Book A Free Math Class # Equilateral Triangle: Definition, Properties, and Formulas ## TL;DR An equilateral triangle is a triangle with all three sides equal and all three angles equal to 60°. This article covers its definition, properties, the area \( \left( \frac{\sqrt{3}}{4}a^2 \right) \), perimeter \( (3a) \) and height \( \left( \frac{\sqrt{3}}{2}a \right) \) formulas with derivations, six worked examples, and common mistakes. **BT** Last updated on July 13, 2026 8 min read 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. ## 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°≠90°, so there is no room for a right angle. ## 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: \[ h^2 + \left( \frac{a}{2} \right)^2 = a^2 \] Solving gives: \[ h = \frac{\sqrt{3}}{2} a \] ### Area Area of any triangle is \( \frac{1}{2} \times \text{base} \times \text{height} \). Using the base \( a \) and the height we found: \[ \text{Area} = \frac{1}{2} \times a \times \frac{\sqrt{3}}{2} a = \frac{\sqrt{3}}{4} a^2 \] | Quantity | Formula | What the variable means | | --- | --- | --- | | Perimeter | \( P = 3a \) | \( a \) is one side length | | Height | \( h = \frac{\sqrt{3}}{2} a \) | \( a \) is one side length | | Area | \( A = \frac{\sqrt{3}}{4} a^2 \) | \( a \) 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. Find the actual side length.** \[ 3a = 60 \Rightarrow a = \frac{60}{3} = 20 \text{ cm} \] **Final answer:** each side is 20 cm. ### Example 3 **Find the area of an equilateral triangle with side 20 inches.** \[ A = \frac{\sqrt{3}}{4} (20)^2 = \frac{\sqrt{3}}{4} \times 400 = 100\sqrt{3} \text{ square inches} \] **Final answer:** \( 100\sqrt{3} \text{ in}^2 \) (about 173.2 in²). ### Example 4 **Find the height of an equilateral triangle whose side is 40 inches.** \[ h = \frac{\sqrt{3}}{2} \times 40 = 20\sqrt{3} \text{ inches} \] **Final answer:** \( 20\sqrt{3} \text{ inches} \) (about 34.6 inches). ### Example 5 **An equilateral triangle has an area of 36√3 cm². Find its side length.** Set the area formula equal to the given value: \[ \frac{\sqrt{3}}{4} a^2 = 36\sqrt{3} \Rightarrow a^2 = 144 \Rightarrow a = 12 \text{ cm} \] **Final answer:** 12 cm. ### Example 6 **A triangular garden bed is to be built as an equilateral triangle with each side 6 m. Find both area and height.** Area: \[ A = \frac{\sqrt{3}}{4} (6)^2 = 9\sqrt{3} \approx 15.59 \text{ m}^2 \] Height: \[ h = \frac{\sqrt{3}}{2} \times 6 = 3\sqrt{3} \approx 5.20 \text{ m} \] **Final answer:** area \( \approx 15.59 \text{ m}^2 \); path \( \approx 5.20 \, ext{m} \). ## 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. - **Structural strength.** Distributes load evenly across its three sides. - **Tiling and packing.** Tiles a flat surface perfectly with no gaps. - **The unifying idea.** The first regular polygon; studying it reveals principles of _regularity_. ## Where Students Slip With Equilateral Triangles ### Mistake 1: Forgetting the 3√3 in the area and height ### Mistake 2: Multiplying instead of dividing to recover a side ### Mistake 3: Assuming "equilateral" allows a right or obtuse version ## Key Takeaways - An **equilateral triangle** has three equal sides and three 60° angles. - Perimeter is 3a, height is \( \frac{\sqrt{3}}{2}a \), and area is \( \frac{\sqrt{3}}{4} a^2 \). - Its special lines coincide on each side; it is always acute. - Its symmetry makes it the strongest and most efficient triangle.