can see that the angle is more closed:

While in drawing B, the angle is more open:

In other words, in diagram (a) the angle is more acute, while in diagram (b) the angle is more obtuse.

Remember that the more obtuse an angle is, the larger it is.

Therefore, the larger of the two angles appears in diagram (b).

Answer:

Video Solution

Exercise #4

What is the size of angle ABC?

Step-by-Step Solution

In order to calculate the value of angle ABC, we must calculate the sum of all the given angles.

That is:

$ABC=60+50$

$ABC=110$

Answer:

110

Video Solution

Exercise #5

Find the measure of the angle $\alpha$

Step-by-Step Solution

It is known that the sum of angles in a triangle is 180 degrees.

Since we are given two angles, we can calculate $a$

$94+92=186$

We should note that the sum of the two given angles is greater than 180 degrees.

Therefore, there is no solution possible.

Answer:

There is no possibility of resolving

Video Solution

Frequently Asked Questions

Everything you need to know about Sum of Angles in a Polygon

How do you find the sum of interior angles in any polygon?

+

Use the formula 180×(n-2) where n is the number of sides. For example, a pentagon has 5 sides, so the sum is 180×(5-2) = 540 degrees.

What is the difference between convex and concave polygons?

+

In convex polygons, all interior angles are less than 180 degrees and any line segment between two points stays inside the polygon. Concave polygons have at least one interior angle greater than 180 degrees with some connecting segments extending outside.

Why do exterior angles of any polygon always sum to 360 degrees?

+

Exterior angles represent the total rotation needed to traverse the polygon's perimeter once. This complete rotation always equals 360 degrees regardless of the polygon's shape or number of sides.

How do you calculate each angle in a regular polygon?

+

First find the total sum using 180×(n-2), then divide by the number of angles. For a regular hexagon: 180×(6-2) = 720 degrees total, so each angle is 720÷6 = 120 degrees.

What are the most common mistakes when using polygon angle formulas?

+

Students often forget to follow order of operations by subtracting 2 first before multiplying by 180. Another mistake is confusing interior and exterior angles or miscounting polygon sides.

Can you use the angle sum formula for irregular polygons?

+

Yes, the formula 180×(n-2) works for any polygon regardless of shape - regular, irregular, convex, or concave. Only the number of sides matters for calculating the total interior angle sum.

How do you identify the number of sides in complex polygons?

+

Count each straight edge systematically, often numbering them to avoid confusion. Include all sides even if the polygon has unusual shapes or indentations that make counting challenging.

What real-world applications use polygon angle calculations?

+

Architecture and construction use these formulas for building designs, land surveying for property boundaries, computer graphics for 3D modeling, and engineering for structural analysis of polygonal frameworks.

More Sum of Angles in a Polygon Questions

Click on any question to see the complete solution with step-by-step explanations