How do you find the apothem if you are only given the side length?

You need additional information like the radius or the number of sides. With the radius and side length, use the Pythagorean theorem: a right triangle forms with the apothem, half the side length, and the radius as its hypotenuse, giving a = \sqrt{r^2 - (s/2)^2}.

What if your polygon is not regular (sides are not all equal)?

This formula only works for regular polygons where all sides and angles are equal. For irregular polygons, you must divide them into simpler shapes like triangles and sum their areas.

Can you use this formula for a circle?

No; a circle is not a polygon and cannot be divided into triangles with an apothem. Use the area formula for a circle, A = \pi r^2, instead.

Why multiply the apothem by the perimeter specifically?

A regular polygon can be divided into congruent triangles from the center, each with the apothem as height and one side as the base. The total area is \frac{1}{2} times height times (sum of all bases), which equals \frac{1}{2}ap.

Area of a Regular Polygon

Find the area of any regular polygon by multiplying half the apothem by the perimeter—the quickest route when you know those two measurements.

A = \frac{1}{2}ap

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What each symbol means

What Area of a Regular Polygon takes

Symbol	Meaning
$A$	The area of the regular polygon, measured in square units (like cm² or m²); misreading it as a side length or diagonal is a common error.
$a$	The apothem, the perpendicular distance from the center of the polygon to the midpoint of any side, in the same linear units as the perimeter; this is not a side length.
$p$	The perimeter, the total distance around the polygon by adding all its sides; if you measure from center to vertex (the radius) instead, the formula breaks.

When to use it

Use this formula when you know both the apothem and perimeter of a regular polygon.

Level

Usually taught in: Geometry

Worked examples

1. Find a regular hexagon's area with known apothem and perimeter

Problem

A regular hexagon has an apothem of 4 cm and a perimeter of 24 cm. Find its area.

$A = \frac{1}{2}ap$
This is the formula we use for any regular polygon.
$A = \frac{1}{2} \cdot 4 \cdot 24$
Substitute the given values for the apothem $a = 4$ and perimeter $p = 24$ .
$A = 2 \cdot 24$
Calculate $\frac{1}{2} \cdot 4 = 2$ to simplify.
$A = 48 \text{ cm}^2$
Multiply to get the final area.

Answer: $A = 48 \text{ cm}^2$

Since all we had to do was substitute and multiply, this is the most straightforward type of problem—you can use the formula directly when both the apothem and perimeter are already known.

2. Find area of a regular pentagon from side length and apothem

Problem

A regular pentagon has a side length of 8 mm and an apothem of 5.5 mm. Find its area.

$p = 5 \cdot 8 = 40 \text{ mm}$
Calculate the perimeter by multiplying the number of sides (5) by the side length.
$A = \frac{1}{2}ap$
Apply the formula for area of a regular polygon.
$A = \frac{1}{2} \cdot 5.5 \cdot 40$
Substitute $a = 5.5$ and $p = 40$ into the formula.
$A = 2.75 \cdot 40$
Calculate $\frac{1}{2} \cdot 5.5 = 2.75$ to simplify.
$A = 110 \text{ mm}^2$
Multiply to get the final area.

Answer: $A = 110 \text{ mm}^2$

This type requires an extra step because you must calculate the perimeter from the side length first; always make sure you multiply by the number of sides, not just use one side length.

3. Find the area of a hexagonal garden bed

Problem

A city plans to install a decorative regular hexagonal garden bed. The apothem is 3 meters and each side is 3.46 meters. How much area will the garden bed occupy?

$p = 6 \cdot 3.46 = 20.76 \text{ m}$
Calculate the perimeter by multiplying the number of sides (6) by the side length.
$A = \frac{1}{2}ap$
Use the formula for area of a regular polygon.
$A = \frac{1}{2} \cdot 3 \cdot 20.76$
Substitute $a = 3$ and $p = 20.76$ into the formula.
$A = 1.5 \cdot 20.76$
Calculate $\frac{1}{2} \cdot 3 = 1.5$ to simplify.
$A = 31.14 \text{ m}^2$
Multiply the simplified coefficient by the perimeter to find the area.

Answer: $A = 31.14 \text{ m}^2$

Real-world problems often require you to set up the perimeter calculation first from a diagram or description, then apply the area formula; this mimics how you would approach an actual design or construction scenario.

Common mistakes

Where Area of a Regular Polygon usually goes wrong

The mistake	Why it is wrong	The fix
Confusing the apothem with the radius	The apothem is the perpendicular distance to the midpoint of a side, while the radius goes to a vertex; using the radius instead gives a wrong area because it changes what 'half' means in the formula.	Always check that your $a$ measurement is from the center perpendicular to a side, not from the center to a corner.
Using the side length as the perimeter	The perimeter is the sum of all sides; if you multiply just one side length by 0.5a, you're only finding the area of one triangle, not the whole polygon.	Count the number of sides, multiply by the side length to get perimeter first, then apply the formula.
Forgetting the $\frac{1}{2}$ or applying it wrong	The formula is $A = \frac{1}{2}ap$ not $A = ap$ ; forgetting the factor of one-half exactly doubles your answer.	Always write out the full formula $A = \frac{1}{2}ap$ before substituting numbers, or calculate $\frac{1}{2} \cdot a$ as a first step to avoid skipping the division.

Tips and when to use something else

If you know the side length $s$ and number of sides $n$ , calculate perimeter as $p = n \cdot s$ before using the area formula.
For a regular polygon inscribed in a circle, the apothem is always smaller than the radius; if your value seems too large, you may have mixed them up.
This formula is equivalent to dividing the polygon into congruent triangles from the center, each with base $s$ (one side) and height $a$ (apothem)—so the total area is $\frac{1}{2} \cdot a \cdot (\text{sum of all bases}) = \frac{1}{2}ap$ .
For a square or rectangle, use Area of a Rectangle instead—it is simpler and faster than finding the apothem.

Related formulas

Area of a TriangleCalculates the area enclosed by a triangle given its base and perpendicular height; essential for geometry, SAT, and ACT math problems involving triangles.Pythagorean TheoremFind the length of any side of a right triangle given the other two sides, or check whether a triangle is a right triangle.Area of a CircleThe area enclosed by a circle, calculated using its radius; needed for geometry and real-world problems involving circles like pools, wheels, or pizza.Sum of Interior AnglesDetermines the sum of all interior angles in a polygon based on the number of sides; critical for solving angle problems in geometry.

Frequently asked questions

How do you find the apothem if you are only given the side length?: You need additional information like the radius or the number of sides. With the radius and side length, use the Pythagorean theorem: a right triangle forms with the apothem, half the side length, and the radius as its hypotenuse, giving $a = \sqrt{r^2 - (s/2)^2}$ .
What if your polygon is not regular (sides are not all equal)?: This formula only works for regular polygons where all sides and angles are equal. For irregular polygons, you must divide them into simpler shapes like triangles and sum their areas.
Can you use this formula for a circle?: No; a circle is not a polygon and cannot be divided into triangles with an apothem. Use the area formula for a circle, $A = \pi r^2$ , instead.
Why multiply the apothem by the perimeter specifically?: A regular polygon can be divided into congruent triangles from the center, each with the apothem as height and one side as the base. The total area is $\frac{1}{2}$ times height times (sum of all bases), which equals $\frac{1}{2}ap$ .

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Reviewed 2026-09-18