Sum and Difference of Angles Practice Problems with Solutions

Master polygon angle calculations with step-by-step practice problems. Learn interior and exterior angle formulas, convex vs concave polygons, and solve real examples.

📚Master Angle Sum Calculations Through Interactive Practice
  • Calculate interior angle sums using the formula 180×(n-2) for any polygon
  • Distinguish between convex and concave polygons based on angle properties
  • Find individual angle measures in regular polygons by dividing total sum
  • Apply exterior angle theorem knowing sum always equals 360 degrees
  • Solve complex polygon problems with step-by-step systematic approach
  • Master order of operations when using polygon angle formulas

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 Angles in a Polygon Equation

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

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

  1. Count how many sides it has.
  2. 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 2 from the number of edges and only then multiply by 180º 180º .

First of all, observe how many sides the given polygon has and write it as =n =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 360º degrees.
However, when it comes to a polygon of 7 7 sides, the sum of its angles will be 900º 900º degrees. 

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

Detailed explanation

Practice Sum of Angles in a Polygon

Test your knowledge with 27 quizzes

Find the size of angle \( \alpha \).

27.727.727.7AAABBBCCC41

Examples with solutions for Sum of Angles in a Polygon

Step-by-step solutions included
Exercise #1

What type of angle is α \alpha ?

αα

Step-by-Step Solution

Remember that an acute angle is smaller than 90 degrees, an obtuse angle is larger than 90 degrees, and a straight angle equals 180 degrees.

Since the lines are perpendicular to each other, the marked angles are right angles each equal to 90 degrees.

Answer:

Straight

Exercise #2

What is the size of the missing angle?

80

Step-by-Step Solution

To find the size of the missing angle, we will use the property that the sum of angles on a straight line is 180∘180^\circ. Given that one angle is 80∘80^\circ, we can calculate the missing angle using the following steps:

  • Step 1: Recognize that the given angle α=80∘\alpha = 80^\circ and the missing angle β\beta form a straight line.
  • Step 2: Use the angle sum property for a straight line: α+β=180∘ \alpha + \beta = 180^\circ
  • Step 3: Substitute the known value: 80∘+β=180∘ 80^\circ + \beta = 180^\circ
  • Step 4: Solve for the missing angle β\beta: β=180∘−80∘=100∘ \beta = 180^\circ - 80^\circ = 100^\circ

Therefore, the size of the missing angle is 100∘100^\circ.

Answer:

100°

Video Solution
Exercise #3

Indicates which angle is greater

Step-by-Step Solution

Note that in drawing B, the two lines form a right angle, which is an angle of 90 degrees:

While the angle in drawing A is greater than 90 degrees:

Therefore, the angle in drawing A is larger.

Answer:

Video Solution
Exercise #4

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 #5

Which angle is greater?

Step-by-Step Solution

The angle in diagram (a) is more acute, meaning it is smaller:

Conversely, the angle in diagram (b) is more obtuse, making it larger.

Answer:

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

Sum and Difference of Angles

Triangle Angle Analysis: Can 30°, 60°, and 90° Form a Valid Triangle? Triangle Angle Verification: Do 56°, 89°, and 17° Form a Valid Triangle? Triangle Angle Problem: Can 90, 115, and 35 Degrees Form a Triangle? Triangle Classification: Analyzing Angles 53°, 117°, and 21° Calculate Angle ABC in Parallelogram with Given Angles 40° and 44°

Continue Your Math Journey

Master Sum of Angles in a Polygon first, then advance to these related topics that build on your fraction skills

Suggested Topics to Practice in Advance

Right angleAcute AnglesObtuse AnglePlane angleTypes of angles (right, acute, obtuse, straight)Angle NotationAngle Bisector

Topics Learned in Later Sections

Sides, Vertices, and AnglesTypes of AnglesSum and Difference of AnglesThe Sum of the Interior Angles of a TriangleExterior angles of a triangle

Practice by Question Type

Choose your preferred learning style and practice format for fraction problems
Angle bisector Applying the formula Calculate Angles in Quadrilaterals Finding the size of angles in a triangle Generate a random angle Identifying and defining elements Identify the greater value Is it possible...? Isosceles triangle Using angles in a triangle Using quadrilaterals Using variables