Regular Tessellations worksheet with exercises on polygon interior angles and tessellation properties.

Problem Analysis

The task involves understanding regular tessellations and working with regular polygons. Let's break it down into sections:

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#### Section A: Regular Tessellations
We need to complete the table for regular polygons, determining:
1. The size of each interior angle.
2. Whether the polygon tessellates (i.e., can cover a plane without gaps or overlaps).

##### Formula for Interior Angle of a Regular Polygon:
The formula to calculate the interior angle of a regular polygon with \( n \) sides is:
\[
\text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n}
\]

##### Tessellation Condition:
A regular polygon can tessellate if its interior angle is a divisor of \( 360^\circ \). This means that multiple copies of the polygon can fit around a single point without leaving any gaps.

---

Step-by-Step Solution for Section A

1. Equilateral Triangle (\( n = 3 \)):
- Interior Angle:
\[
\frac{(3-2) \times 180^\circ}{3} = \frac{1 \times 180^\circ}{3} = 60^\circ
\]
- Tessellation: Yes, because \( 6 \times 60^\circ = 360^\circ \).

2. Square (\( n = 4 \)):
- Interior Angle:
\[
\frac{(4-2) \times 180^\circ}{4} = \frac{2 \times 180^\circ}{4} = 90^\circ
\]
- Tessellation: Yes, because \( 4 \times 90^\circ = 360^\circ \).

3. Regular Pentagon (\( n = 5 \)):
- Interior Angle:
\[
\frac{(5-2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = 108^\circ
\]
- Tessellation: No, because \( 108^\circ \) is not a divisor of \( 360^\circ \).

4. Regular Hexagon (\( n = 6 \)):
- Interior Angle:
\[
\frac{(6-2) \times 180^\circ}{6} = \frac{4 \times 180^\circ}{6} = 120^\circ
\]
- Tessellation: Yes, because \( 3 \times 120^\circ = 360^\circ \).

5. Regular Heptagon (\( n = 7 \)):
- Interior Angle:
\[
\frac{(7-2) \times 180^\circ}{7} = \frac{5 \times 180^\circ}{7} \approx 128.57^\circ
\]
- Tessellation: No, because \( 128.57^\circ \) is not a divisor of \( 360^\circ \).

6. Regular Octagon (\( n = 8 \)):
- Interior Angle:
\[
\frac{(8-2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = 135^\circ
\]
- Tessellation: No, because \( 135^\circ \) is not a divisor of \( 360^\circ \).

---

Completed Table for Section A:

| Regular Polygon | Size of Each Interior Angle | Does This Polygon Tessellate? |
|-----------------------|-----------------------------|--------------------------------|
| Equilateral Triangle | \( 60^\circ \) | Yes |
| Square | \( 90^\circ \) | Yes |
| Regular Pentagon | \( 108^\circ \) | No |
| Regular Hexagon | \( 120^\circ \) | Yes |
| Regular Heptagon | \( 128.57^\circ \) | No |
| Regular Octagon | \( 135^\circ \) | No |

---

Explanation for Why Only Some Regular Polygons Tessellate:

Only polygons whose interior angles are divisors of \( 360^\circ \) can tessellate. This is because, at each vertex where the polygons meet, the sum of the angles must equal \( 360^\circ \). For example:
- Triangles (\( 60^\circ \)): \( 6 \times 60^\circ = 360^\circ \).
- Squares (\( 90^\circ \)): \( 4 \times 90^\circ = 360^\circ \).
- Hexagons (\( 120^\circ \)): \( 3 \times 120^\circ = 360^\circ \).

Polygons like pentagons, heptagons, and octagons do not tessellate because their interior angles cannot be combined to sum to \( 360^\circ \) at a vertex.

---

#### Section B: Missing Angles
We need to find the missing angles in the given diagrams. Each diagram involves regular polygons, so we use the interior angle formulas and properties of regular polygons.

