What Is a Concave Shape?
A concave shape is a shape that has at least one part pushed inward, so it is not fully "bulging out" on every side. The clearest test is the angle: a concave shape has at least one interior angle greater than 180°, called a reflex angle. That inward dent is what gives the shape its cave-like look, which is where the name comes from.
There is a second test that always agrees with the first. In a concave shape, you can pick two points inside it and draw a straight line between them that slips outside the shape somewhere along the way. If every such line stays fully inside, the shape is not concave - it is convex. The opposite of a concave shape is a convex shape, where every interior angle is under 180° and every point bulges outward.
For Example: The Telescope Mirror That Had to be Curved Inward, and Got it Wrong
When engineers built the Hubble Space Telescope, its main mirror had to be ground into a precise concave curve so it would bow inward and focus starlight to a single point. A tiny error in that inward curve - less than the width of a hair - blurred every early image the telescope sent back. That is the power of a concave shape: because it curves in on itself, it collects and redirects whatever meets it. Get the "caving in" exactly right and you see distant galaxies; get it slightly wrong and you see fog.
What Is the Difference Between Concave and Convex Shapes?
The dividing line is a single reflex angle. A convex shape has none; a concave shape has at least one.
Feature | Convex shape | Concave shape |
|---|---|---|
Interior angles | All less than 180° | At least one greater than 180° (reflex) |
Line between inside points | Always stays inside | Can pass outside |
Look | Bulges outward everywhere | Caves inward at one or more vertices |
Diagonals | All lie inside | At least one lies partly outside |
Examples | Square, regular pentagon, circle | Star, arrowhead, letter L |
Every polygon is either convex or concave. A concave polygon is just a polygon with at least one reflex angle; when you want the polygon-specific rules and diagonal counts, see concave polygons. The idea of "caving inward," though, is broader than polygons - it also applies to curves and to 3D solids.
How Do You Identify a Concave Shape?
Two quick tests settle it, and either one is enough.
The reflex-angle test. Check every interior angle. If even one is greater than 180°, the shape is concave. If all are under 180°, it is convex.
The diagonal test. Draw the diagonals. If any diagonal lies partly outside the shape, it is concave. In a convex shape, every diagonal stays inside.
For a curve rather than a straight-sided shape, use the "line between two points" version: if a straight segment joining two interior points ever leaves the region, the shape is concave.
Why Does a Reflex Angle Make a Shape Concave?
Because a reflex angle is the geometric name for an inward dent. An interior angle of exactly 180° would be a straight edge with no corner at all. Push that corner outward and the angle drops below 180° (convex). Push it inward, past the straight line, and the angle opens beyond 180° into the reflex range - the vertex now points into the shape instead of away from it. That inward-pointing vertex is precisely the "cave." A reflex angle is any angle between 180° and 360°, and a single one anywhere in a shape is enough to make the whole shape concave.
What Are the Types of Concave Shapes?
Concavity turns up in three families of shape, and the same "caves inward" idea drives all of them.
Concave polygons. Straight-sided shapes with at least one reflex vertex: darts (arrowheads), star polygons, block letters such as L and T, and plus or cross shapes. The inward corner is the reflex angle.
Concave curves. Smooth shapes that bow inward, such as a crescent moon or a kidney-bean outline. They have no corners, so the line-between-points test, not the angle test, exposes the concavity.
Concave solids (3D). Three-dimensional objects with a scooped-out or dented region: a bowl, the front of a spoon, a bitten apple, or a block with a notch cut from one edge.
Across all three families the defining mark is identical: somewhere a straight segment joining two interior points slips outside the shape.
Examples of Concave Shapes
Six examples, from checking a single angle to spotting concavity in three dimensions.
Example 1
Is a five-pointed star a concave shape?
Look at the vertices. The five outer points each have a small angle, but the five "notches" between the points each open past 180°.
Those five inward notches are reflex angles, and a straight line drawn from one point tip to another passes outside the star through the empty notch.
Final answer: yes, a five-pointed star is concave. Any inward notch is a reflex angle, and one is enough.
Example 2
A student says "a crescent moon curves inward, so it must be convex, because a smooth curve has no big angles." Where does this go wrong?
The tempting move ties concavity to sharp corners only and assumes a smooth curve cannot cave in.
The crescent has no vertices to measure, so the angle test seems to fail - but the line test still works. Pick two points near the two tips of the crescent and join them: the straight segment passes through the empty bite of the moon, outside the shape entirely.
The correct method uses the line-between-points test for curves:
If a segment between two interior points leaves the region, the shape is concave.
Final answer: a crescent moon is concave. Concavity is about caving inward, not about having corners.
Example 3
Is the letter L (as a block shape) concave or convex?
