Circular Triangle — Definition, Formula & Examples

A circular triangle is a closed shape formed by three circular arcs that meet at three vertices. Unlike a standard triangle with straight sides, each side of a circular triangle curves inward or outward along part of a circle.

A circular triangle is a region in the plane bounded by exactly three arcs of circles (or lines, which may be treated as circles of infinite radius), where each pair of consecutive arcs shares a common endpoint. The interior angles at each vertex are measured between the tangent lines to the two arcs meeting at that vertex.

How It Works

To construct a circular triangle, choose three circles (or use parts of three circles) so that each pair intersects. Select one arc from each circle between two intersection points to form a closed region with three vertices. The angle at each vertex is the angle between the tangent lines to the two arcs at that point, not the angle between radii. A classic example is the Reuleaux triangle, formed from three arcs of circles centered at the opposite vertices of an equilateral triangle.

Worked Example

Problem: An equilateral triangle has side length 6 cm. A Reuleaux triangle is formed by drawing a circular arc centered at each vertex through the other two vertices. Find the perimeter and area of this Reuleaux triangle.

Identify the arcs: Each arc is part of a circle with radius 6 cm, centered at one vertex, spanning the 60° angle at that vertex.

Find the perimeter: Each arc subtends a central angle of 60°. The arc length is (60/360) × 2πr = πr/3. With three such arcs, the total perimeter is 3 × πr/3 = πr.

P = \pi r = \pi(6) = 6\pi \approx 18.85 \text{ cm}

Find the area: The area of a Reuleaux triangle equals the area of the equilateral triangle plus three circular segments. Each circular segment has area (πr²/6 − r²√3/4). So the total area is:

A = \frac{\pi r^2}{2}(1) - r^2\left(\frac{\sqrt{3}}{2}\right) + r^2\frac{\sqrt{3}}{4} \cdot 3 \;\;\Rightarrow\;\; A = \frac{r^2}{2}(\pi - \sqrt{3}) = \frac{36}{2}(\pi - \sqrt{3}) \approx 25.11 \text{ cm}^2

Answer: The Reuleaux triangle has a perimeter of

6\pi \approx 18.85

cm and an area of

18(\pi - \sqrt{3}) \approx 25.11

cm².

Why It Matters

Circular triangles appear in engineering and design — the Reuleaux triangle is used in drill bits that cut nearly square holes and in rotary engines like the Wankel engine. Understanding them also builds intuition for non-Euclidean geometry, where all triangles have curved sides.

Common Mistakes

Mistake: Measuring the vertex angle as the angle between the radii of the two arcs rather than between their tangent lines.

Correction: The interior angle at a vertex of a circular triangle is defined by the tangent lines to the two arcs at that point. These tangent angles generally differ from the central angles of the arcs.