Apothem: Definition, Formula, and Examples

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 — an irregular polygon has no single centre distance, since its sides sit at different distances. The apothem is what powers the area of shapes like the hexagon and octagon; for the polygon family itself.

By the end you will compute the apothem from the side length, use it to find any regular polygon's area, and know why an irregular polygon has none. The angles this construction relies on are the same ones in interior angles.

Apothem Versus Radius: Do Not Confuse Them

A regular polygon has two centre distances, and mixing them up is the single most common apothem mistake.

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. The apothem is the in-radius (inscribed circle); the radius is the circumradius (circumscribed circle).

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:

$$\tan\left(\frac{180°}{n}\right) = \frac{s/2}{a}$$

Solving for $a$ gives the apothem formula:

$$a = \frac{s}{2\tan(180°/n)}$$

Here $s$ is the side length, $n$ is the number of sides, and $\dfrac{180°}{n}$ is the half-central-angle. (Working in radians, the angle is $\dfrac{\pi}{n}$.)

From Apothem to Area of a Regular Polygon

The apothem exists mainly to make area effortless. Each of the $n$ triangles from the centre has base $s$ and height $a$, so its area is $\tfrac{1}{2},s,a$. Adding all $n$:

$$A = n \times \tfrac{1}{2},s,a = \tfrac{1}{2},(n s),a = \tfrac{1}{2},P,a$$

where $P = ns$ is the perimeter. So the area of any regular polygon is:

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

This single formula handles the equilateral triangle, square, pentagon, hexagon, octagon, and every regular $n$-gon — you only need the perimeter and the apothem. It works because the apothem is the height of every triangle the polygon decomposes into.

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

$$a = \frac{6}{2 \times 0.577} = \frac{6}{1.155} \approx 5.2 \text{ cm}$$

The apothem is about $5.2$ cm. (For a hexagon this also equals $\tfrac{\sqrt{3}}{2},s = \tfrac{\sqrt{3}}{2}\times 6 \approx 5.2$ cm — a useful shortcut.)

Example 2

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

A natural first move is to measure centre-to-corner, since the corner is easy to spot. Try it on a square of side $4$ cm: the centre-to-corner distance is $\dfrac{4\sqrt{2}}{2} \approx 2.83$ cm.

But that is the radius, not the apothem. The apothem reaches the midpoint of a side, not a corner.

The correct apothem of a square is half the side, $\dfrac{4}{2} = 2$ cm. Reaching the side gives $2$ cm; reaching the corner gives $2.83$ cm — confirming the apothem is the shorter of the two.

Example 3

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

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

Perimeter:

$$P = 6 \times 6 = 36 \text{ cm}$$

Area:

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

The hexagon's area is about $93.5$ cm².

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 $\tan 22.5° \approx 0.414$:

$$a = \frac{5}{2 \times 0.414} = \frac{5}{0.828} \approx 6.04 \text{ cm}$$

The octagon's apothem is about $6.04$ cm. This is the value used to find the octagon's area by $A = \tfrac{1}{2}Pa$.

Example 5

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

Apply the area formula directly:

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

The pentagon's area is $58$ cm². The pentagon's properties are covered in pentagon shape.

Example 6

A hexagonal paving stone has a side of $20$ cm. A landscaper needs its area to estimate how many stones cover a path. Find the area.

Apothem of a hexagon, using the shortcut $a = \tfrac{\sqrt{3}}{2},s$:

$$a = \frac{\sqrt{3}}{2}\times 20 = 10\sqrt{3} \approx 17.32 \text{ cm}$$

Perimeter:

$$P = 6 \times 20 = 120 \text{ cm}$$

Area:

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

Each stone covers about $1039$ cm² ($\approx 0.10$ m²), so the landscaper divides the path area by this to estimate the count.

Where the Apothem Earns its Keep

The apothem is more than a homework device — it is the practical bridge between a regular polygon and the circle hidden inside it.

• Area without trigonometry tables. Builders and tilers who know a regular tile's side and apothem get its area from one multiplication, $\tfrac{1}{2}Pa$, no per-shape formula needed. The apothem turns every regular polygon into "half perimeter times height."

• The inscribed circle. The apothem is exactly the radius of the biggest circle that fits inside the polygon. Engineers use it to size the largest round hole, pipe, or bolt head that a regular-polygon frame can hold.

• Why it links to the circle. The destination is a deep idea: as a regular polygon gains more sides, its apothem and its radius converge, and the polygon's area $\tfrac{1}{2}Pa$ approaches the circle's area $\tfrac{1}{2}(2\pi r)(r) = \pi r^2$. The apothem is how Archimedes squeezed $\pi$ between inscribed and circumscribed polygons.

That approach — bounding the circle between regular polygons of more and more sides — is Archimedes' method for estimating $\pi$, and the apothem is the very distance that makes the inner bound work.

Mistakes To Watch For

Mistake 1: Confusing the apothem with the radius

Where it slips in: Measuring or computing centre-to-vertex when the problem needs centre-to-side.

Don't do this: Use the circumradius (corner distance) in the area formula $\tfrac{1}{2}Pa$.

The correct way: The apothem reaches the midpoint of a side, perpendicular to it; the radius reaches a vertex. The apothem is always the shorter one. The student who grabs whichever centre distance is easiest to draw inflates the area every time, because the corner distance is too long.

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.

Don't do this: Compute "the apothem" of a scalene quadrilateral or a lopsided pentagon.

The correct way: Only regular polygons have a single apothem, because only they have one centre equidistant from every side. For an irregular polygon, split it into triangles and add the areas. The memoriser who learned "apothem gives area" without the "regular" condition applies it to the first uneven plot and gets nonsense.

Mistake 3: Using degrees in a calculator set to radians

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

The correct way: The formula $a = \dfrac{s}{2\tan(180°/n)}$ uses degrees; if your calculator is in radians, use $\dfrac{\pi}{n}$ instead of $\dfrac{180°}{n}$. Mismatching the mode gives a wildly wrong apothem.

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 $A = \tfrac{1}{2},P,a$, that is, half the perimeter times the apothem.

• The apothem is always shorter than the radius (centre-to-vertex), and only regular polygons have one.

• As sides increase, the apothem approaches the radius and the polygon's area approaches $\pi r^2$.

A Practical Next step

Work through the exercises below. For each polygon, find the apothem first, then use it to compute the area.

1. Find the apothem of a regular pentagon with side $8$ cm. (Answer to Question 1: $a = \dfrac{8}{2\tan 36°} \approx 5.5$ cm.)

2. A regular octagon has perimeter $48$ cm and apothem $7.2$ cm. Find its area. (Answer to Question 2: $\tfrac{1}{2}\times 48 \times 7.2 = 172.8$ cm².)

To work through apothem and area problems with a teacher, explore Bhanzu's geometry tutor, our high school math tutor sessions, or math classes online. To see a trainer derive the apothem formula live, you can book a free demo class.

Read More

• Types of polygon — only regular polygons, one class here, have an apothem.

• Convex polygon — the shape family every regular apothem-bearing polygon falls in.

• Area of a circle — the limit the polygon area approaches as sides increase.

• Quadrilaterals — including the square, whose apothem is half its side.

• Geometric shapes — the broader 2-D and 3-D shape family.