Question

question 24 of 25

what kinds of regular polygons can be used for regular tessellations?

check all that apply.

a. five-sided
b. four-sided
c. six-sided
d. three-sided

Explanation:

Define regular tessellation condition

Using the Regular Tessellation knowledge point

For a regular polygon to form a regular tessellation, the interior angle of the polygon must divide \(360^\circ\) exactly without leaving a remainder. This ensures that the polygons can meet at a vertex with no gaps or overlaps.

Calculate interior angles

Using the Interior Angle Formula knowledge point

The interior angle \(\theta\) of a regular \(n\)-sided polygon is given by:

$$ \theta = \frac{(n-2) \times 180^\circ}{n} $$

We evaluate this formula for each given option:

• For three-sided (equilateral triangle, \(n=3\)):

$$ \theta = \frac{(3-2) \times 180^\circ}{3} = 60^\circ $$

• For four-sided (square, \(n=4\)):

$$ \theta = \frac{(4-2) \times 180^\circ}{4} = 90^\circ $$

• For five-sided (regular pentagon, \(n=5\)):

$$ \theta = \frac{(5-2) \times 180^\circ}{5} = 108^\circ $$

• For six-sided (regular hexagon, \(n=6\)):

$$ \theta = \frac{(6-2) \times 180^\circ}{6} = 120^\circ $$

Test divisibility of 360 degrees

Using the Regular Tessellation knowledge point

We check if \(360^\circ\) is divisible by each interior angle \(\theta\):

• Three-sided (\(\theta = 60^\circ\)):

$$ \frac{360^\circ}{60^\circ} = 6 \quad (\text{Integer}) $$

• Four-sided (\(\theta = 90^\circ\)):

$$ \frac{360^\circ}{90^\circ} = 4 \quad (\text{Integer}) $$

• Five-sided (\(\theta = 108^\circ\)):

$$ \frac{360^\circ}{108^\circ} = 3.33 \quad (\text{Not an integer}) $$

• Six-sided (\(\theta = 120^\circ\)):

$$ \frac{360^\circ}{120^\circ} = 3 \quad (\text{Integer}) $$

Therefore, three-sided, four-sided, and six-sided regular polygons can be used for regular tessellations.

Answer:

• A. Five-sided
• B. Four-sided (Correct answer)
• C. Six-sided (Correct answer)
• D. Three-sided (Correct answer)