Solved Exercises on Polygons in Grade 7

Master polygons: properties, classification, and problem solving through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Polygon Classification
Exercise 1
A polygon has 8 sides. Name the polygon and find the sum of its interior angles.
Definition:

Octagon: A polygon with eight sides and eight angles.

Method for naming polygons:
  1. Count the number of sides
  2. Use the appropriate Greek prefix
  3. Apply the polygon formula for interior angles
Number of sides
n = 8
Polygon name
Octagon
Sum of angles
1080°
Step 1: Identify the polygon

A polygon with 8 sides is called an octagon

Step 2: Apply the interior angle sum formula

Sum of interior angles = (n - 2) × 180°

Where n = number of sides

Step 3: Substitute and calculate

Sum = (8 - 2) × 180°

Sum = 6 × 180° = 1080°

Step 4: Verify with alternative method

An octagon can be divided into 6 triangles by drawing diagonals from one vertex

Sum = 6 × 180° = 1080° ✓

Octagon, Sum of interior angles = 1080°
Final answer:

The polygon is an octagon with a sum of interior angles of 1080°.

Applied rules:

• Interior Angle Sum: (n - 2) × 180°

• Polygon Naming: Based on Greek prefixes

• Triangulation: Any polygon can be divided into (n-2) triangles

2 Regular Polygon
Exercise 2
A regular hexagon has a perimeter of 42 cm. Find the length of each side and the measure of each interior angle.
Definition:

Regular Polygon: A polygon with all sides equal and all angles equal.

Given
Regular hexagon, Perimeter = 42cm
Sides
6 sides
Results
Side = 7cm, Angle = 120°
Step 1: Find the length of each side

Regular hexagon has 6 equal sides

Perimeter = 6 × side length

42 = 6 × side length

Side length = 42 ÷ 6 = 7 cm

Step 2: Calculate sum of interior angles

Sum = (n - 2) × 180°

Sum = (6 - 2) × 180° = 4 × 180° = 720°

Step 3: Find measure of each interior angle

In a regular polygon, all angles are equal

Each angle = Sum ÷ number of angles

Each angle = 720° ÷ 6 = 120°

Step 4: Verify with regular polygon formula

Each interior angle = [(n - 2) × 180°] ÷ n

Each angle = [(6 - 2) × 180°] ÷ 6 = 720° ÷ 6 = 120° ✓

Side length = 7 cm, Each interior angle = 120°
Final answer:

Each side measures 7 cm and each interior angle measures 120°.

Applied rules:

• Regular Polygon: All sides and angles equal

• Perimeter: Sum of all sides

• Interior Angle: [(n-2) × 180°] ÷ n for regular polygons

3 Exterior Angles
Exercise 3
Find the measure of each exterior angle of a regular decagon and verify that the sum of all exterior angles is 360°.
Definition:

Exterior Angle: The angle formed by extending one side of a polygon. At each vertex, the interior and exterior angles are supplementary (sum to 180°).

Decagon
n = 10
Exterior Angle
360°/n
Result
36°
Step 1: Apply the exterior angle sum theorem

The sum of exterior angles of any polygon is always 360°

Step 2: Calculate each exterior angle for regular decagon

In a regular polygon, all exterior angles are equal

Each exterior angle = 360° ÷ n

Each exterior angle = 360° ÷ 10 = 36°

Step 3: Verify the sum

Sum = 10 × 36° = 360° ✓

Step 4: Find each interior angle

Interior + Exterior = 180°

Interior angle = 180° - 36° = 144°

Step 5: Verify with interior angle formula

Each interior angle = [(n - 2) × 180°] ÷ n

Each interior angle = [(10 - 2) × 180°] ÷ 10 = 1440° ÷ 10 = 144° ✓

Each exterior angle = 36°, Sum = 360°
Final answer:

Each exterior angle measures 36° and the sum of all exterior angles is 360°.

Applied rules:

• Exterior Angle Sum: Always 360° for any polygon

• Regular Polygon: Each exterior angle = 360°/n

• Linear Pair: Interior + Exterior = 180°

Polygon Properties and Formulas
\(\text{Sum of interior angles} = (n-2) \times 180°\)
Interior Angle Sum Formula
Triangle
n=3, Sum=180°
Basic polygon
Quadrilateral
n=4, Sum=360°
Four-sided polygon
Pentagon
n=5, Sum=540°
Five-sided polygon
Key definitions:

Polygon: A closed plane figure made up of line segments connected end-to-end.

Regular Polygon: A polygon with all sides and all angles equal.

Irregular Polygon: A polygon with sides or angles that are not all equal.

Convex Polygon: A polygon where all interior angles are less than 180° and no sides bend inward.

Concave Polygon: A polygon with at least one interior angle greater than 180°.

