TutorialKart.comAn equilateral triangle is a triangle in which all three sides have the same length. Because the sides are equal, all three interior angles are also equal, and each angle measures 60°.
Equilateral Triangle Definition
A triangle is called an equilateral triangle when its three sides are equal in length. If the vertices are A, B, and C, then:
AB = BC = CA
An equilateral triangle is completely determined by the length of one side. Once the side length is known, its perimeter, height, area, inradius, and circumradius can be calculated.
Equal Sides and 60-Degree Angles in an Equilateral Triangle
All Three Sides Are Equal
The defining property of an equilateral triangle is that all three sides have equal lengths.
AB = BC = CA
All Three Interior Angles Measure 60°
The sum of the interior angles of any triangle is 180°. Since all three angles of an equilateral triangle are equal, each angle is:
180° ÷ 3 = 60°
Therefore:
∠A = ∠B = ∠C = 60°
Altitudes, Medians, and Angle Bisectors in an Equilateral Triangle
A line drawn from any vertex to the opposite side has several roles in an equilateral triangle. The same line is an altitude, median, angle bisector, and perpendicular bisector.
Each Altitude Is Perpendicular to the Opposite Side
An altitude drawn from a vertex meets the opposite side at a right angle. If AD is the altitude from vertex A to side BC, then:
AD ⟂ BC
Each Altitude Bisects the Opposite Side
The altitude divides the opposite side into two equal segments. If D lies on BC, then:
BD = DC
Each Altitude Bisects the Vertex Angle
The altitude also divides the 60° vertex angle into two equal angles of 30° each.
∠BAD = ∠DAC = 30°
Centers of an Equilateral Triangle Coincide
In an equilateral triangle, the centroid, circumcenter, incenter, and orthocenter are located at the same point.
- Centroid: the point where the three medians meet.
- Circumcenter: the center of the circle passing through all three vertices.
- Incenter: the center of the circle touching all three sides.
- Orthocenter: the point where the three altitudes meet.
This common point lies two-thirds of the distance from a vertex to the midpoint of the opposite side.
Lines and Rotational Symmetry of an Equilateral Triangle
An equilateral triangle has three lines of symmetry. Each line passes through a vertex and the midpoint of the opposite side.
It also has rotational symmetry of order 3. Rotating the triangle by 120° or 240° about its center places it in the same position.
Equilateral Triangle Perimeter Formula
Let the length of each side be a. Since all three sides are equal, the perimeter is:
Perimeter = 3a
Equilateral Triangle Perimeter Example
If each side is 8 cm, then:
Perimeter = 3 × 8 = 24 cm
Equilateral Triangle Height Formula
Drawing an altitude divides an equilateral triangle into two congruent 30°-60°-90° right triangles. If the side length is a, the height is:
Height = (√3/2)a
Deriving the Equilateral Triangle Height
The altitude divides the base into two parts of length a/2. Applying the Pythagorean theorem gives:
h² + (a/2)² = a²
h² = a² − a²/4 = 3a²/4
h = (√3/2)a
Equilateral Triangle Area Formula
The area of any triangle is one-half of the product of its base and height:
Area = 1/2 × base × height
For an equilateral triangle, substitute the base a and height (√3/2)a:
Area = (√3/4)a²
Equilateral Triangle Area Example
For an equilateral triangle with side length 6 cm:
Area = (√3/4) × 6²
Area = (√3/4) × 36 = 9√3 cm²
The approximate area is 15.59 cm².
Inradius and Circumradius of an Equilateral Triangle
Because the incenter and circumcenter are the same point, the inradius and circumradius can be expressed directly in terms of the side length.
- Inradius: r = (√3/6)a
- Circumradius: R = (√3/3)a
- Relationship: R = 2r
The inradius is the perpendicular distance from the center to any side. The circumradius is the distance from the center to any vertex.
Equilateral Triangle Formula Summary
| Measurement | Formula |
|---|---|
| Perimeter | 3a |
| Height | (√3/2)a |
| Area | (√3/4)a² |
| Inradius | (√3/6)a |
| Circumradius | (√3/3)a |
In these formulas, a represents the length of one side.
How to Identify an Equilateral Triangle
A triangle is equilateral when any of the following equivalent conditions is established:
- All three sides are equal.
- All three interior angles are equal.
- Each interior angle measures 60°.
- All three medians are also altitudes and angle bisectors.
Showing only two equal sides is not enough to prove that a triangle is equilateral. That information proves only that the triangle is isosceles unless the third side or third angle is also shown to be equal.
Equilateral Triangle and Isosceles Triangle Difference
| Feature | Equilateral Triangle | Isosceles Triangle |
|---|---|---|
| Equal sides | All three sides | At least two sides |
| Equal angles | All three angles | At least two angles |
| Angle measures | 60°, 60°, 60° | Depends on the triangle |
| Lines of symmetry | Three | Usually one |
Under the inclusive definition of an isosceles triangle, an equilateral triangle is a special type of isosceles triangle because it has at least two equal sides.
Equilateral Triangle Frequently Asked Questions
What is the angle of an equilateral triangle?
Each interior angle of an equilateral triangle measures 60°.
What is the area formula for an equilateral triangle?
If the side length is a, the area is (√3/4)a².
How many lines of symmetry does an equilateral triangle have?
An equilateral triangle has three lines of symmetry, one through each vertex and the midpoint of the opposite side.
Is every equilateral triangle also isosceles?
Yes, under the definition that an isosceles triangle has at least two equal sides. An equilateral triangle has three equal sides, so it satisfies that definition.
Can an equilateral triangle have a right angle?
No. Every angle in an equilateral triangle is 60°, so it cannot contain a 90° right angle.