What Are Concave And Convex | Clear Examples In Math

Students meet these two words in geometry, on coordinate graphs, and in mirrors and lenses. The tough part is that the “inward vs outward” idea stays the same, but the thing you’re judging changes: a polygon, a curve, a surface, or a function.

If you searched “what are concave and convex,” you probably want a clean way to label shapes fast, plus a few checks that still work when the picture looks confusing. Let’s lock in the meaning, then practice it with simple tests you can use on quizzes.

Concave And Convex At A Glance

Where You See It	Concave Means	Convex Means
Polygon (2D shape)	At least one corner points “in,” making a dent	No dents; all corners point “out”
Interior angles (polygons)	At least one interior angle is more than 180°	Every interior angle is 180° or less
Line segment test (polygons)	Some segment between two points leaves the shape	Every segment between two points stays inside
Curve on a graph (calc/algebra)	Bends downward like a frown	Bends upward like a smile
Function values (calc)	Slope is getting smaller as x increases	Slope is getting larger as x increases
Mirror or lens surface	Caves in (like the inside of a bowl)	Bulges out (like the outside of a ball)
Everyday quick picture	“Scooped in” shape	“Puffed out” shape
Common student trap	Mixing “concave” in optics with “concave” on graphs	Assuming “convex” always means “bigger”

What Are Concave And Convex

In plain terms, these words tell you the direction of a bend.

• Concave means the figure bends inward somewhere. You can spot a “dent,” an inward dip, or a part that caves in.
• Convex means the figure bends outward only. No dents. It’s “pushed out” all the way around.

That’s the core idea. The next step is picking the right test for the type of math problem you’re solving. A polygon test is not the same as a calculus test, even though the words match.

Concave Versus Convex Shapes In Geometry Class

In most school geometry units, “concave vs convex” shows up with polygons. A polygon is a 2D shape with straight sides. The easiest way to think about it is this: convex polygons have no “caved-in” corner, and concave polygons have at least one.

Fast visual cue for polygons

Scan the outline. If you see a dent that points into the shape, call it concave. If every corner points outward, call it convex.

The reflex angle check

A more precise test uses angles. A polygon is concave if it has at least one reflex angle, meaning an interior angle that measures more than 180°. If all interior angles are 180° or less, the polygon is convex.

If you want a clean reference with diagrams, Khan Academy’s lesson on convex and concave polygons shows the “dent” idea next to the angle idea.

The line segment test

This one feels almost like a magic trick, and it’s hard to unsee once you learn it:

• Pick any two points inside a polygon.
• Draw the straight line segment between them.
• If the segment stays inside the shape every time, the polygon is convex.
• If you can find a pair of points where the segment cuts outside the shape, it’s concave.

Wolfram MathWorld states the same idea in a formal way for polygons: a concave polygon is one that is not convex, and a quick marker is having an interior angle greater than 180° on a simple polygon. See Concave Polygon for the definition and picture pair.

Why triangles never show up as concave

A triangle can’t have a dent. With only three sides, every simple triangle is convex. So if a worksheet shows “concave triangle,” it’s either a trick, a misprint, or it’s not a triangle at all.

Quick Checks That Stop Mix-Ups

When a shape looks busy, students often label it based on a vibe. That’s when points get lost. Use a check that forces a yes-or-no decision.

Use the “dent hunt” first

Look for one inward notch. If you find it, you’re done: concave. If you don’t, move to the angle check.

Use one angle you trust

If you can spot a corner that opens wider than a straight line, that’s a reflex angle. One reflex angle is enough to call the whole polygon concave.

Don’t confuse “regular” with “convex”

A regular polygon has all sides and angles equal. Regular polygons are convex, but a polygon can be convex without being regular. A lopsided pentagon can still be convex if it has no dent.

Concave And Convex On Graphs

On coordinate graphs, you’ll hear the same pair of words when you study curves. Here, you’re judging the bend of a curve, not a straight-edged shape.

