Question

question 7 of 10

if a tessellation is regular, how many sides can the tessellating regular polygon have?

check all that apply.

a. 6

b. 4

c. 3

d. 8

e. 12

f. 10

Explanation:

Define regular tessellations

Using the Regular Tessellation and Tessellation Definition knowledge points

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

$$ k \times \theta = 360^\circ \implies k \times \frac{(n-2) \times 180^\circ}{n} = 360^\circ $$

Solve for possible values of n

Using the Regular Tessellation knowledge point

$$ \frac{k(n-2)}{n} = 2 \implies k(n-2) = 2n \implies kn - 2k = 2n \implies n(k-2) = 2k $$

$$ n = \frac{2k}{k-2} = \frac{2(k-2) + 4}{k-2} = 2 + \frac{4}{k-2} $$

Since \(k\) must be an integer greater than or equal to 3:

• If \(k = 3\): \(n = 2 + \frac{4}{1} = 6\) (regular hexagons)
• If \(k = 4\): \(n = 2 + \frac{4}{2} = 4\) (squares)
• If \(k = 6\): \(n = 2 + \frac{4}{4} = 3\) (equilateral triangles)

Match with given options

Using the Regular Tessellation knowledge point

• Option A (6): Yes, equilateral hexagons tessellate.
• Option B (4): Yes, squares tessellate.
• Option C (3): Yes, equilateral triangles tessellate.
• Options D (8), E (12), F (10): No, these do not form regular tessellations.

Answer:

• A. 6 (Correct answer)
• B. 4 (Correct answer)
• C. 3 (Correct answer)
• D. 8
• E. 12
• F. 10