Questions: Area of Regular Polygons

Question 1

A regular hexagon has side length 6 cm. The circumscribed radius (center to vertex) is also 6 cm. What is the apothem?

• A. 6 cm — the apothem equals the radius in a regular hexagon
• B. 12 cm — the apothem is the full diameter
• C. Approximately 5.2 cm — the perpendicular distance from center to the midpoint of a side
• D. 3 cm — the apothem is half the side length

Answer: (C)

The apothem is the perpendicular distance from the center to the midpoint of a side — it is NOT the radius (center to vertex). For a regular hexagon with side length 6, the apothem = 6 × (√3/2) ≈ 5.196 cm, which is shorter than the radius of 6 cm. The apothem is always shorter than the radius because the perpendicular to the side is shorter than the line to the corner. Confusing them gives an incorrect area calculation.

Question 2

A student calculates the area of a regular octagon using A = (1/2) × r × P, where r is the radius (center to vertex) instead of the apothem. How does her answer compare to the correct area?

• A. Her answer is too small, because the radius is shorter than the apothem
• B. Her answer is too large, because the radius is longer than the apothem
• C. Her answer is correct — radius and apothem are equal for regular polygons
• D. It depends on the number of sides — for some polygons they are equal

Answer: (B)

The radius (center to vertex) is always longer than the apothem (center to midpoint of side) for any regular polygon. Using the larger radius instead of the apothem inflates the height of each constituent triangle, producing an area that is too large. The apothem is specifically the perpendicular height of those triangles — using any other measurement (like the slant radius) overestimates that height.

Question 3

The formula A = (1/2) × apothem × perimeter is a special rule unique to regular polygons, derived from principles unrelated to triangle area.

• T. True
• F. False

Answer: False

The formula is derived directly from triangle area. Divide a regular n-gon into n congruent triangles by connecting the center to each vertex. Each triangle has base = side length s and height = apothem a. Area of one triangle = (1/2) × s × a. Multiply by n triangles: n × (1/2) × s × a = (1/2) × a × (n × s) = (1/2) × a × P. The polygon formula IS triangle area applied n times and consolidated — it is not a separate formula but a direct consequence of the triangular decomposition.

Question 4

As the number of sides of a regular polygon increases without bound, the formula A = (1/2) × apothem × perimeter converges to the circle area formula πr².

• T. True
• F. False

Answer: True

As n → ∞, the apothem approaches the radius r (the perpendicular to the midpoint of an increasingly short side converges toward the vertex distance), and the perimeter approaches the circumference 2πr. Substituting into the polygon formula: (1/2) × r × 2πr = πr². The circle is the limiting case of a regular polygon, and the circle area formula emerges naturally from the polygon formula taken to its limit.

Question 5

Explain why the apothem — not the radius — appears in the area formula for regular polygons, and describe the geometric role the apothem plays in the derivation.

Model answer: When a regular n-gon is divided into n congruent triangles (by connecting the center to each vertex), the apothem is the height of each triangle. Triangle area = (1/2) × base × height, where height is always the perpendicular distance from the apex to the base. The base of each triangle is one side of the polygon; the perpendicular from the center to the midpoint of that base is the apothem. The radius (center to vertex) is the slant side of each triangle, not its height — using the radius would not give the triangle's height and would overestimate the area. The apothem is the right measurement precisely because it is the perpendicular height.

A useful check: for a regular hexagon with side length s, the triangles are equilateral. The apothem is the altitude of an equilateral triangle = s√3/2. Using A = (1/2) × a × P = (1/2) × (s√3/2) × (6s) = (3s²√3)/2, which matches the known formula for a regular hexagon. The triangular decomposition is both the derivation and the verification.