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Concave vs. convex are terms that describe two types of curvatures or shapes that are omnipresent in our daily lives and in the natural world. These two words can easily be mixed up because of their similarity: both their form and meaning are similar. Our understanding of these terms does not only enrich our vocabulary but is also critical when we delve into physics, especially when discussing lenses and light behavior.

## Concave vs. Convex

### Key Takeaways

- These words have a similar meaning: they talk about the type of curvature an object has.
**Convex**means “having an outline or surface that curves outwards”, while**concave**means “having an outline or surface that curves inwards”.

### Concave vs. Convex: The Definition

**What Does “Concave” Mean?**

A **concave** surface curves inward, much like the inside of a spoon or a bowl. When we refer to something as concave, we’re noting that it is thinner in the middle than on the edges—creating a recessed or hollowed shape.

**What Does “Convex” Mean?**

In contrast, a **convex** surface bulges outwards, resembling the exterior of a ball. Items with a convex shape have a surface that protrudes outward from the center, often forming a dome-like structure.

### Concave vs. Convex: Usage and Examples

#### When to Use Concave

If you drop a plate but it doesn’t break fully, it’s just missing a piece, then the shape of the plate would be a concave shape. If you have a ping pong ball and you step on it, denting it in the process, then the shape of the ball would also be concave. So concave is always used for curves that go inwards. In math, a function is concave if no line segment joining two points on the graph lies above the graph at any point (there are formulas for this, I just gave you the definition).

**Examples:**

*Did you know that your bathtub is actually***concave**?*If the second derivative of the function is negative, then the function is***concave**.

#### When to Use Convex

If your soccer ball or tire has a bulge, then you can call them convex. If you ever see a square you can call it convex too. So convex means that there exists a curve that goes outwards. A convex function is similar to a concave function, but this time there should be no segment joining two points on the graph that lies below the graph.

**Examples:**

*The doctor told me to use***convex**lenses to improve my sight.**Convex**functions are cool, they look like a bucket.

### Tips for Using Concave vs. Convex

There is no obvious reason for why concave is concave and convex is convex, so you can get them mixed up really easily. I did that in math class all the time. I figured out a trick that saved my life (and grades). Imagine a mountain with a cave at the bottom, just like in cartoons. Now, only draw the outlines. What you get is a concave shape. Simple, right?

## Concave vs. Convex Examples

### Examples of “Concave”

*The***concave**design of the satellite dish helps to focus the signals at a central point.*The architect incorporated a***concave**ceiling in the lobby to enhance the acoustics of the space.*The bowl was beautifully crafted with a smooth***concave**interior, perfect for holding fruit.*For the experiment, they used a***concave**lens to spread the beam of light.*The***concave**face of the spoon cradled the soup, preventing spills as she ate.*The***concave**depression in the landscape was the result of a meteor impact millions of years ago.

### Examples of “Convex”

*The***convex**hull of the boat was designed to cut through the water more efficiently.*The security peephole in the door was***convex**to give a wider field of view.*The***convex**surface of the dome created an interesting reflection of the city skyline.*To demonstrate optical principles, the teacher used a***convex**mirror in the classroom.*The designer preferred using***convex**shapes to add a sense of depth to her artwork.*The***convex**curvature of the car’s fender gave it a sleek and aerodynamic look.*The bubble’s***convex**form shimmered with iridescent colors before it burst.

## Concave vs. Convex: **Practice and Exercises**

**Practice Exercise: Understanding Concave vs. Convex**

Below are a list of subjects. For each subject, decide whether it is described by the term ‘concave’ or ‘convex’.

- A spoon’s inner surface where you place food.
- The outer surface of a soccer ball.
- A cave entrance curving inwards.
- The lens of magnifying glass that makes things look bigger.
- A valley between two mountains.
- The exterior of a bowl.
- A bumpy road surface.
- The inner part of a skateboard ramp where skaters slide down.
- The shape of a person’s back when they hunch over.
- A mirror that causes light to diverge.

**Answers:**

- Concave
- Convex
- Concave
- Convex
- Concave
- Convex
- Convex
- Concave
- Concave
- Concave

## Frequently Asked Questions

**What is the difference between concave and convex?**

Concave shapes curve inward, like the inside of a bowl, while convex shapes curve outward, similar to the exterior of a sphere.

**How can we tell if a function is concave or convex?**

By examining the sign of a function’s second derivative. If it’s positive, the function is convex, and if it’s negative, the function is concave.

**Can a polygon be both concave and convex?**

No, these properties are mutually exclusive. A concave polygon has at least one interior angle greater than 180 degrees, creating an indentation. A convex polygon’s interior angles are all less than 180 degrees, without any indentations.

**How do concave and convex properties affect the function graph?**

A function’s concavity determines the direction it curves. For intervals where it is concave, it curves downwards and for intervals where it is convex, it curves upwards.

**What is a point of inflexion?**

It is a point on a graph where the function changes curvature from concave to convex or vice versa. At this point, the second derivative is often zero.

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