Diagonal: A line segment connecting two non-adjacent vertices.

Complete methodology:
  1. Identify Polygon Type: Determine number of sides and regularity
  2. Select Appropriate Formula: Use correct formula for the property needed
  3. Substitute Known Values: Plug in the given information
  4. Solve Systematically: Perform calculations step by step
  5. Verify Solution: Check using alternative methods
  6. State Answer: Provide final answer with proper units
Tip 1: Remember: exterior angles of any polygon sum to 360°.
Tip 2: The sum of interior angles increases by 180° for each additional side.
Tip 3: In regular polygons, divide the total sum by the number of sides to find each angle.
Tip 4: Draw the polygon to visualize and label vertices when solving complex problems.
Common errors: Confusing interior and exterior angles, miscounting sides, using wrong formulas for regular vs irregular polygons.
Exam preparation: Memorize polygon names up to 10 sides, practice with both regular and irregular polygons, master the formulas.
Formulas to know by heart:

• Interior Angle Sum: (n - 2) × 180°

• Each Interior Angle (regular): [(n - 2) × 180°] ÷ n

• Exterior Angle Sum: 360° (always)

• Each Exterior Angle (regular): 360° ÷ n

• Number of Diagonals: [n(n - 3)] ÷ 2

• Perimeter (regular): n × side length

Solution: Exercises 4 to 5
4 Diagonal Count
Exercise 4
How many diagonals does a heptagon have? Verify your answer by explaining the logic behind the formula.
Definition:

Diagonal: A line segment connecting two non-adjacent vertices of a polygon.

Heptagon
n = 7
Formula
[n(n-3)]/2
Result
14 diagonals
Step 1: Identify the polygon

A heptagon has 7 sides and 7 vertices

Step 2: Apply the diagonal formula

Number of diagonals = [n(n - 3)] ÷ 2

Number of diagonals = [7(7 - 3)] ÷ 2

Number of diagonals = [7 × 4] ÷ 2 = 28 ÷ 2 = 14

Step 3: Explain the formula logic

From each vertex, you can draw lines to (n - 1) other vertices

Of these, 2 are sides of the polygon (adjacent vertices)

So from each vertex, (n - 3) diagonals can be drawn

Total connections = n(n - 3), but this counts each diagonal twice

So actual diagonals = n(n - 3) ÷ 2

Step 4: Verify with manual counting approach

From vertex A: can connect to D, E, F, G (4 diagonals)

From vertex B: can connect to E, F, G, A (but A-B is a side, so E, F, G, A are 4 diagonals)

Wait, let me be more systematic:

From each of 7 vertices: 4 possible diagonals = 28

But each diagonal connects 2 vertices, so we counted each twice

Actual diagonals = 28 ÷ 2 = 14 ✓

Heptagon has 14 diagonals
Final answer:

A heptagon has 14 diagonals.

Applied rules:

• Diagonal Formula: [n(n - 3)] ÷ 2

• Combinatorial Logic: Each vertex connects to (n-3) others

• Counting Principle: Avoid double counting

5 Complex Polygon Problem
Exercise 5
In a regular polygon, each interior angle measures 150°. How many sides does the polygon have? What is the measure of each exterior angle?
Definition:

Interior and Exterior Angles: At each vertex, the interior and exterior angles are supplementary (sum to 180°).

Given
Interior angle = 150°
Find n
[(n-2)×180]/n = 150
Results
n=12, Exterior=30°
Step 1: Set up the equation

For a regular polygon: Each interior angle = [(n - 2) × 180°] ÷ n

150° = [(n - 2) × 180°] ÷ n

Step 2: Solve for n

150n = (n - 2) × 180

150n = 180n - 360

150n - 180n = -360

-30n = -360

n = 12

Step 3: Verify the solution

Each interior angle = [(12 - 2) × 180°] ÷ 12

= (10 × 180°) ÷ 12 = 1800° ÷ 12 = 150° ✓

Step 4: Find each exterior angle

Interior + Exterior = 180°

Exterior angle = 180° - 150° = 30°

Step 5: Verify exterior angle

Each exterior angle = 360° ÷ n = 360° ÷ 12 = 30° ✓

Polygon has 12 sides, Each exterior angle = 30°
Final answer:

The polygon has 12 sides (dodecagon) and each exterior angle measures 30°.

Applied rules:

• Regular Polygon Angle: [(n - 2) × 180°] ÷ n

• Supplementary Angles: Interior + Exterior = 180°

• Algebraic Solution: Solve equations with variables

Complete Guide: Polygons and Their Properties
\(\text{Number of diagonals} = \frac{n(n-3)}{2}\)
Diagonal Formula
Key definitions:

Polygon: A two-dimensional shape made with straight lines that connect to form a closed figure. The word "polygon" comes from Greek meaning "many angles."

