Apothem: Definition, Formula, and Examples

Apothem: Definition, Formula, and Examples

The apothem of a regular polygon is the perpendicular distance from its centre to the midpoint of any side. It is found with a = \dfrac{s}{2\tan(180°/n)} and is the key to the area formula A = \tfrac{1}{2},P,a. This article defines the apothem, derives both formulas, and works through examples for the hexagon, octagon, and pentagon.

The Shortest Line From A Polygon's Heart To Its Edge

There is exactly one shortest path from the centre of a regular polygon to its boundary, and it always lands on the middle of a side, never on a corner. That special distance has a name — the apothem — and it unlocks the area of every regular polygon.

The apothem of a regular polygon is the line segment (and its length) drawn from the centre perpendicular to a side, meeting that side at its midpoint. Because a regular polygon is symmetric, this distance is the same to every side, and it equals the radius of the largest circle that fits inside the polygon (the inscribed circle). Only regular polygons have an apothem.

Apothem Versus Radius: Do Not Confuse Them

Distance Goes from centre to Symbol Relative size
Apothem Midpoint of a side (perpendicular) a Shorter
Radius (circumradius) A vertex R Longer

The apothem always reaches a side; the radius always reaches a corner. Since the perpendicular distance to a side is shorter than the slanted distance to a corner, the apothem is always less than the radius.

Deriving the Apothem Formula

Build the formula instead of memorising it. Take a regular polygon with n sides of length s, and draw segments from the centre to each vertex. This splits the polygon into n identical isosceles triangles.

Now focus on one triangle. Drop the apothem from the centre to the midpoint of that triangle's base. The apothem is perpendicular to the base and bisects it, creating a right triangle whose:

In that right triangle, the tangent of the centre angle is opposite over adjacent:

tan(180°/n) = \frac{s/2}{a}

Solving for a gives the apothem formula:

a = \dfrac{s}{2\tan(180°/n)}.

From Apothem to Area of a Regular Polygon

The area of any regular polygon is:

A = \frac{1}{2},P,a = \frac{1}{2}\times(\text{perimeter})\times(\text{apothem}).

Examples of Apothem

Example 1

Find the apothem of a regular hexagon with side 6 cm.

Use the formula with n=6:

a = \frac{s}{2\tan(180°/6)} = \frac{6}{2\tan 30°}.

Since \tan 30° = \dfrac{1}{\sqrt{3}} \approx 0.577:

a = \frac{6}{1.155} \approx 5.2 \text{ cm}.

Example 2

A student finds a square's apothem by computing the distance from the centre to a corner. Find the error.

The correct apothem of a square is half the side, \dfrac{4}{2} = 2 \text{ cm}.

Example 3

Find the area of a regular hexagon with side 6 cm using its apothem.

From Example 1, the apothem is \approx 5.2 cm.

Area:

A = \frac{1}{2},P,a = \frac{1}{2}\times 36 \times 5.2 \approx 93.5 \text{ cm}^2.

Example 4

A regular octagon has a side of 5 cm. Find its apothem.

Use n=8:

a = \frac{s}{2\tan(180°/8)} = \frac{5}{2\tan 22.5°}.

Since \tan 22.5° \approx 0.414:

a \approx 6.04 \text{ cm}.

Example 5

A regular pentagon has an apothem of 4 cm and a perimeter of 29 cm. Find its area.

A = \frac{1}{2},P,a = \frac{1}{2}\times 29 \times 4 = 58 \text{ cm}^2.

Example 6

A hexagonal paving stone has a side of 20 cm. Find the area.

Apothem of a hexagon:

a = \frac{\sqrt{3}}{2}\times 20 \approx 17.32 \text{ cm}.

Area:

A = \frac{1}{2},P,a = \frac{1}{2}\times 120 \times 17.32 \approx 1039 \text{ cm}^2.

Mistakes To Watch For

Mistake 1: Confusing the apothem with the radius

Where it slips in: Measuring or computing centre-to-vertex when centre-to-side is needed.

Mistake 2: Trying to find the apothem of an irregular polygon

Where it slips in: Applying the apothem idea to a polygon whose sides are not all equal.

Mistake 3: Using degrees in a calculator set to radians

Where it slips in: Evaluating \tan(180°/n) with the calculator in radian mode.

Key Takeaways