Blank chart for students to fill in the number of sides and angle sums for various regular polygons.

Let’s solve this step by step.

We are given a table with different polygons: triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, and n-gon (a polygon with any number of sides).

We need to fill in 5 rows for each shape:

1. Number of sides
2. Sum of interior angles
3. Each interior angle (if regular)
4. Sum of exterior angles
5. Each exterior angle (if regular)

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Step 1: Number of sides

This is easy — just count the sides or use the name:

- Triangle → 3
- Quadrilateral → 4
- Pentagon → 5
- Hexagon → 6
- Heptagon → 7
- Octagon → 8
- n-gon → n

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Step 2: Sum of interior angles

Formula: (n - 2) × 180°

Let’s calculate:

- Triangle (n=3): (3-2)×180 = 1×180 = 180°
- Quadrilateral (n=4): (4-2)×180 = 2×180 = 360°
- Pentagon (n=5): (5-2)×180 = 3×180 = 540°
- Hexagon (n=6): (6-2)×180 = 4×180 = 720°
- Heptagon (n=7): (7-2)×180 = 5×180 = 900°
- Octagon (n=8): (8-2)×180 = 6×180 = 1080°
- n-gon: (n - 2) × 180°

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Step 3: Each interior angle (if regular)

For a regular polygon, all interior angles are equal. So divide the sum by the number of sides:

Formula: Sum ÷ n or [(n - 2) × 180] ÷ n

Calculate:

- Triangle: 180 ÷ 3 = 60°
- Quadrilateral: 360 ÷ 4 = 90°
- Pentagon: 540 ÷ 5 = 108°
- Hexagon: 720 ÷ 6 = 120°
- Heptagon: 900 ÷ 7 ≈ 128.57° (we’ll write as fraction or decimal? Let’s keep it exact: 900/7 °)
But since this is for students, maybe round to two decimals? Actually, let’s keep it precise but simple.

Wait — better to leave as fractions if not whole numbers? But for clarity, we can write decimals rounded to nearest hundredth.

Actually, let’s do exact values where possible, and approximate only when needed.

Heptagon: 900 ÷ 7 = 128.571... → ≈128.57°

Octagon: 1080 ÷ 8 = 135°

n-gon: [(n - 2) × 180] ÷ n

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Step 4: Sum of exterior angles

Important fact: For ANY convex polygon, the sum of exterior angles (one at each vertex) is always 360°, no matter how many sides!

So:

- All shapes from triangle to octagon → 360°
- n-gon → 360°

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Step 5: Each exterior angle (if regular)

Since the polygon is regular, all exterior angles are equal. So divide 360° by number of sides.

Formula: 360° ÷ n

Calculate:

- Triangle: 360 ÷ 3 = 120°
- Quadrilateral: 360 ÷ 4 = 90°
- Pentagon: 360 ÷ 5 = 72°
- Hexagon: 360 ÷ 6 = 60°
- Heptagon: 360 ÷ 7 ≈ 51.43°
- Octagon: 360 ÷ 8 = 45°
- n-gon: 360° ÷ n

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Now let’s put everything into the table format.

Final Answer:

| | triangle | quadrilateral | pentagon | hexagon | heptagon | octagon | n-gon |
|------------------|----------|---------------|----------|---------|------------|---------|-------------------|
| number of sides | 3 | 4 | 5 | 6 | 7 | 8 | n |
| sum of interior angles | 180° | 360° | 540° | 720° | 900° | 1080° | (n-2)×180° |
| each interior angle (regular) | 60° | 90° | 108° | 120° | ≈128.57° | 135° | [(n-2)×180]/n |
| sum of exterior angles | 360° | 360° | 360° | 360° | 360° | 360° | 360° |
| each exterior angle (regular) | 120° | 90° | 72° | 60° | ≈51.43° | 45° | 360/n |

Note: For heptagon, we used approximations for decimal places. If your teacher wants exact fractions, you can write:

- Heptagon interior: 900/7 °
- Heptagon exterior: 360/7 °

But for most school levels, rounding to two decimal places is acceptable.

✔ Double-checked all calculations — they are correct.