Isosceles Triangle Formula

Area, perimeter, height, and base-angle relations

Reviewed by [email protected], Geometry Calculator Developer & Online Math Educator Last updated May 12, 2026

An isosceles triangle has two sides of equal length (the legs a) and one different side (the base b). The two angles opposite the equal sides — the base angles — are always equal. These symmetry properties give very clean formulas for area, perimeter, and height.

The Formulas

Name	Formula	Notes
Area (base × height)	`A = ½ × b × h`	b = base, h = altitude from the apex perpendicular to the base.
Altitude from leg + base	`h = √(a² − b²/4)`	a = equal leg length, b = base. Drop a perpendicular from the apex; it bisects the base.
Area from legs + base	`A = (b / 4) × √(4a² − b²)`	Pure-side form combining the altitude into the area formula.
Perimeter	`P = 2a + b`	Two equal legs plus the base.
Base Angles Theorem	`∠B = ∠C`	The angles opposite the equal sides are themselves equal.
Apex Angle from Base Angle	`∠A = 180° − 2·∠B`	Sum of triangle angles is 180°.
Area (sides + apex angle)	`A = ½ × a² × sin(∠A)`	∠A is the apex angle between the two equal legs.

Worked Examples

Example 1: Isosceles triangle with legs 5 cm and base 6 cm

Altitude h = √(5² − 6²/4) = √(25 − 9) = √16 = 4 cm
Area A = ½ × 6 × 4 = 12 cm²
Perimeter P = 2(5) + 6 = 16 cm

Example 2: Find missing leg given base 10 and altitude 12

h² = a² − (b/2)² → a² = h² + (b/2)²
a² = 144 + 25 = 169 → a = 13
P = 2(13) + 10 = 36; A = ½ × 10 × 12 = 60

Example 3: Apex angle 40° → base angles?

∠B + ∠C = 180° − 40° = 140°
Since ∠B = ∠C: each base angle = 70°

Frequently Asked Questions

What is the area formula of an isosceles triangle?

Area = ½ × base × height. If you only know the legs (a) and base (b), first compute the altitude using h = √(a² − (b/2)²), then A = ½·b·h. Alternatively, with two legs and the apex angle, A = ½·a²·sin(∠apex).

How do I find the height of an isosceles triangle?

Drop a perpendicular from the apex to the midpoint of the base. By the Pythagorean theorem: h = √(leg² − (base/2)²). If the leg is shorter than half the base, no valid isosceles triangle exists.

What does the Isosceles Triangle Theorem say?

The base angles opposite the two equal legs are themselves equal. The converse also holds: if two angles of a triangle are equal, the sides opposite them are equal, making the triangle isosceles.

How do I calculate base angles from the apex angle?

Each base angle = (180° − apex angle) / 2. Example: apex 40° → base angles = (180 − 40)/2 = 70° each.

Is an equilateral triangle also isosceles?

Yes — an equilateral triangle is a special case where all three sides (and angles) are equal. It satisfies the isosceles definition (at least two equal sides) trivially.