TutorialKart.comAn isosceles triangle is a triangle with at least two sides of equal length. The angles opposite those equal sides are also equal. This symmetry gives an isosceles triangle several useful properties for calculating angles, height, area, perimeter, and other measurements.
Isosceles Triangle Definition
A triangle is called an isosceles triangle when two of its sides have the same length. In the diagram below, sides AC and BC are equal, so triangle ABC is an isosceles triangle.
Parts of an Isosceles Triangle
- Legs: The two equal sides are called the legs. In the figure above, AC and BC are the legs.
- Base: The third side is called the base. In the figure above, AB is the base.
- Vertex angle: The angle formed by the two equal sides is called the vertex angle. In the figure, this is angle C.
- Base angles: The two angles at the endpoints of the base are called the base angles. In the figure, these are angles A and B.
Because AC = BC, the opposite angles are equal. Therefore, angle A = angle B.
Properties of an Isosceles Triangle
Two Sides of an Isosceles Triangle Are Equal
The defining property of an isosceles triangle is that two sides have equal lengths. If AC and BC are the equal sides, then:
AC = BC
Base Angles of an Isosceles Triangle Are Equal
The angles opposite the two equal sides are equal. These angles are called the base angles. If AC = BC, then angle A and angle B are equal.
∠A = ∠B
The converse is also true: if two angles of a triangle are equal, then the sides opposite those angles are equal, and the triangle is isosceles.
Altitude from the Vertex Is Perpendicular to the Base
Draw a line from the vertex angle to the base. When this line is the altitude, it meets the base at a right angle. If CD is the altitude to base AB, then:
CD ⟂ AB
Altitude from the Vertex Bisects the Base
The altitude drawn from the vertex angle divides the base into two equal segments. If D is the point where the altitude meets AB, then:
AD = DB
Altitude from the Vertex Bisects the Vertex Angle
The same line that forms the altitude also divides the vertex angle into two equal angles. Therefore, if CD is drawn from vertex C to base AB, then:
∠ACD = ∠BCD
As a result, the altitude from the vertex is also the median, perpendicular bisector, and angle bisector of the isosceles triangle.
Symmetry of an Isosceles Triangle
An isosceles triangle has one line of symmetry. This line passes through the vertex angle and the midpoint of the base. Folding the triangle along this line makes its two halves coincide.
Triangle Centers in an Isosceles Triangle
The line of symmetry contains several important centers of an isosceles triangle. The circumcenter, incenter, centroid, and orthocenter all lie on the altitude drawn from the vertex to the base.
Circumcenter of an Isosceles Triangle
The circumcenter lies on the altitude to the base. The circumcenter is the center of the circumcircle, which passes through all three vertices of the triangle.
Incenter of an Isosceles Triangle
The incenter also lies on the altitude to the base. The incenter is the center of the incircle, which touches all three sides of the triangle.
Centroid and Orthocenter of an Isosceles Triangle
The centroid, where the three medians meet, lies on the altitude drawn to the base. The orthocenter, where the three altitudes meet, lies on the same line.
Isosceles Triangle Angle Formulas
The sum of the three interior angles of any triangle is 180°. Since the two base angles of an isosceles triangle are equal, its unknown angles can often be calculated directly.
If the vertex angle is V and each base angle is B, then:
- V + 2B = 180°
- B = (180° − V) ÷ 2
- V = 180° − 2B
Example: Find the Base Angles
Suppose the vertex angle of an isosceles triangle is 40°. The two base angles are equal.
- Subtract the vertex angle from 180°: 180° − 40° = 140°.
- Divide the remaining angle equally between the two base angles: 140° ÷ 2 = 70°.
Therefore, the angles of the triangle are 70°, 70°, and 40°.
Isosceles Triangle Perimeter, Height, and Area
Let each equal side have length a, let the base have length b, and let the height to the base be h.
Perimeter of an Isosceles Triangle
The perimeter is the sum of all three sides:
Perimeter = 2a + b
Height of an Isosceles Triangle
The altitude divides the base into two equal parts, each measuring b/2. Applying the Pythagorean theorem to either right triangle gives:
h = √(a² − b²/4)
Area of an Isosceles Triangle
The standard triangle area formula is:
Area = 1/2 × b × h
Substituting the height formula gives an area formula based only on the side lengths:
Area = b/4 × √(4a² − b²)
Example: Find the Area and Perimeter
Consider an isosceles triangle with equal sides of 5 units and a base of 6 units.
- The altitude divides the base into two segments of 3 units each.
- Height = √(5² − 3²) = √(25 − 9) = 4 units.
- Area = 1/2 × 6 × 4 = 12 square units.
- Perimeter = 5 + 5 + 6 = 16 units.
Special Types of Isosceles Triangles
Isosceles Right Triangle
An isosceles right triangle has one 90° angle and two equal sides. Its other two angles are both 45°, so its angles are 45°, 45°, and 90°.
Equilateral Triangle as an Isosceles Triangle
Under the inclusive definition, an equilateral triangle is also an isosceles triangle because it has at least two equal sides. Some elementary classifications use the term isosceles only for triangles with exactly two equal sides, so the convention should be checked when solving a classification question.
How to Identify an Isosceles Triangle
A triangle can be identified as isosceles when any one of the following is established:
- Two sides have equal lengths.
- Two interior angles are equal.
- An altitude from a vertex also bisects the opposite side.
- A median from a vertex is also perpendicular to the opposite side.
- A line from a vertex acts as both an angle bisector and a perpendicular bisector of the opposite side.
Isosceles Triangle Frequently Asked Questions
How many equal sides does an isosceles triangle have?
An isosceles triangle has at least two equal sides. In many school-level definitions, it is described as having exactly two equal sides.
Which angles are equal in an isosceles triangle?
The angles opposite the equal sides are equal. These are called the base angles.
Can an isosceles triangle have a right angle?
Yes. An isosceles right triangle has one 90° angle and two 45° angles.
Can an isosceles triangle be obtuse?
Yes. The vertex angle may be greater than 90°, but each of the two equal base angles must then be less than 45°.
Is the altitude always a median in an isosceles triangle?
The altitude drawn from the vertex between the equal sides to the base is also a median, angle bisector, and perpendicular bisector. An altitude drawn from a base angle does not generally have all these properties.