An equilateral triangle is a triangle in which all three sides have the same length. The name says it: equi means "equal" and lateral means "sided." Because equal sides force equal angles, all three interior angles are equal too, and since a triangle's angles add to $180°$, each one is exactly $\frac{180°}{3} = 60°$. So an equilateral triangle is also equiangular.
The equal-side requirement makes it the simplest regular polygon — a regular polygon is one with all sides equal and all angles equal, and the equilateral triangle is the three-sided case. By the end you will know its properties cold and be able to derive (not just recite) its area, perimeter, and height. It sits within the wider family covered in types of triangle, alongside the isosceles triangles it generalises.
Properties of an Equilateral Triangle
Three equal sides. All sides have the same length $a$. This is the defining property.
Three equal angles. Each interior angle is $60°$.
It is equiangular and regular. Equal sides and equal angles together make it a regular polygon.
Three lines of symmetry. A line from each vertex to the midpoint of the opposite side is an axis of symmetry.
The special lines coincide. For each side, the median (vertex to midpoint), the altitude (vertex perpendicular to the side), the angle bisector, and the perpendicular bisector are all the same line. In other triangles these are usually four different lines; the equilateral triangle's symmetry collapses them into one.
It cannot be a right triangle. A natural question: can an equilateral triangle have a right angle? No. Every angle is $60°$, and $60° \neq 90°$, so there is no room for a right angle. Three angles that must sum to $180°$ and are all equal are forced to $60°$ each.
Equilateral Triangle Formulas
Let the side length be $a$. Three formulas do most of the work — and each one comes from somewhere, so it is worth seeing why rather than memorising.
Perimeter
The perimeter is the total distance around the triangle. With three sides each of length $a$:
$$P = a + a + a = 3a$$
Height (altitude)
Drop a perpendicular from one vertex to the opposite side. By symmetry it hits the midpoint, splitting the equilateral triangle into two identical right triangles. Each right triangle has hypotenuse $a$ (the original side) and base $\frac{a}{2}$ (half the bottom side). Call the height $h$ and use the Pythagorean relation:
$$h^2 + \left(\frac{a}{2}\right)^2 = a^2$$ $$h^2 = a^2 - \frac{a^2}{4} = \frac{3a^2}{4}$$ $$h = \frac{\sqrt{3}}{2},a$$
That $\sqrt{3}$ is not arbitrary — it falls straight out of the Pythagorean step. (If the $\sqrt{}$ and right-triangle steps are new, the Pythagorean relation for right triangles is the tool being used here.)
Area
Area of any triangle is $\frac{1}{2} \times \text{base} \times \text{height}$. Use base $a$ and the height we just found:
$$\text{Area} = \frac{1}{2} \times a \times \frac{\sqrt{3}}{2},a = \frac{\sqrt{3}}{4},a^2$$
So the famous $\frac{\sqrt{3}}{4}a^2$ is just "half base times height" with the height filled in. The dedicated height of equilateral triangle article walks the altitude derivation in more depth.
Quantity | Formula | What the variable means |
|---|---|---|
Perimeter | $P = 3a$ | $a$ is one side length |
Height | $h = \dfrac{\sqrt{3}}{2},a$ | $a$ is one side length |
Area | $A = \dfrac{\sqrt{3}}{4},a^2$ | $a$ is one side length |
Examples of Equilateral Triangle
Example 1
Find the perimeter of an equilateral triangle with side 9 cm.
$$P = 3a = 3 \times 9 = 27 \text{ cm}$$
Final answer: 27 cm.
Example 2
An equilateral triangle has a perimeter of 60 cm. A student says each side is $60 \times 3 = 180$ cm. Find the actual side length.
The instinct to multiply is the slip. Perimeter is the sum of three equal sides, so to go from perimeter to one side you divide, not multiply.
$$3a = 60$$ $$a = \frac{60}{3} = 20 \text{ cm}$$
The "multiply by 3" student confused the direction: $3a$ builds the perimeter up from a side, so recovering the side undoes that with division.
Final answer: each side is 20 cm.
Example 3
Find the area of an equilateral triangle with side 20 inches. Leave the answer in exact form.
$$A = \frac{\sqrt{3}}{4}a^2 = \frac{\sqrt{3}}{4},(20)^2 = \frac{\sqrt{3}}{4}\times 400 = 100\sqrt{3} \text{ square inches}$$
Final answer: $100\sqrt{3} \text{ in}^2$ (about $173.2 \text{ in}^2$).
Example 4
Find the height of an equilateral triangle whose side is 40 inches.
$$h = \frac{\sqrt{3}}{2}a = \frac{\sqrt{3}}{2}\times 40 = 20\sqrt{3} \text{ inches}$$
Final answer: $20\sqrt{3}$ inches (about $34.6$ inches).
Example 5
An equilateral triangle has an area of $36\sqrt{3} \text{ cm}^2$. Find its side length.
