Concave — Definition, Meaning & Examples
Concave
Non-Convex
A shape or solid which has an indentation or "cave". Formally, a geometric figure is concave if there is at least one line segment connecting interior points which passes outside of the figure.
See also
Key Formula
A polygon is concave if any interior angle θi>180°
Where:
- θi = An interior angle of the polygon at vertex i
- 180° = The threshold angle; any interior angle exceeding this value (a reflex angle) makes the polygon concave
Worked Example
Problem: A quadrilateral has interior angles of 30°, 60°, 40°, and 230°. Determine whether this quadrilateral is concave or convex.
Step 1: Verify the angles form a valid quadrilateral by checking that they sum to 360°.
30°+60°+40°+230°=360°✓
Step 2: Check each interior angle to see if any exceeds 180°.
30°<180°,60°<180°,40°<180°,230°>180°
Step 3: Since the angle 230° is a reflex angle (greater than 180°), the polygon has an indentation at that vertex. By the concavity test, the quadrilateral is concave.
θ4=230°>180°⟹concave
Answer: The quadrilateral is concave because it has one reflex interior angle of 230°.
Another Example
This example uses the formal line-segment definition of concavity with coordinates, rather than checking interior angles directly.
Problem: A pentagon has vertices at A(0, 0), B(4, 0), C(4, 3), D(2, 1), and E(0, 3). Determine whether the pentagon is concave by using the line segment test.
Step 1: Plot the vertices and connect them in order: A → B → C → D → E → A. Notice that vertex D at (2, 1) appears to dip inward relative to the other vertices.
Step 2: Pick two interior points near vertices C(4, 3) and E(0, 3), and draw the line segment between them. This segment passes through the region near D.
Segment from (3.5,2.8) to (0.5,2.8)
Step 3: Check whether this segment stays entirely inside the pentagon. Near D(2, 1), the boundary of the pentagon dips below y = 1, so the edge CD and DE create an indentation. The horizontal segment at y = 2.8 passes through a region that is outside the polygon boundary near the indent.
Step 4: Since we found a line segment connecting two interior points that passes outside the figure, the pentagon is concave.
Answer: The pentagon is concave because a line segment between two interior points passes outside the polygon near the indentation at vertex D.
Frequently Asked Questions
What is the difference between concave and convex?
A convex shape has no indentations — every line segment between two points inside the shape stays entirely inside the shape. A concave shape has at least one indentation, meaning you can find a line segment between two interior points that passes outside the shape. For polygons, a concave polygon has at least one interior angle greater than 180°, while all interior angles in a convex polygon are 180° or less.
How do you tell if a polygon is concave?
The quickest method is to check the interior angles. If any interior angle is greater than 180° (a reflex angle), the polygon is concave. Alternatively, you can try drawing line segments between pairs of points inside the polygon. If any such segment passes outside the polygon, it is concave.
Can a triangle be concave?
No. Every triangle has three interior angles that sum to 180°, so no single angle can exceed 180°. This means every triangle is convex. In fact, the triangle is the only polygon that is always convex regardless of its shape.
Concave vs. Convex
| Concave | Convex | |
|---|---|---|
| Definition | Has at least one indentation; a line segment between two interior points can pass outside the figure | No indentations; every line segment between two interior points stays inside the figure |
| Interior angles (polygons) | At least one interior angle > 180° | All interior angles ≤ 180° |
| Visual cue | Has a "caved-in" region or dent | Bulges outward everywhere; no dents |
| Diagonals | At least one diagonal lies partly outside the polygon | All diagonals lie entirely inside the polygon |
| Simplest example | An arrowhead-shaped quadrilateral | A regular hexagon or any rectangle |
Why It Matters
Recognizing concave vs. convex shapes is essential in geometry courses when classifying polygons, computing areas, and working with diagonals. In computer science and physics, concavity affects collision detection, rendering algorithms, and structural analysis — concave shapes are harder to process than convex ones. Standardized tests often ask students to identify concave polygons or use the reflex angle criterion.
Common Mistakes
Mistake: Confusing concave with convex because the words sound similar.
Correction: Remember that "concave" contains the word "cave" — a concave shape has a cave-like indentation. Convex shapes bulge outward with no dents.
Mistake: Thinking a shape must look dramatically dented to be concave.
Correction: Even a tiny indentation qualifies. If just one interior angle exceeds 180° by even a fraction, or if just one line segment between interior points exits the figure, the shape is concave.
Related Terms
- Convex — The opposite property — no indentations
- Geometric Figure — General term for shapes that can be concave
- Line Segment — Used in the formal definition of concavity
- Interior — The inside region referenced in the definition
- Point — Interior points are used in the line segment test
- Polygon — Common shape type classified as concave or convex
- Reflex Angle — An angle > 180° that indicates concavity
- Diagonal — In concave polygons, some diagonals lie outside