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Textbook
1. Welcome
2. Vocabulary approach
3. Quantitative reasoning
3.1 Quant intro
3.2 Arithmetic & algebra
3.3 Statistics and data interpretation
3.4 Geometry
3.4.1 Angles
3.4.2 Triangle basics
3.4.3 Sum of interior angles
3.4.4 Pythagorean theorem
3.4.5 Right triangles (45-45-90)
3.4.6 Right triangles (30-60-90)
3.4.7 Triangle inequality theorem
3.4.8 Coordinate plane
3.4.9 Equation for a line
3.4.10 Graphing inequalities
3.4.11 Graphing parabolas
3.4.12 Graphing circles
3.4.13 Parallel and perpendicular lines
3.4.14 Quadrilaterals
3.4.15 Circles
3.4.16 3D shapes
3.4.17 Polygons
3.4.18 Regular polygons
3.4.19 Shaded region problems
3.5 Strategies
4. Verbal reasoning
5. Analytical writing
6. Wrapping up
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3.4.3 Sum of interior angles
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3. Quantitative reasoning
3.4. Geometry

Sum of interior angles

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Every shape has a unique sum of its interior (inside) angles:

  • Triangle (3 sides): 180 degrees
  • Square (4 sides): 360 degrees
  • Pentagon (5 sides): 540 degrees
  • Hexagon (6 sides): 720 degrees
  • Heptagon (7 sides): 900 degrees
  • Octagon (8 sides): 1080 degrees

For example, imagine a triangle with two interior angles of 50 and 70. That’s enough information to know that the third angle will be 60, since 50+70+60=180. Here’s the math:

50+70+x120+xx​=180=180=60​

But what if the shape had four, five, or even six sides? Would those interior angles add up to 180 as well? No… they would add up to a different number, but you don’t need to memorize the list above, since there’s a simple equation to find the sum of the interior angles of any shape.

Definitions
Sum of interior angles for an n-sided shape
(n−2)× 180 degrees

For instance, a triangle (3 sides) has interior angles with a sum of (n−2)×180=(3−2)×180=1×180=180 degrees.

A square (4 sides) has interior angles with a sum of (n−2)×180=(4−2)× 180=2×180=360 degrees.

Since n in this equation represents the number of sides, if we had 5 sides, the sum of interior angles would be 540:

sum of interior angles​=(n−2)×180=(5−2)× 180=3× 180=540degrees​

And this works for any 5-sided pentagon, regardless of the shape! The individual angles might be different, but the total sum will always be 540.

Interior angles of irregular polygons

Let’s try an example that’s a bit more difficult, where we need to find the angles for an irregular polygon.

Solve for the variable x in the figure below.

Irregular hexagon sum of interior angles 720 degrees

Take a moment to try it yourself, then continue reading to check your work.

The first step to solving for x is to determine the sum of the interior angles of this shape. Counting them up, we can see that the shape has 6 sides and a few different angles, making this an irregular hexagon.

Regardless of what it’s called, we now know we can plug in 6 for the 6 sides into the interior angles formula:

sum of interior angles​=(n−2)×180=(6−2)× 180=4× 180=720degrees​

The sum of the interior angles in this hexagon must be 720. Now, we can write an equation that equates all the interior angles to 720, remembering that right angles are 90 degrees, and solve for x.

90+90+4x+x+2x+2x180+9x9xx​=720=720=540=60​

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