Apothem

Apothem

The line segment from the center of a regular polygon to the midpoint of a side, or the length of this segment. Same as the inradius; that is, the radius of a regular polygon's inscribed circle.

Note: Apothem is pronounced with the emphasis on the first syllable with the a pronounced as in apple (A-puh-thum).

A regular hexagon with a vertical line segment from its center to the midpoint of the bottom side, labeled "apothem.

Key Formula

A = \frac{1}{2} \times p \times a

Where:

$A$ = Area of the regular polygon
$p$ = Perimeter of the regular polygon
$a$ = Apothem (perpendicular distance from center to midpoint of a side)

Worked Example

Problem: A regular hexagon has a side length of 6 cm. Find its apothem and use it to calculate the hexagon's area.

Step 1: Identify the number of sides and the side length. A regular hexagon has n = 6 sides, each of length s = 6 cm.

Step 2: Use the apothem formula. The apothem of any regular polygon can be found using trigonometry: a = s / (2 tan(π/n)).

a = \frac{s}{2\tan\!\left(\dfrac{\pi}{n}\right)} = \frac{6}{2\tan\!\left(\dfrac{\pi}{6}\right)}

Step 3: Evaluate the tangent. Since tan(π/6) = tan(30°) = 1/√3 ≈ 0.5774, substitute this value.

a = \frac{6}{2 \times \frac{1}{\sqrt{3}}} = \frac{6}{\frac{2}{\sqrt{3}}} = \frac{6\sqrt{3}}{2} = 3\sqrt{3} \approx 5.196 \text{ cm}

Step 4: Find the perimeter of the hexagon.

p = 6 \times 6 = 36 \text{ cm}

Step 5: Calculate the area using the apothem formula A = (1/2) × p × a.

A = \frac{1}{2} \times 36 \times 3\sqrt{3} = 54\sqrt{3} \approx 93.53 \text{ cm}^2

Answer: The apothem is 3√3 ≈ 5.20 cm, and the area of the hexagon is 54√3 ≈ 93.53 cm².

Another Example

Problem: A regular pentagon has a side length of 10 cm. Find its apothem.

Step 1: Identify the values: n = 5 sides, s = 10 cm.

Step 2: Apply the apothem formula.

a = \frac{s}{2\tan\!\left(\dfrac{\pi}{n}\right)} = \frac{10}{2\tan\!\left(\dfrac{\pi}{5}\right)}

Step 3: Evaluate: tan(π/5) = tan(36°) ≈ 0.7265.

a = \frac{10}{2 \times 0.7265} = \frac{10}{1.4530} \approx 6.88 \text{ cm}

Answer: The apothem of the regular pentagon is approximately 6.88 cm.

Frequently Asked Questions

How do you find the apothem of a regular polygon?

Use the formula a = s / (2 tan(π/n)), where s is the side length and n is the number of sides. You can also find it by drawing a right triangle from the center to a vertex and then to the midpoint of a side, then using trigonometry or the Pythagorean theorem if you know the circumradius.

What is the difference between the apothem and the radius of a regular polygon?

The apothem goes from the center to the midpoint of a side (perpendicular to that side), while the radius (circumradius) goes from the center to a vertex. The apothem is always shorter than the circumradius. They are related by a = R cos(π/n), where R is the circumradius and n is the number of sides.

Apothem (Inradius) vs. Circumradius

The apothem is the distance from the center of a regular polygon to the midpoint of a side, which equals the inradius (radius of the inscribed circle). The circumradius is the distance from the center to a vertex, which equals the radius of the circumscribed circle. For any regular polygon, the apothem is always less than the circumradius. They are connected by the formula a = R cos(π/n).

Why It Matters

The apothem is essential for computing the area of any regular polygon, since the area formula A = (1/2) × perimeter × apothem works for all regular polygons regardless of the number of sides. It also appears in architecture, tiling, and engineering wherever regular polygonal shapes are used. As the number of sides increases, the apothem approaches the circumradius, illustrating how regular polygons approximate circles.

Common Mistakes

Mistake: Confusing the apothem with the circumradius (the distance from center to a vertex).

Correction: The apothem reaches the midpoint of a side, not a vertex. It is always perpendicular to the side and is shorter than the circumradius.

Mistake: Trying to use the apothem formula on an irregular polygon.

Correction: The apothem is only defined for regular polygons (all sides and angles equal). Irregular polygons do not have a single, well-defined apothem.

Related Terms

Regular Polygon — Apothem is defined only for regular polygons
Inradius — The apothem equals the inradius
Inscribed Circle — Apothem is the radius of this circle
Radius — Circumradius is related but distinct from apothem
Midpoint — Apothem meets a side at its midpoint
Line Segment — The apothem is a specific line segment
Side of a Polygon — Apothem is perpendicular to a side