A concave quadrilateral is a four-sided shape where one interior angle is greater than 180 degrees, creating a “dent” or inward notch in the outline. This single feature separates it from convex quadrilaterals, where every angle is smaller than 180 degrees. Think of an arrowhead or a dart shape: one vertex points inward rather than outward, giving the figure its distinctive caved-in appearance.
The Reflex Angle That Defines the Shape
Every quadrilateral has four interior angles. In a convex quadrilateral like a rectangle or trapezoid, all four angles are less than 180 degrees. A concave quadrilateral breaks that pattern by having exactly one angle that exceeds 180 degrees. This oversized angle is called a reflex angle (sometimes called a reentrant angle), and it’s the reason one vertex appears to fold inward.
A four-sided shape can have at most one reflex angle. If two or more angles exceeded 180 degrees, the sides would cross each other, and the figure would no longer be a simple (non-self-intersecting) quadrilateral. So when you see the term “concave quadrilateral,” you can always expect exactly one inward-pointing vertex and three outward-pointing ones.
How It Compares to a Convex Quadrilateral
The easiest way to tell a convex quadrilateral from a concave one is to look at its outline. If every vertex points outward and the shape bulges in all directions, it’s convex. If one vertex dips inward, it’s concave. Here are the key differences:
- Interior angles: All angles in a convex quadrilateral measure less than 180 degrees. A concave quadrilateral has one angle greater than 180 degrees.
- Diagonals: Both diagonals of a convex quadrilateral stay entirely inside the shape. In a concave quadrilateral, one diagonal lies partly or entirely outside the figure.
- Parallel sides: A concave quadrilateral cannot have any pair of parallel sides. That means it can never be a parallelogram, rectangle, rhombus, or square.
That last point surprises many people. Because one vertex folds inward, the sides adjacent to that vertex angle in a way that prevents any two sides from running parallel. Every named “special” quadrilateral you learn in school (parallelogram, rectangle, trapezoid with one pair of parallel sides) is convex by definition.
Interior Angles Still Add Up to 360°
One common question is whether the reflex angle changes the total sum of the interior angles. It doesn’t. The formula for the sum of interior angles in any simple polygon with n sides is (n − 2) × 180°. For a quadrilateral, that gives (4 − 2) × 180° = 360°, regardless of whether the shape is convex or concave.
What changes is how that 360° is distributed. In a convex quadrilateral, you might have four angles near 90°. In a concave quadrilateral, one angle might be 270° while the other three are small, say 40°, 30°, and 20°, still totaling 360°. The reflex angle eats up a large share of the budget, forcing the remaining three angles to be relatively small.
Spotting It by Its Diagonals
Drawing the two diagonals is the quickest geometric test. Connect each pair of opposite vertices with a straight line. In a convex quadrilateral, both diagonals fall completely inside the shape and cross each other. In a concave quadrilateral, one diagonal passes through the interior normally, but the other one extends outside the boundary of the figure to connect its two vertices. If any diagonal lands outside the shape, you’re looking at a concave quadrilateral.
Everyday Examples
The most familiar concave quadrilateral is the arrowhead, sometimes called a dart. Picture a simple arrow tip: two sides angle outward to a point, and the other two sides angle back inward to form the notch at the base. That notch is where the reflex angle lives. Boomerang shapes, chevron patches, and certain decorative tiles also form concave quadrilaterals.
You’ll also see the shape in architectural floor plans and land surveys. An L-shaped room, when simplified to four vertices, often forms a concave quadrilateral. Recognizing the shape matters in those contexts because calculating area and perimeter requires accounting for the inward vertex.
Calculating the Area
The area of a concave quadrilateral can be found several ways. The most versatile approach is to split the shape into two triangles by drawing the diagonal that stays inside the figure, then add the areas of the two triangles. Standard triangle area formulas (half the base times the height, or using coordinates) work for each piece.
There’s also a general formula that works for any quadrilateral, concave or convex: the area equals half the product of the two diagonals multiplied by the sine of the angle where they intersect. In notation, that’s ½ × d₁ × d₂ × sin(θ), where d₁ and d₂ are the diagonal lengths and θ is the angle between them. For concave quadrilaterals, one diagonal extends outside the shape, so measuring that angle requires extending the lines until they meet, but the formula still holds.
Other Names You Might See
Concave quadrilaterals are sometimes called non-convex quadrilaterals or reentrant quadrilaterals. The term “reentrant” comes from the same root as the reflex angle’s alternate name (reentrant angle) and simply means the boundary “reenters” itself at one vertex. All three terms describe the same shape. In older geometry texts, you may also see the Latin-influenced term “quadrangle” used interchangeably with “quadrilateral,” since one emphasizes the four sides and the other the four angles.