Calculate area, height, perimeter, and base angles of any isosceles triangle. Enter the equal side length and base, or the equal sides and height, for instant complete results.
✓ Verified: Euclidean Geometry — NIST Digital Library of Mathematical Functions — April 2026
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Length of the unequal (base) side
Calculate area, height, perimeter, and base angles of any isosceles triangle. Enter the equal side length and base, or the equal sides and height, for instant complete results.
Isosceles Triangle Results
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Sources & Methodology
✓ Formulas verified against authoritative sources listed below.
Original source of the isosceles triangle base angle theorem (pons asinorum)
Methodology: Height h = sqrt(a^2 - (b/2)^2) from Pythagorean theorem on the half-triangle. Area = (1/2) x b x h. Perimeter = 2a + b. Base angle = arccos((2a^2 - b^2) / (2a^2)) using law of cosines. Apex angle = 180 - 2 x base angle. Triangle inequality requires b < 2a.
⏱ Last reviewed: April 2026
How to Calculate an Isosceles Triangle
An isosceles triangle has two equal sides (legs) of length a and a base of length b. The two base angles are always equal to each other. The height of an isosceles triangle bisects both the base and the apex angle, creating two congruent right triangles. This property makes all isosceles triangle calculations straightforward using the Pythagorean theorem and basic trigonometry.
Height of an Isosceles Triangle
The height (altitude from apex to base) bisects the base exactly. Using the Pythagorean theorem on the resulting right triangle: h = sqrt(a² − (b/2)²). For a = 10, b = 8: h = sqrt(100 − 16) = sqrt(84) = 9.165. This height formula works because the altitude divides the isosceles triangle into two congruent right triangles.
Area and Perimeter
Area = (1/2) x base x height = (1/2) x b x sqrt(a² − (b/2)²). Perimeter = 2a + b (two equal sides plus the base). For a = 10, b = 8: Area = (1/2) x 8 x 9.165 = 36.66 sq units. Perimeter = 2(10) + 8 = 28 units.
Base Angles and Apex Angle
Both base angles are equal: Base angle = arccos((2a² − b²) / (2a²)). Alternatively: base angle = arcsin(h/a) or arctan((2h)/b). Apex angle = 180° − 2 x base angle. All three angles sum to 180°. For an equilateral triangle (a = b), all angles equal 60°.
Special Cases of Isosceles Triangles
Equilateral triangle: a = b, all sides equal, all angles 60°. Right isosceles triangle: apex angle = 90°, base angles = 45° each, base = a√2. Golden gnomon: apex angle = 36°, base angles = 72° (related to golden ratio). Golden triangle: apex angle = 108°, base angles = 36°. These special cases appear in regular pentagons and the Fibonacci spiral.
h = sqrt(a^2 - (b/2)^2) | Area = b*h/2 | Perimeter = 2a + b | Base angle = arccos((2a^2 - b^2) / (2a^2))
a = equal side length, b = base, h = height. Triangle inequality: b must be less than 2a. Apex angle = 180 - 2 x base angle. All angles sum to 180 degrees. For equilateral triangle: a = b, all angles = 60 degrees.
Isosceles Triangle Quick Reference
Equal Side (a)
Base (b)
Height (h)
Area
Base Angle
Apex Angle
5
6
4.000
12.000
53.13°
73.74°
5
8
3.000
12.000
36.87°
106.26°
10
8
9.165
36.661
54.90°
70.19°
10
10
8.660
43.301
60.00°
60.00° (equilateral)
10
14
5.745
40.211
34.92°
110.16°
13
10
12.000
60.000
67.38°
45.24°
💡 Geometry Tip: The isosceles triangle theorem (Euclid Book I, Prop. 5) states that the base angles are always equal. This is one of the oldest theorems in geometry, sometimes called the pons asinorum (bridge of asses) because it was historically the proposition that separated students who understood geometry from those who did not. The converse is also true: if two angles of a triangle are equal, the triangle is isosceles.
Frequently Asked Questions
Height h = sqrt(a^2 - (b/2)^2) where a is the equal side and b is the base. Example: a=10, b=8: h = sqrt(100 - 16) = sqrt(84) = 9.165 units. The height bisects the base, creating two right triangles.
Area = (1/2) x base x height = (b/2) x sqrt(a^2 - (b/2)^2). Example: a=10, b=8, h=9.165: Area = (1/2) x 8 x 9.165 = 36.66 square units.
Base angles are equal and calculated as arccos((2a^2 - b^2) / (2a^2)) where a is the equal side and b is the base. For a=10, b=8: arccos((200-64)/200) = arccos(0.68) = 47.16 degrees each.
Apex angle = 180 degrees - 2 x base angle. Since angles sum to 180 and both base angles are equal, the apex angle is the remainder. For base angles of 47.16 each: apex = 180 - 94.32 = 85.68 degrees.
A right isosceles triangle has one 90-degree angle (apex) and two 45-degree base angles. The base (hypotenuse) = a x sqrt(2) where a is the equal side. Area = a^2/2. This is the most common special right triangle alongside the 30-60-90 triangle.
Perimeter = 2a + b where a is the equal side and b is the base. For a=10, b=8: Perimeter = 20 + 8 = 28 units.
A triangle is isosceles if at least two of its three sides are equal. As a consequence, the angles opposite the equal sides are also equal (the base angles). An equilateral triangle is a special case where all three sides and all three angles are equal.
Yes. An isosceles triangle is obtuse when its apex angle exceeds 90 degrees. This happens when b > a x sqrt(2) (the base is longer than the equal sides times sqrt(2)). The base angles will each be less than 45 degrees.
A golden triangle is an isosceles triangle where the ratio of the equal side to the base equals the golden ratio phi (approximately 1.618). It has apex angle 36 degrees and base angles 72 degrees each. It appears in regular pentagons and is related to the Fibonacci sequence.
If you know one angle and one side: use the law of sines or basic trig. For apex angle A and equal side a: base b = 2a x sin(A/2). Height h = a x cos(A/2). If you know the base angle B and base b: equal side a = b / (2 x cos(B)).