The block letter L has six corners. Five of them are ordinary right angles of 90°, but the inner corner where the two arms meet opens to 270°.
$270° > 180°$, so that inner corner is a reflex angle.
Final answer: the block letter L is concave. The single 270° inner corner makes it so.
Example 4
A hexagon has interior angles 100°, 120°, 140°, 200°, 90°, and 70°. Is it concave?
Scan the list for any angle over 180°.
The fourth angle is $200°$, which is greater than $180°$.
Final answer: the hexagon is concave. One reflex angle is all it takes; the other five angles do not matter.
Example 5
A quadrilateral has all four interior angles equal to 90°. Is it concave?
Check each angle against 180°.
Every angle is $90°$, and $90° < 180°$, so none is reflex.
Final answer: it is convex, not concave. This is just a rectangle; a shape with no reflex angle cannot be concave.
Example 6
Can a 3D shape be concave? Consider a solid block with a rectangular notch cut out of one edge.
Apply the line test in three dimensions: pick two points inside the solid.
Choose one point on each side of the notch. The straight line joining them passes through the empty notch, outside the solid.
Final answer: yes, the notched block is a concave 3D shape. The same "line exits the shape" rule that works for flat shapes works for solids - a dented can, a bowl, or a 3D shape with a scooped-out region is concave. The reasoning step is applying the line test in space, not just on paper.
Why Concave Shapes Matter: They Collect and Redirect
Concave shapes are not just the "odd ones out" next to neat convex shapes. Their inward curve is exactly what makes them useful in the physical world.
They focus energy. A concave mirror or dish curves inward so it gathers light or signal onto one point - the principle behind telescopes, satellite dishes, and solar cookers.
They fit and interlock. Concave notches let shapes lock together, from jigsaw pieces to gears to the arms of a letter in a font.
They need careful measuring. Because a concave shape's diagonals and lines can leave the region, area and containment calculations take more care than for convex shapes.
What Are the Most Common Mistakes With Concave Shapes?
Three errors account for most wrong answers, and the first comes from trusting the eyes instead of the test.
Mistake 1: Judging by "looks pointy" instead of measuring the angle
Where it slips in: Deciding a shape is concave because it has sharp points, or convex because it looks smooth.
Don't do this: Calling a five-pointed star concave "because of the spikes" — the spikes are not the reason.
The correct way: A star is concave because of the reflex angles in the notches between the points, not the sharp points themselves. Check the actual interior angles or use the line test. The rusher who reads the silhouette misses that the concavity lives in the dents, not the spikes.
Mistake 2: Thinking a shape needs several reflex angles to be concave
Where it slips in: Assuming one reflex angle is not "enough" to change the whole shape.
Don't do this: Calling a block letter L convex because five of its six angles are fine.
The correct way: A single interior angle over 180° makes the entire shape concave. The definition is "at least one" reflex angle. The second-guesser who keeps looking for more inward dents already had the answer at the first one.
Mistake 3: Believing only flat shapes can be concave
Where it slips in: Restricting the idea to 2D polygons.
Don't do this: Saying a bowl or a dented can "can't be concave because it's a solid."
The correct way: Concavity applies in three dimensions too. If a straight line between two interior points of a solid leaves the solid, it is concave — a bowl, a spoon's front, and a notched block all qualify. The confusion between "concave shape" as flat-only versus general is a common source of wrong answers.
Conclusion
A concave shape caves inward and has at least one interior angle greater than 180°, called a reflex angle.
Two tests identify it: the reflex-angle test and the line-between-inside-points test.
A single reflex angle anywhere is enough to make the whole shape concave.
Concavity is not limited to flat polygons; 3D solids like bowls and notched blocks can be concave too.
To build shape-and-space confidence with a teacher, explore Bhanzu's geometry tutor, a middle school math tutor, or math classes online.
Practice These to Solidify Your Understanding
Work through these, then check your answers:
A pentagon has interior angles 100°, 90°, 220°, 80°, and 50°. Concave or convex? (Answer to Question 1: concave — the 220° angle is reflex.)
Is a regular hexagon concave or convex? (Answer to Question 2: convex — every interior angle is 120°, all under 180°.)
Is a plus sign (+) shape concave or convex? (Answer to Question 3: concave — its four inner corners are reflex angles of 270°.)
If Question 3 tripped you, revisit Example 3 on the block letter L. Want a trainer to walk shapes through with your child? Book a free demo class.
Read More
Convex polygon — the opposite case, where every interior angle stays under 180°.
Quadrilaterals — four-sided shapes that can be convex or concave.
Types of quadrilaterals — the full family of four-sided shapes.
2D shapes — the flat shapes concavity first shows up in.
Diagonal — the segment whose position gives the diagonal test for concavity.
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