# 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:

- vertical leg is the apothem a,
- horizontal leg is half a side, \dfrac{s}{2},
- angle at the centre is half the full central angle, \dfrac{1}{2}\cdot\dfrac{360°}{n} = \dfrac{180°}{n}.

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

- The **apothem** is the perpendicular distance from a regular polygon's centre to the midpoint of a side.
- It is found with a = \dfrac{s}{2\tan(180°/n)} and equals the inscribed-circle radius.
- The area of any regular polygon is A = \frac{1}{2},P,a.
- The apothem is always shorter than the radius (centre-to-vertex).
- As sides increase, the apothem approaches the radius and the polygon's area approaches \pi r^2.