The terms “concave” and “convex” describe the curvature of objects or mathematical functions. They’re ubiquitous in a range of disciplines, including optics, mathematics, engineering, and everyday life. Here are the definitions of concave and convex, everyday examples, and how to remember which is which.

### Concave vs Convex: Definitions and Everyday Examples

#### Concave

**Definition:** An object or a function is concave if it curves inward. In simple terms, it’s hollow or bowed in, much like a cave.

**Everyday Examples:**

- A bowl.
- A satellite dish.
- A spoon’s interior.
- Skateboard ramps.
- A pie with a slice taken out of it.

#### Convex

**Definition:** An object or function is convex if it curves outward, or in other words, bulges out.

**Everyday Examples:**

- An eye
- A speed bump.
- A magnifying glass.
- A globe.
- A triangle.

#### Ways to Remember

**Concave**: Think “Con-cave”; it has a “cave” or an inward dip.**Convex**: Think of “Con-vex” as “convicts” trying to escape, bulging outwards.

### Concave and Convex Lenses

Concave lenses focus light inside the curve of the lens, while convex lenses focus light using the outer curve. A lens that curves like a “C” (no flat side) is both concave or convex, depending on which side of the lens you view from.

#### Concave Lenses

**Examples:** Eyeglasses for nearsighted people, cameras, telescopic sights.

**Properties:**

- Diverge light rays.
- Produce a virtual, diminished image.

**Ways to Remember:** A concave lens is “caving in”, so it spreads light away or diverges it.

#### Convex Lenses

**Examples:** Magnifying glasses, eyeglasses for farsighted people, microscopes.

**Properties:**

- Converge light rays.
- Produce both real and virtual images.

**Ways to Remember:** Convex lenses “converge” light, making things appear larger or closer.

### Concave and Convex Mirrors

#### Concave Mirrors

**Examples:** Shaving mirrors, car headlights, astronomical telescopes.

**Properties:**

- Converge light rays.
- Can produce both real and virtual images.

**Ways to Remember:** Concave mirrors “converge” light, much like they’re collecting it into a focus.

#### Convex Mirrors

**Examples:** Security mirrors in stores, car side mirrors, rear-view mirrors.

**Properties:**

- Diverge light rays.
- Produce only virtual, diminished images.

**Ways to Remember:** Convex mirrors “diverge” light, spreading it out to give a wider field of view.

### Concave and Convex Polygons

#### Concave Polygons

**Examples:** Star-shaped figures, snowflakes, some irregular polygons.

**Properties:**

- Have at least one interior angle greater than 180°.
- Have vertices that “cave” inward.

**Ways to Remember:** “Concave” has a “cave” or an inward dip; you can imagine it as a polygon that’s been “pinched” inward.

#### Convex Polygons

**Examples:** Squares, rectangles, triangles, regular hexagons.

**Properties:**

- All interior angles are less than 180°.
- No vertices “cave” inward.

**Ways to Remember:** Convex polygons “bulge” outward, with no indents or “caves.”

### Concave and Convex Functions

#### Concave Functions

**Examples:** Exponential functions, quadratic functions with a positive leading coefficient.

**Properties:**

- Slope increases as you move along the function.
- Holds the property
*f*(*tx*+(1−*t*)*y*) ≥*tf*(*x*) + (1−*t*)*f*(*y*) for 0 ≤*t*≤1.

**Ways to Remember:** The shape of a concave function resembles the entrance to a cave or a hill.

#### Convex Functions

**Examples:** Logarithmic functions, negative exponential functions.

**Properties:**

- Slope decreases as you move along the function.
- Holds the property
*f*(*tx*+ (1−*t*)*y*) ≤*tf*(*x*) + (1−*t*)*f*(*y*) for 0 ≤*t*≤1.

**Ways to Remember:** A convex function looks like a valley.

By understanding the basic differences and applications of concave and convex lenses, mirrors, polygons, and functions, you gain a deeper insight into how these concepts shape our world, from the glasses we wear to the roads we drive on.

### References

- Hass, Joel; Heil, C.; Weir, M. (2017).
*Thomas’ Calculus*(14th ed.). Pearson. ISBN 978-0-13-443898-6. - Hecht, Eugene (2002).
*Optics*(4th ed.). Addison Wesley. ISBN 978-0-321-18878-6. - Nayak, Sanjay K.; Bhuvana, K.P. (2012).
*Engineering Physics*. New Delhi: Tata McGraw-Hill Education. ISBN 9781259006449. - Sines, George; Sakellarakis, Yannis A. (1987). “Lenses in antiquity”.
*American Journal of Archaeology*. 91 (2): 191–196. doi:10.2307/505216