Introduction
Interior Angles of Polygons is an important Grade 8 math skill because students are moving from simple answers toward explaining how the math works.
In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with interior angles of polygons.
What Is Interior Angles of Polygons?
Interior Angles of Polygons means looking at attributes such as sides, angles, equal parts, and shape categories.
The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.
Understanding Interior Angles of Polygons
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Look for attributes such as side count, equal sides, angles, and parallel sides.
- Classify shapes using evidence instead of only how the shape looks.
- Remember that one shape can belong to more than one category.
- Use drawings to test whether the attributes really match the name.
Visual Models
Visual Model 1
Question: The figure shows a pentagon with interior angles. If \(\angle A = 100°\), \(\angle B = 120°\), \(\angle C = 110°\), and \(\angle D = 130°\), what is \(\angle E\)?
- A. 80°
- B. 100°
- C. 120°
- D. 140°
Why it works: The sum of angles in a pentagon is 540°. So \(\angle E = 540 - 100 - 120 - 110 - 130 = 80°\).
Answer: 80°
Visual Model 2
Question: A regular hexagon is shown with one interior angle marked. What is the measure of the marked interior angle?
- A. 100°
- B. 110°
- C. 120°
- D. 130°
Why it works: Each interior angle of a regular hexagon is \(\frac{(6-2)\times180}{6}=\frac{720}{6}=120°\).
Answer: 120°
Worked Examples
Example 1
Question: A hexagon has interior angles: \(85°, x, 125°, 140°, 110°, 98°\). Find \(x\).
- A. 162°
- B. 227°
- C. 232°
- D. 237°
- The sum of interior angles in a hexagon is 720°.
- The known angles add to \(85+125+140+110+98=558°\), so \(x=720-558=162°\).
Answer: 162°
Example 2
Question: A hexagon has five known interior angles: \(120°, 100°, 90°, 115°, 130°\). What is the sixth angle?
- A. 145°
- B. 155°
- C. 165°
- D. 175°
- A hexagon's interior angles add to 720°.
- The known angles add to \(120+100+90+115+130=555°\), so the missing angle is \(720-555=165°\).
Answer: 165°
Example 3
Question: A quadrilateral has angles \(A = 85°\), \(B = 95°\), \(C = 100°\). What is angle \(D\)?
- A. 70°
- B. 80°
- C. 90°
- D. 100°
- The sum of interior angles in a quadrilateral is 360°.
- So \(D = 360 - 85 - 95 - 100 = 80°\).
Answer: 80°
Real-World Word Problems
Problem 1
Question: A student claims the sum of interior angles of a nonagon is 1260°. Is this correct?
- A. Yes, 1260° is correct.
- B. No, it is 1080°.
- C. No, it is 1440°.
- D. No, it is 1620°.
Why it works: For a nonagon (\(n=9\)): \((9-2)\times180=7\times180=1260°\). The student is correct.
Answer: Yes, 1260° is correct
Problem 2
Question: What is the sum of the interior angles of a hexagon?
- A. 900°
- B. 540°
- C. 360°
- D. 720°
Why it works: The sum of interior angles of an \(n\)-gon is \((n-2)\times180°\). For a hexagon, \((6-2)\times180=720°\).
Answer: 720°
Common Mistakes
- Naming a shape from appearance instead of attributes.
- Forgetting that squares are also rectangles and quadrilaterals.
- Mixing up sides and angles.
- Assuming a rotated shape changed its category.
Strategy Tips
- List attributes before naming the shape.
- Use examples and non-examples to test a category.
- Look for shared attributes across shape groups.
- Draw a quick sketch when the wording feels abstract.
Practice Questions
Question 1
A regular pentagon has all sides equal and all angles equal. What is the measure of each interior angle?
- A. 120°
- B. 100°
- C. 72°
- D. 108°
Question 2
What is the sum of the interior angles of an octagon?
- A. 1350°
- B. 900°
- C. 1260°
- D. 1080°
Question 3
In a regular quadrilateral (square), what is the measure of each interior angle?
- A. 135°
- B. 60°
- C. 120°
- D. 90°
Question 4
What is the sum of the interior angles of a decagon (10-sided polygon)?
- A. 1800°
- B. 1260°
- C. 1620°
- D. 1440°
Question 5
Consider a regular heptagon (7-sided polygon). Which is closest to the measure of each interior angle?
- A. \(\approx 147°\)
- B. \(\approx 120°\)
- C. \(\approx 138°\)
- D. \(\approx 129°\)
Question 6
A polygon has an interior angle sum of 1260°. How many sides does it have?
- A. \(9\)
- B. \(8\)
- C. \(7\)
- D. \(10\)
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: 108°
The sum for a pentagon is \((5-2)\times180=540°\). Each angle of a regular pentagon is \(\frac{540}{5}=108°\).
Question 2
Answer: 1080°
For an octagon (\(n=8\)): \((8-2)\times180=6\times180=1080°\).
Question 3
Answer: 90°
The sum for a quadrilateral is \((4-2)\times180=360°\). Each angle of a square is \(\frac{360}{4}=90°\).
Question 4
Answer: 1440°
For a decagon (\(n=10\)): \((10-2)\times180=8\times180=1440°\). ✓
Question 5
Answer: \(\approx 129°\) (rounded from 128.6°)
The sum for a heptagon is \((7-2)\times180=900°\). Each angle is \(\frac{900}{7}=128.57\ldots°\), which rounds to \(\approx 129°\).
Question 6
Answer: \(9\) sides
Set \((n-2)\times180=1260\). Then \(n-2=7\), so \(n=9\) (a nonagon).
Connection to Standards
This lesson supports Grade 8 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.
Summary
Interior Angles of Polygons becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Attributes prove the shape name.
