Isosceles triangle
An isosceles triangle has at least two congruent sides. The congruent sides are commonly called the legs, the remaining side is the base, and the angle between the legs is the vertex angle. This symmetry forces the two base angles to be equal and creates a single axis of reflection through the vertex and the midpoint of the base.
Core properties
- Equal base angles: congruent legs imply congruent base angles.
- Converse statement: congruent base angles imply congruent opposite sides, producing an isosceles triangle.
- Symmetry axis: the line from the vertex to the base midpoint is simultaneously an altitude, a median, and an angle bisector.
- Equilateral inclusion: an equilateral triangle satisfies the “at least two equal sides” condition and is therefore isosceles under that definition.
Labeled diagram
Standard naming and constraints
Notation commonly assigns the equal sides the same symbol \(a\) and the base the symbol \(b\). The triangle inequality becomes a single nontrivial condition:
\[ b < 2a \quad \text{and} \quad a>0,\ b>0. \]
Angle relationships
Let the base angles be \(\alpha\) and \(\beta\) and the vertex angle be \(\gamma\). In an isosceles triangle the base angles are equal, so \(\alpha=\beta\). The interior-angle sum gives
\[ \alpha + \beta + \gamma = 180^\circ \quad \Longrightarrow \quad 2\alpha + \gamma = 180^\circ, \]
so \(\alpha = \beta = \tfrac{180^\circ - \gamma}{2}\). The symmetry axis through the vertex bisects \(\gamma\).
Height, area, and perimeter formulas
The altitude to the base splits the isosceles triangle into two congruent right triangles, each with hypotenuse \(a\) and one leg \(b/2\). The Pythagorean theorem yields
\[ h^2 + \left(\frac{b}{2}\right)^2 = a^2 \quad \Longrightarrow \quad h = \sqrt{a^2 - \left(\frac{b}{2}\right)^2}. \]
The area and perimeter follow from standard triangle formulas:
\[ A = \frac{1}{2}bh, \qquad P = 2a + b. \]
Reference table
| Quantity | Formula | Notes on variables |
|---|---|---|
| Perimeter | \(P = 2a + b\) | \(a\): congruent leg length, \(b\): base length |
| Height to base | \(h = \sqrt{a^2 - (b/2)^2}\) | Requires \(b<2a\) to keep the radicand positive |
| Area | \(A = \tfrac12 bh\) | Height \(h\) is perpendicular distance to the base |
| Angle sum | \(\alpha + \beta + \gamma = 180^\circ\) | \(\alpha=\beta\) for an isosceles triangle (base angles) |
| Base angles from vertex angle | \(\alpha = \beta = \tfrac{180^\circ - \gamma}{2}\) | Degree measure shown; in radians replace \(180^\circ\) with \(\pi\) |
Worked numerical example
Leg length \(a=10\) and base length \(b=12\) produce the height
\[ h=\sqrt{10^2-\left(\frac{12}{2}\right)^2} =\sqrt{100-6^2} =\sqrt{64} =8. \]
The area and perimeter follow immediately:
\[ A=\frac12\cdot 12 \cdot 8 = 48, \qquad P=2\cdot 10 + 12 = 32. \]
Common pitfalls
- “At least two equal sides” convention: equilateral triangles satisfy the definition in many textbooks; a stricter “exactly two equal sides” convention excludes them.
- Height placement: the altitude from the vertex meets the base at its midpoint; an arbitrary segment to the base is not generally a height.
- Triangle inequality neglect: the condition \(b<2a\) prevents degenerate configurations where the three points become collinear.
- Angle labeling errors: the equal angles are the base angles, not the vertex angle.