Worksheets for learning about interior angles of regular polygons, including diagrams, tables, and problem-solving activities.

It looks like you've uploaded a worksheet titled "The Interior Angles of Regular Polygons" with three sections. The goal is to explore the interior angles of regular polygons (shapes with equal sides and angles), using both visual analysis and mathematical formulas.

Let’s go through each section step by step, solve the problems, and explain the reasoning.

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🔷 Section 1: Basic Exploration

Task:
- Identify shapes (triangle, square, pentagon, hexagon).
- Fill in a table with:
- Number of angles
- Interior angle (in degrees)
- Total of all interior angles

#### ✔ Step-by-step Solution:

We know that:
- A regular polygon has all sides and angles equal.
- The sum of interior angles of an *n*-gon is:
$$
\text{Sum} = (n - 2) \times 180^\circ
$$
- Each interior angle in a regular *n*-gon is:
$$
\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}
$$

Let’s fill in the table:

| Shape | Number of Angles | Interior Angle | Total of All Interior Angles |
|-------------------|------------------|----------------|-------------------------------|
| Equilateral Triangle | 3 | 60° | 180° |
| Square | 4 | 90° | 360° |
| Regular Pentagon | 5 | 108° | 540° |
| Regular Hexagon | 6 | 120° | 720° |

> ✔ Calculations:
> - Triangle: $(3 - 2) \times 180 = 180^\circ$, so each angle = $180 / 3 = 60^\circ$
> - Square: $(4 - 2) \times 180 = 360^\circ$, each = $360 / 4 = 90^\circ$
> - Pentagon: $(5 - 2) \times 180 = 540^\circ$, each = $540 / 5 = 108^\circ$
> - Hexagon: $(6 - 2) \times 180 = 720^\circ$, each = $720 / 6 = 120^\circ$

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🔷 Section 2: Extending to More Polygons

Task:
- Draw regular polygons (heptagon, octagon, etc.) and complete a similar table.

Let’s add more rows:

| Shape | Number of Angles | Interior Angle | Total of All Interior Angles |
|-------------------|------------------|----------------|-------------------------------|
| Regular Heptagon | 7 | ~128.57° | 900° |
| Regular Octagon | 8 | 135° | 1080° |
| Regular Nonagon | 9 | 140° | 1260° |
| Regular Decagon | 10 | 144° | 1440° |

> ✔ Calculations:
> - Heptagon: $(7 - 2) \times 180 = 900^\circ$, $900 / 7 \approx 128.57^\circ$
> - Octagon: $(8 - 2) \times 180 = 1080^\circ$, $1080 / 8 = 135^\circ$
> - Nonagon: $(9 - 2) \times 180 = 1260^\circ$, $1260 / 9 = 140^\circ$
> - Decagon: $(10 - 2) \times 180 = 1440^\circ$, $1440 / 10 = 144^\circ$

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🔷 Section 3: General Formula & Application

This section asks deeper questions:

#### ❓ "What is the interior angle for any regular polygon with n sides?"
Answer:
$$
\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}
$$

You can also write it as:
$$
= 180^\circ - \frac{360^\circ}{n}
$$
(This comes from the fact that exterior angles sum to $360^\circ$.)

#### ❓ "How would you calculate the interior angle from the angle of turn?"
- When walking around a polygon, the turn at each vertex is the exterior angle.
- Exterior angle = $360^\circ / n$
- So, Interior angle = $180^\circ - \text{Exterior angle}$

✔ Example: For a hexagon:
- Exterior angle = $360 / 6 = 60^\circ$
- Interior angle = $180 - 60 = 120^\circ$

#### ❓ "Can you use the formula to find the interior angle of any regular polygon?"
Yes! Use:
$$
\text{Interior Angle} = \frac{(n - 2) \times 180}{n}
$$

Try it for a 12-gon:
- $(12 - 2) \times 180 = 1800^\circ$
- $1800 / 12 = 150^\circ$

So, each interior angle of a regular dodecagon is 150°.

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🔷 Final Summary & Key Concepts

| Concept | Formula |
|--------|--------|
| Sum of interior angles | $(n - 2) \times 180^\circ$ |
| Each interior angle (regular) | $\frac{(n - 2) \times 180^\circ}{n}$ |
| Each exterior angle | $\frac{360^\circ}{n}$ |
| Interior + Exterior = | $180^\circ$ (at each vertex) |

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✔ How to Complete the Worksheet

1. Fill in the tables using the formulas above.
2. Draw shapes on the grid paper — ensure they are regular (equal sides/angles).
3. Describe how the shapes look: e.g., “A regular pentagon has five equal sides and angles; it looks like a star base.”
4. Use the formula to predict angles for larger polygons.

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🌟 Tips for Students:

- Practice drawing regular polygons with a protractor.
- Remember: More sides → angles get closer to 180°.
- As $n \to \infty$, the polygon becomes a circle, and interior angles approach $180^\circ$.

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