Vertex (plural: vertices): A corner point where two sides of a polygon meet.

Side: A line segment that forms part of the boundary of a polygon.

Interior Angle: The angle formed inside the polygon at each vertex.

Exterior Angle: The angle formed outside the polygon when one side is extended.

Regular Polygon: A polygon with all sides equal and all angles equal.

Irregular Polygon: A polygon with sides and/or angles that are not all equal.

Complete methodology:
  1. Identify the Polygon: Determine the number of sides and whether it's regular or irregular
  2. Select the Appropriate Formula: Choose the correct formula for the property you need to find
  3. Substitute Known Values: Replace variables with given information
  4. Perform Calculations: Execute mathematical operations carefully
  5. Verify Results: Check your answer using alternative methods or formulas
  6. Express Answer Clearly: Provide the final answer with proper units
Tip 1: Remember that the sum of exterior angles is always 360°, regardless of the number of sides.
Tip 2: For regular polygons, all interior angles are equal and all exterior angles are equal.
Tip 3: Interior and exterior angles at each vertex are supplementary (add up to 180°).
Tip 4: Draw and label polygons to visualize problems and verify your solutions.
Common errors: Mixing up interior and exterior angles, miscounting the number of sides, using the wrong formula for regular vs irregular polygons, arithmetic mistakes in algebraic equations.
Exam preparation: Memorize polygon names up to 12 sides (triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon, hendecagon, dodecagon), practice with both regular and irregular polygons, master all the key formulas.
Formulas to know by heart:

• Sum of Interior Angles: (n - 2) × 180°

• Each Interior Angle (regular): [(n - 2) × 180°] ÷ n

• Sum of Exterior Angles: 360° (constant for any polygon)

• Each Exterior Angle (regular): 360° ÷ n

• Number of Diagonals: [n(n - 3)] ÷ 2

• Perimeter (regular): n × side length

• Area formulas vary by polygon type

Visualizing Polygon Properties: Interior Angle Sums
Exercise 6: Polygon Angle Relationships
Consider how interior angle sums change with the number of sides:
Triangle (n=3) to Dodecagon (n=12)
Showing the linear relationship

Analysis: The chart shows how interior angle sums increase linearly with the number of sides.

  • Each additional side adds 180° to the angle sum
  • The relationship follows the formula (n-2) × 180°
  • As n increases, the polygon approaches a circle
  • Regular polygons have equal angles that increase toward 180°

Questions & Answers

Question: Why is the sum of exterior angles always 360°, no matter how many sides the polygon has?

Answer: This is a beautiful geometric property! Think of walking around the perimeter of any polygon:

Intuitive explanation:

  • Imagine you're walking along the edges of the polygon
  • At each vertex, you turn through the exterior angle
  • After completing one full trip around the polygon, you've turned a complete rotation
  • A complete rotation is always 360°, regardless of the path shape

Mathematical reasoning:

  • At each vertex: Interior angle + Exterior angle = 180°
  • Sum of all interior + exterior angles = n × 180°
  • Sum of interior angles = (n - 2) × 180°
  • Therefore: Sum of exterior angles = n × 180° - (n - 2) × 180° = 360°

Question: How do I remember the names of polygons with many sides?

Answer: Here are some memory aids for polygon names:

Common polygons (memorize these):

  • Triangle (3 sides) - "tri" = three
  • Quadrilateral (4 sides) - "quad" = four
  • Pentagon (5 sides) - "pent" = five
  • Hexagon (6 sides) - "hex" = six
  • Heptagon (7 sides) - "hept" = seven
  • Octagon (8 sides) - "oct" = eight
  • Nonagon (9 sides) - "non" = nine
  • Decagon (10 sides) - "dec" = ten

For higher numbers:

  • Hendecagon (11 sides) - think "end" as in ending the teens
  • Dodecagon (12 sides) - "do" = two, "dec" = ten, so 12
  • For n > 12, we often just say "n-gon"

Focus on memorizing up to decagon as these are most commonly used in grade 7.

Question: What's the difference between a convex and concave polygon? How can I tell them apart?

Answer: Here's how to distinguish between convex and concave polygons:

Convex Polygon:

  • All interior angles are less than 180°
  • No sides bend inward
  • Any line segment between two points inside the polygon stays entirely inside
  • Examples: triangles, squares, regular pentagons

Concave Polygon:

  • At least one interior angle is greater than 180°
  • At least one side bends inward (creates a "dent")
  • You can draw a line segment between two points that goes outside the polygon
  • Examples: star-shaped polygons, some irregular polygons

Quick identification: If you can "see into" the polygon from the outside at any point, it's concave. If you can't, it's convex.