Interarc Triangle Calculator

Name: Interarc Triangle - Geometry Calculator
Author: Jürgen Kummer

Calculations at an interarc triangle. This is formed, when two equal sized circles touch, between the arcs starting at the point of contact and a straight line from circle to circle.
Enter the circle radius and the base height. Choose the number of decimal places, then click Calculate. The height must not be larger than the circle radius. The base together with both triangle chords forms an isosceles triangle, between this and the interarc triangle are two circular segments.

The construction of an interarc triangle.

Formulas:

a = 2 \cdot (r - \sqrt{r^{2} - h^{2}})

α = asin (\frac{h}{r}), α in rad

b = α \cdot r

i = r \cdot [1 - \cos (\frac{α}{2})]

s = 2 \cdot \sqrt{2 \cdot r \cdot i - i^{2}}

p = a + 2 \cdot b

A = \frac{h \cdot a}{2} - r \cdot b + s \cdot (r - i)

Radius, height, lengths and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

An interarc triangle is a non-convex shape with three vertices, one straight side and two equal curved sides that are circular arcs. Its area is smaller and its perimeter is larger than that of the surrounding isosceles triangle. All three vertices are acute. However, the angle at the vertex situated between the two circular arcs measures zero degrees, as both circles share a common tangent at this point. If the height is equal to the radius, then the angles at the tips of the other two vertices also measure zero degrees. In this specific case, the interarc triangle attains its maximum size and its straight side serves as a common tangent to both circles.
The interarc triangle possesses the same symmetry properties as the isosceles triangle. It is axially symmetric with respect to a line running along its height.

Shapes constructed according to similar principles include the inner circular arc triangle situated between three identical circles, as well as the interarc quadrangle and the intercircle quadrangle located between two identical circles that do not touch. A tricorn is also situated between two identical circles, but is enclosed by a third circle positioned in their middle.

Calculation of area and other dimensions: interarc triangle

Last updated on 05/31/2026. Author: Jürgen Kummer

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