Set the area formula equal to the given value and solve for $a$:
$$\frac{\sqrt{3}}{4}a^2 = 36\sqrt{3}$$ $$a^2 = 36\sqrt{3} \times \frac{4}{\sqrt{3}} = 144$$ $$a = 12 \text{ cm}$$
Final answer: 12 cm.
Example 6
A triangular garden bed is to be built as an equilateral triangle with each side 6 m. The gardener needs the area to order soil and the height to plan a central path from one corner. Find both.
Area:
$$A = \frac{\sqrt{3}}{4}(6)^2 = \frac{\sqrt{3}}{4}\times 36 = 9\sqrt{3} \approx 15.59 \text{ m}^2$$
Height (the path length from a vertex to the opposite midpoint):
$$h = \frac{\sqrt{3}}{2}\times 6 = 3\sqrt{3} \approx 5.20 \text{ m}$$
Final answer: area $\approx 15.59 \text{ m}^2$; path $\approx 5.20$ m.
Why The Equilateral Triangle Matters: Maximum Strength, Maximum Symmetry
The equilateral triangle is not just the prettiest triangle. Its three-fold symmetry makes it the most efficient and stable triangle, which is why it keeps appearing in design and nature.
Structural strength. Of all triangles, the equilateral distributes a load most evenly across its three sides — no side or angle is weaker than another. This is why geodesic domes, the Eiffel Tower's lattice, and many trusses lean on equilateral or near-equilateral triangles.
Tiling and packing. Equilateral triangles tile a flat surface perfectly with no gaps, and six of them meet around a point to form a regular hexagon — the shape bees use for honeycomb because it stores the most with the least material.
The unifying idea. The equilateral triangle is the first regular polygon. Studying it is the cleanest entry to regularity — the principle that equal sides plus equal angles produce maximum symmetry, the same principle behind the square, the regular pentagon, and on up.
The equilateral triangle's symmetry has carried cultural weight for millennia; it appears in the Sierpiński triangle fractal, where the same equilateral shape repeats at every scale — a single triangle generating infinite structure.
Where Students Slip With Equilateral Triangles
Mistake 1: Forgetting the $\sqrt{3}$ in the area and height
Where it slips in: A student uses $\frac{1}{2} a \times a$ for the area, treating a side as the height.
Don't do this: Write $\text{Area} = \frac{1}{2}a^2$ as if the side were the altitude.
The correct way: The height is not a side — it is $\frac{\sqrt{3}}{2}a$, shorter than the side. The area is $\frac{\sqrt{3}}{4}a^2$. The memorizer who recalls "half base times height" but plugs the side in for height drops the $\sqrt{3}$ and overstates the area by about 15%.
Mistake 2: Multiplying instead of dividing to recover a side
Where it slips in: Given the perimeter, the student multiplies by 3 instead of dividing.
Don't do this: Compute side $= 3 \times \text{perimeter}$.
The correct way: Perimeter is $3a$, so $a = \frac{\text{perimeter}}{3}$. The rusher reaches for the same "$\times 3$" used to build the perimeter and runs it the wrong way.
Mistake 3: Assuming "equilateral" allows a right or obtuse version
Where it slips in: A student answers "yes" to whether an equilateral triangle can be right-angled or obtuse.
Don't do this: Treat the angle type as free to vary.
The correct way: Every equilateral triangle is acute — all angles are exactly $60°$, so it can never be right or obtuse. The angle type is fixed the moment the sides are equal. A learner who hasn't connected equal sides to equal $60°$ angles treats the two classifications as independent when, for this triangle, they are locked together.
Key Takeaways
An equilateral triangle has three equal sides and three $60°$ angles, making it the simplest regular polygon.
Perimeter is $3a$, height is $\frac{\sqrt{3}}{2}a$, and area is $\frac{\sqrt{3}}{4}a^2$, each derivable, not just memorised.
Its median, altitude, angle bisector, and perpendicular bisector all coincide on each side.
It is always acute; it can never be a right or obtuse triangle.
Its symmetry makes it the strongest and most efficient triangle, central to structures and tiling.
A Practical Next Step
Practice these problems to solidify your understanding. For each, write the formula first, substitute, then simplify — keep $\sqrt{3}$ in exact form unless a decimal is asked for.
Find the area of an equilateral triangle with side 8 cm in exact form. (Answer to Question 1: $\frac{\sqrt{3}}{4}(8)^2 = 16\sqrt{3} \text{ cm}^2$.)
An equilateral triangle has height $5\sqrt{3}$ cm. Find its side. (Answer to Question 2: $\frac{\sqrt{3}}{2}a = 5\sqrt{3} \Rightarrow a = 10$ cm.)
To build this further with a teacher, explore Bhanzu's geometry tutor, a middle school math tutor, or math classes online. To see a trainer derive the area formula live, you can book a free demo class.
Read More
Properties of a triangle — the general properties every triangle shares.
Isosceles triangles formula — area and height formulas for the related isosceles case.
Isosceles triangle theorem — equal sides force equal angles, the basis of the 60° result.
Triangles — the full overview of triangle parts, types, and properties.
Scalene triangle — the no-equal-sides triangle at the opposite extreme.
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