In geometry, concave describes a shape that has at least one interior angle greater than 180 degrees, creating an indentation or “caved-in” section. The word itself comes from the Latin “concavus,” meaning hollow. Whether you’re talking about polygons, curves, or functions, the core idea is the same: part of the shape curves or angles inward.
Concave Polygons
A concave polygon is any polygon where at least one interior angle exceeds 180 degrees. That angle points inward, creating a notch or dent in the outline of the shape. A simple test: if you can draw a straight line between two points inside the polygon and that line passes outside the shape, the polygon is concave. Think of a star, an arrow shape, or an L-shaped figure.
The opposite is a convex polygon, where every interior angle is less than 180 degrees and no line segment between interior points ever leaves the shape. A regular hexagon or a square is convex. A star is concave.
Concave polygons have a few distinctive geometric properties:
- Diagonals extend outside the shape. In a convex polygon, every diagonal stays inside. In a concave polygon, one or more diagonals will lie partly or fully outside the boundary.
- At least one reflex angle. A reflex angle is any angle greater than 180 degrees. Every concave polygon has at least one.
- No minimum side count. A triangle can never be concave because its angles always sum to 180 degrees, making it impossible for any single angle to exceed 180. Concave polygons start at four sides.
You might also see concave polygons called “re-entrant polygons” or “nonconvex polygons” in older or more formal textbooks. These all refer to the same thing, though “nonconvex” is considered the more precise mathematical term.
How Concavity Works in Curves and Functions
Concavity also applies to curves and mathematical functions. A concave curve bows upward like the inside of a bowl turned upside down. More precisely, a function is concave if every straight line segment connecting two points on its graph stays on or below the curve itself. The graph always sits above any chord drawn between two of its points.
A convex curve is the mirror concept: it bows downward, and any line segment between two points on the graph stays on or above the curve. Picture the shape of a standard cereal bowl viewed from the side.
If you’ve taken calculus, there’s a quick shortcut. When the second derivative of a function is negative over an interval, the function is concave on that interval. When the second derivative is positive, the function is convex. The point where concavity switches direction is called an inflection point.
Concave vs. Convex at a Glance
The easiest way to remember the difference: “concave” has the word “cave” in it. A concave shape has a cave-like indentation. A convex shape bulges outward with no indentations. For polygons, the test is whether any interior angle exceeds 180 degrees. For curves, the test is whether a line segment between two points on the curve dips below (convex) or rises above (concave) the curve itself.
Calculating the Area of Concave Polygons
Finding the area of a concave polygon is trickier than measuring a convex one because standard formulas for regular shapes don’t apply directly. The most common approach is decomposition: you break the concave polygon into smaller triangles, calculate each triangle’s area, and add them together. Any concave polygon with N sides can be split into N triangles this way.
Another method works by subtraction. You draw a larger, simpler shape (like a big triangle or rectangle) that fully contains the concave polygon, calculate its area, then subtract the regions that fall outside the polygon’s actual boundary. Both approaches produce correct results for concave shapes, including complex ones like stars.
Where Concave Shapes Appear in the Real World
Concave geometry shows up constantly in architecture and engineering. The Hearst Tower in New York uses a diagrid structural system built from repeating concave polyhedral forms. The Guangzhou Opera House, designed by Zaha Hadid, features sweeping concave surfaces that define its exterior. Other examples include the Bloomberg Pavilion in Tokyo and the Baku Crystal Hall in Azerbaijan, all of which rely on concave geometric structures for both visual impact and structural efficiency.
Beyond architecture, concave shapes are everywhere: satellite dishes, the inside surface of contact lenses, spoons, and acoustic mirrors all use concave geometry to focus or redirect energy toward a central point. In optics, concave mirrors and lenses converge light, which is why they’re used in telescopes and flashlight reflectors. The inward curve gathers parallel rays and directs them to a single focal point.