9.1652
units
36.6606
sq units
28
units
66.4218
deg
47.1564
deg
2.6186
units
14
units
1
9.1652
units
36.6606
sq units
28
units
66.4218
deg
47.1564
deg
2.6186
units
14
units
1
The Isosceles Triangle Calculator determines all geometric properties of an isosceles triangle given the length of the two equal sides and the base. An isosceles triangle has at least two sides of equal length, with the angles opposite those equal sides also being equal. This symmetry makes isosceles triangles one of the most commonly encountered triangle types in mathematics, architecture, and design.
The word "isosceles" comes from the Greek isoskeles, meaning "equal legs" (from isos = equal and skelos = leg). The study of isosceles triangles dates back to ancient Greece, where Euclid proved in Elements Book I, Proposition 5, that the base angles of an isosceles triangle are equal — a result sometimes called the Pons Asinorum ("Bridge of Asses") because it was considered the first serious test for geometry students.
Isosceles triangles appear throughout architecture and engineering. Gothic arches, A-frame buildings, roof trusses, and gable ends are often isosceles in cross-section. The pediment of a Greek temple is an isosceles triangle, designed so that the apex angle creates a visually pleasing proportion. Modern suspension bridges use isosceles triangular configurations in their cable supports.
In nature, isosceles triangular forms appear in crystal structures, butterfly wing patterns, and the cross-sections of certain plant stems. The Sierpinski triangle, one of the most famous fractals, is typically constructed using isosceles or equilateral triangles.
The key property of the isosceles triangle is its axis of symmetry: the altitude from the apex (the vertex between the two equal sides) to the base is also the median and the angle bisector. This line of symmetry divides the isosceles triangle into two congruent right triangles, which is the basis for all calculations in this tool.
The height is computed by applying the Pythagorean theorem to one of these right triangles: with the equal side as hypotenuse and half the base as one leg, the height is the other leg. The base angles are found using the inverse cosine function, and the apex angle is simply 180° minus twice the base angle.
Enter the equal side length and the base length below. The equal side must be greater than half the base for a valid triangle to exist. The calculator returns the area, perimeter, height, both angle types, and the inradius.
For an isosceles triangle with equal sides of length $$a$$ and base $$b$$, the axis of symmetry creates two right triangles with hypotenuse $$a$$ and base $$b/2$$.
Height (Altitude from Apex):
$$h = \sqrt{a^2 - \left(\frac{b}{2}\right)^2}$$
Area:
$$A = \frac{1}{2} \cdot b \cdot h = \frac{b}{2} \sqrt{a^2 - \frac{b^2}{4}}$$
Perimeter:
$$P = 2a + b$$
Base Angles: Each base angle (at the ends of the base) is:
$$\theta_{\text{base}} = \arccos\left(\frac{b}{2a}\right) \cdot \frac{180}{\pi}$$
This follows from the right triangle where $$\cos(\theta) = \frac{b/2}{a}$$.
Apex Angle:
$$\theta_{\text{apex}} = 180° - 2 \cdot \theta_{\text{base}}$$
Inradius: The inradius equals the area divided by the semi-perimeter:
$$r = \frac{A}{s} = \frac{A}{\frac{2a + b}{2}}$$
The Area gives the surface enclosed by the triangle. For an isosceles triangle, the area depends on both the equal side and the base — a very narrow triangle (small base) or a very flat triangle (base close to 2a) will have a small area.
The Perimeter is the total boundary length: two equal sides plus the base.
The Height is the perpendicular distance from the apex to the base. It coincides with the median and the angle bisector from the apex, due to the triangle's symmetry.
The Base Angles are the two equal angles at the ends of the base. They are always equal, which is the defining angular property of isosceles triangles.
The Apex Angle is the angle at the top, between the two equal sides. If the apex angle equals 90°, you have an isosceles right triangle. If the apex angle exceeds 90°, the triangle is obtuse isosceles.
The Inradius is the radius of the inscribed circle, tangent to all three sides.
Inputs
Results
Height h = sqrt(100 - 16) = sqrt(84) = 9.1652. Area = 0.5 * 8 * 9.1652 = 36.6606. Perimeter = 20 + 8 = 28. Base angle = arccos(8/(2*10)) = arccos(0.4) = 66.42°. Apex angle = 180 - 2*66.42 = 47.16°. Inradius = 36.6606 / 14 = 2.6186.
Inputs
Results
When b = a√2 ≈ 7.071, the apex angle approaches 90°, making this an isosceles right triangle. Height = sqrt(25 - 12.5) = sqrt(12.5) = 3.5356. The base angles are each approximately 45°.
The equal side a must be greater than half the base: a > b/2. This is the triangle inequality for isosceles triangles. If a = b/2, the triangle degenerates into a straight line. If a < b/2, no triangle can be formed.
Yes. By the standard definition, an isosceles triangle has at least two equal sides. Since an equilateral triangle has three equal sides, it is a special case of isosceles. However, in everyday usage, "isosceles" often implies exactly two equal sides.
Use the Pythagorean theorem on the right triangle formed by the height: a = sqrt(h² + (b/2)²). For example, with height 9.165 and base 8, a = sqrt(84 + 16) = sqrt(100) = 10.
The apex angle is uniquely determined by the ratio b/(2a). When this ratio is small (narrow triangle), the apex angle is small. When b approaches 2a, the apex angle approaches 180° and the triangle flattens. At b = a√2, the apex angle is 90°, creating an isosceles right triangle.
Roof gables, A-frame houses, Gothic arches, yield signs, pizza slices, suspension bridge cables, folded paper airplanes, and many architectural pediments are isosceles triangles. The shape provides structural stability with visual symmetry.
The golden gnomon (apex angle 36°) and the golden triangle (apex angle 108°) are isosceles triangles whose side-to-base ratio equals the golden ratio φ = (1+√5)/2 ≈ 1.618. These appear in regular pentagons and pentagrams.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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