Concave Pentagon Calculator

Calculations at a concave or non-convex pentagon. A polygon is called concave here, when at least one of its angles has more than 180°. This special concave pentagon is made from a square, from which a isosceles, right triangle, formed by its diagonals, has been removed. The angle α has 270°.
Enter one value and choose the number of decimal places. Then click Calculate.

	Edge length of the square (a):
	Leg length of the triangle (b):
	Perimeter (p):
	Area (A):
	Round to   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15   decimal places.

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

Concave means curved inwards, though sharply bent inwards would be more appropriate here. Thus, only one vertex, respectively one side of the original square is concave. This side has been replaced by two new edges pointing inwards, forming a reflex interior angle between them. The other three sides are neither concave nor convex but planar, that is flat. The two newly formed acute angles between sides a and b measure 45 degrees. The original square is convex, with four planar sides and four right angles.

This shape is arguably the simplest example of a pentagon with one or more inward-pointing vertices. Other such pentagons, based on more complex quadrilaterals, can be created analogously by removing a corresponding triangle between two vertices of the quadrilateral. Hereby, the perimeter increases by the length of the two new sides minus the removed side, while the area decreases by the area of ​​the triangle.
The pentagon is the non-crossed polygon with the fewest edges which is capable of having more than one concave vertex. While this is not possible if the angle α measures 270 degrees or more, two concave vertices are possible if the angle is smaller. In that case, a convex vertex lies between the two concave vertices on one side, while two convex vertices lie between them on the other side. A pentagon with two concave vertices can be formed from a triangle by indenting two of its sides.

Last updated on 06/19/2026. Author: Jürgen Kummer

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