Visual cue for curves

• Concave up: the curve bends upward, like a cup you could set a marble in.
• Concave down: the curve bends downward, like a cap sitting on top of your head.

Yes, the word “concave” is still about a bend, but you’ll often attach “up” or “down” to say which way the curve opens.

Slope-based cue

If you’re in algebra or calculus and you’ve drawn tangent lines, here’s a quick way to decide without guessing:

• If the slope is getting larger as you move right, the curve is concave up.
• If the slope is getting smaller as you move right, the curve is concave down.

This explains why a parabola like y = x² is concave up: the slopes climb from negative to zero to positive as x increases.

Second derivative cue

In calculus language, concavity links to the second derivative:

• If f”(x) > 0 on an interval, the graph is concave up there.
• If f”(x) < 0 on an interval, the graph is concave down there.

If you’re not using derivatives yet, skip this and stick with the visual and slope checks. They’ll carry you through most classes.

Concave And Convex In Mirrors And Lenses

In optics, concave and convex describe a surface shape. A concave mirror caves inward. A convex mirror bulges outward. Same bend idea, new object.

One-second surface test

• Concave mirror: inside-bowl shape.
• Convex mirror: outside-ball shape.

If your class connects this to images, you may learn that concave mirrors can form both real and virtual images based on where the object sits, while convex mirrors form virtual images that look smaller. If you’re only labeling shapes, the surface test is enough.

Words That Sound Similar But Don’t Match

Some mistakes come from nearby vocabulary, not from the math itself.

Concave vs “cave”

“Concave” contains “cave,” and that’s a handy memory hook: it caves in. Keep it simple.

Convex vs “convenient”

Don’t chase a long memory trick. Just think “convex = bulges out.” One clean image beats a complicated sentence you won’t recall under time pressure.

Convex does not mean “bigger”

A convex shape can be tiny. Convex only tells you the outline has no dents. Size is a separate idea.

Practice Set You Can Do In One Sitting

Try these in your notebook with quick sketches. You’ll feel the pattern after a few rounds.

Polygon practice

1. Draw a pentagon with all corners pointing out. Label it convex.
2. Draw another pentagon, then push one vertex inward to make a notch. Label it concave.
3. Mark the reflex angle at the notch. Write “> 180°” next to it.

Graph practice

1. Sketch y = x². Mark it concave up.
2. Sketch y = -x². Mark it concave down.
3. Sketch a gentle S-curve and mark where it switches between concave up and concave down.

That last sketch is where many students pause. A curve can change concavity. Polygons don’t “switch” like that. This difference is one reason teachers keep repeating the words across units.

Table Of Common Questions And Fast Checks

Task	Fast Check	What To Write
Label a polygon	Spot a dent or a reflex angle	Concave if a dent exists; else convex
Verify with a rule	Try a segment between two interior points	If any segment leaves, it’s concave
Label a curve’s bend	See if it opens like a cup or cap	Concave up or concave down
Use slopes on a curve	Check if slope values rise or fall as x rises	Rising slopes: concave up; falling slopes: concave down
Optics surface label	Inside-bowl vs outside-ball shape	Concave mirror caves in; convex mirror bulges out
Fix a mix-up fast	Ask “What object am I labeling?”	Polygon, curve, or surface

Mini Checklist For Tests And Homework

Before you hand in an answer, run this quick list.

• Am I labeling a polygon, a curve, or a surface?
• If it’s a polygon, did I check for a dent or a reflex angle?
• If it’s a curve, did I label it as concave up or concave down?
• If I used a memory hook, is it the simple “caves in” one?
• Did I avoid mixing polygon rules with graph rules?

If you came in asking “what are concave and convex,” this is the clean takeaway: concave has an inward bend somewhere, convex doesn’t. Then you pick the matching test for the topic in front of you.

One last nudge: write the phrase “what are concave and convex” in your notes next to a two-column sketch—one dented, one smooth. That single picture can save you points when you’re tired and moving fast.