1. Diagram 1 (Equilateral Triangle):
- Each interior angle of an equilateral triangle is \( 60^\circ \).
- Since the triangle is split into two parts by a line, the angle \( a \) is:
\[
a = 60^\circ
\]

2. Diagram 2 (Regular Pentagon):
- Each interior angle of a regular pentagon is \( 108^\circ \).
- The angle \( b \) is the exterior angle, which is:
\[
b = 180^\circ - 108^\circ = 72^\circ
\]

3. Diagram 3 (Regular Pentagon):
- Each interior angle of a regular pentagon is \( 108^\circ \).
- The angle \( c \) is the exterior angle, which is:
\[
c = 180^\circ - 108^\circ = 72^\circ
\]

4. Diagram 4 (Regular Hexagon):
- Each interior angle of a regular hexagon is \( 120^\circ \).
- The angle \( d \) is the exterior angle, which is:
\[
d = 180^\circ - 120^\circ = 60^\circ
\]

5. Diagram 5 (Regular Octagon):
- Each interior angle of a regular octagon is \( 135^\circ \).
- The angle \( e \) is the exterior angle, which is:
\[
e = 180^\circ - 135^\circ = 45^\circ
\]
- The angle \( f \) is also the exterior angle of the octagon:
\[
f = 45^\circ
\]

6. Diagram 6 (Regular Heptagon):
- Each interior angle of a regular heptagon is approximately \( 128.57^\circ \).
- The angle \( g \) is the exterior angle, which is:
\[
g = 180^\circ - 128.57^\circ \approx 51.43^\circ
\]
- The angle \( h \) is also the exterior angle of the heptagon:
\[
h = 51.43^\circ
\]

7. Diagram 7 (Regular Pentagon and Hexagon):
- For the pentagon, each interior angle is \( 108^\circ \).
- For the hexagon, each interior angle is \( 120^\circ \).
- The angle \( i \) is the exterior angle of the pentagon:
\[
i = 180^\circ - 108^\circ = 72^\circ
\]
- The angle \( j \) is the exterior angle of the hexagon:
\[
j = 180^\circ - 120^\circ = 60^\circ
\]

8. Diagram 8 (Regular Hexagon):
- Each interior angle of a regular hexagon is \( 120^\circ \).
- The angle \( k \) is the exterior angle, which is:
\[
k = 180^\circ - 120^\circ = 60^\circ
\]

---

Answers for Section B:

1. \( a = 60^\circ \)
2. \( b = 72^\circ \)
3. \( c = 72^\circ \)
4. \( d = 60^\circ \)
5. \( e = 45^\circ \), \( f = 45^\circ \)
6. \( g = 51.43^\circ \), \( h = 51.43^\circ \)
7. \( i = 72^\circ \), \( j = 60^\circ \)
8. \( k = 60^\circ \)

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#### Extension: Semi-Tessellation
Semi-tessellation refers to patterns where shapes can partially fill a plane but leave gaps or require additional shapes to complete the tessellation. From the diagrams:
- Diagrams involving pentagons (2, 7) and heptagons (6) do not tessellate perfectly but can semi-tessellate with other shapes.

---

Final Answer:

\[
\boxed{
\begin{array}{l}
\text{Section A:} \\
\begin{array}{|c|c|c|}
\hline
\text{Regular Polygon} & \text{Size of Each Interior Angle} & \text{Does This Polygon Tessellate?} \\
\hline
\text{Equilateral Triangle} & 60^\circ & \text{Yes} \\
\text{Square} & 90^\circ & \text{Yes} \\
\text{Regular Pentagon} & 108^\circ & \text{No} \\
\text{Regular Hexagon} & 120^\circ & \text{Yes} \\
\text{Regular Heptagon} & 128.57^\circ & \text{No} \\
\text{Regular Octagon} & 135^\circ & \text{No} \\
\hline
\end{array} \\
\text{Section B:} \\
1. a = 60^\circ \\
2. b = 72^\circ \\
3. c = 72^\circ \\
4. d = 60^\circ \\
5. e = 45^\circ, f = 45^\circ \\
6. g = 51.43^\circ, h = 51.43^\circ \\
7. i = 72^\circ, j = 60^\circ \\
8. k = 60^\circ \\
\text{Extension:} \\
\text{Diagrams 2, 6, and 7 involve shapes that semi-tessellate.}
\end{array}
}
\]