Concave and convex are opposite terms used to describe the shapes of mirrors, lenses, graphs, or slopes. In general, an object is concave if it’s hollow and slopes downward like an empty pool. An object is convex if it’s raised or bulging upward like a dome.

## What’s the difference between concave and convex?

If we were to describe the difference between concave and convex in the most general way possible, we would say any hollow, bowl-like object is concave, while any object resembling a rugby ball or football has a convex shape. But perhaps the reason convex and concave are so commonly misused is because the two words are not as simple as we would like them to be.

Most grammar sources insist on how using mnemonics is possible for deciphering convex and concave through the word cave. While the notion of associating concave with the event of something “caving-in” helps learn about surface features, it’s confusing to relate this idea toward subjects involving graphs.

To understand how concave and convex are different from one another, we need to learn how each term is used in the capacity of mathematics, mirrors, lenses, and, finally, within creative writing. Once you’ve decided on which context you’d like to use words like concave and convex, learning about their technical aspects will allow you to be more confident about using them in everyday speech.

## What does concave mean?

Concave is used as an adjective or noun to describe the shape of an object or surface that is curved inward or hollowed out like a bowl. Examples of concave in a sentence include,

“She scooped the ice cream from the container, leaving a concave footprint.”

According to The Oxford Dictionary of Word Histories, the word concave stems from late Middle English. Concave and concavity render the Latin term concavus (con- and cavus), which translates to ‘together’ and ‘hollow’ (“Concave,” 110).

Synonyms of concave include:

Dented, depressed, dished, hollow, indented, recessed, and sunken.

Antonyms of concave include:

Bulging, cambered, protruding, protrusive, protuberant.

## What does convex mean?

The word convex, or convexity, is an adjective describing an object or surface that is rounded or curved. In regards to objects, the middle surface of a convex object is wider than its outer corner. Examples of convex in a sentence include,

“I fell into poison ivy and now have red, convex bumps all over my skin.”

Synonyms of convex include:

Arched, bent, bulging, raised.

Antonyms of convex include:

Depressed, sinking.

## Concave vs convex functions

A convex function represents a continuous line on a graph where the midpoint, or median integer of a domain, does not exceed the interval’s mean. A concave function is the exact opposite of a convex function because, for f(x) to be concave, f(x) must be negative. To make the differences more clear, here is a quick run-down of how the terms compare:

Concave upwards = convex = convex downward

Concave downward = concave = convex upward

The difference between concave and convex functions is shown more clearly if we look at a graph. Notice how the convex function opens up, while the concave function opens down.

Another way of identifying concave and convex functions is to connect points on the graph along the x-axis. A concave function only connects lines below the graph, while a convex function only produces lines above the graph.

We can additionally use calculus to decipher whether or not a function is convex or concave. If the second derivative of f(x) is greater than zero, then the function is convex. But if the second derivative of f(x) is less than zero, the function is concave.

## Convex vs. concave polygons

Don’t panic, but the words convex and concave are also used for geometry too. Convex and concave shapes are most often discussed in reference to polygons, which are shapes with a minimum of three sides and angles.

Regular polygons exist with equal sides and angles, but convex and concave polygons are a bit more complicated. Convex polygons contain interior angles that are less than 180 degrees, while concave polygons contain one or more internal angles that are greater than 180 degrees.

A second method of identifying concave and convex polygons is to draw two diagonal lines across the shape, starting from the shape’s corners. If every line exists within the shape, the shape is convex. If there’s at least one line that crosses outside of the shape, it’s concave.

## Concave vs. convex mirrors

The words concave and convex are commonly used while discussing optical objects, such as mirrors and lenses. Any convex surface will protrude outward, similar to a bubble, which gives the effect of broadness. Convex mirrors are commonly found inside parking garages, where drivers need a wide view around corners or potential blind spots.

In contrast, a concave mirror is curved inward and produces a magnified reflection that is upside down. As shown by the University of South Wales’ School of Physics, anyone with a shiny spoon can test this observation by looking at their reflection in the spoon. On the concave surface of the spoon, where you pick up your food, your reflection is distorted in a narrow fashion, and you will appear upside down. But if you flip your spoon around to the convex side, your reflection will be upright and smaller.

Reflections vary so greatly between convex and concave surfaces because whether or not the surface curves outward or inward, the mirror itself is part of a sphere. Because the surface of a sphere is not flat, the light reflected from its surface will travel different distances before contacting the mirror.

Depending on which area of the surface the light makes contact, a ray of light may reflect off of nearby surfaces to produce a more focused area of visibility. This is why, while looking at the inside surface of a spoon, one might only see the reflection of what is closest to the spoon’s surface–– even if it still appears smaller. In addition, if you were to place your finger inside the spoon, the surface of the mirror can produce two or three different reflections at the same time.

## Concave lens vs. convex lens

Concave and convex lenses exist in everyday objects such as eyeglasses, contacts, binoculars, and telescopes. Similarly to the science behind light reflection on convex and concave mirrors, there are also patterns for how light passes through concave and convex lenses to produce a visible image.

### Convex lenses

According to Manocha Academy, convex lenses are called converging lenses because of their ability to produce images that are diminished, magnified, or introverted. How accurately an image appears through the convex lens depends on how close the object is to the focal point of the lens. For any convex lens, there are symmetrical focal points on either side of the lens.

How accurately an object is produced through a convex lens is predicted with three principles:

Principle 1: Any ray of light that passes through a convex lens is parallel to the principal axis, which is a central line passing through the absolute center of a spherical lens. Once the light contacts the lens, the light refracts and passes through the focal point on the other side. Passing light rays that are parallel to the principal axis travel above the object in front of the lens.

Principle 2: Any ray of light that passes through the center of a convex lens will continue in a straight line on the other side.

Principle 3: Any ray of light that passed through a focal point on the same side as the object will refract upon contacting the lens and become parallel to the principal axis on the other side.

While there are three principles for predicting image quality through a convex lens, only two principals can apply to an object while drawing a ray diagram. Any two principals will produce an intersection of light on the other side of the lens, which will indicate the area where the object’s image appears.

The farther away an object is to a convex lens, the smaller the image will be on the other side, and the closer an object is to the convex lens, the larger it will appear. An object located past the nearest focal point produces a larger image behind the object. As Manocha Academy points out, this type of lens magnification is used for optical tools such as magnifying glasses.

Depending on which focal point an object is located, it is possible to produce an image that is approximate in size. It is also possible to create real images that aren’t visible if the passing light rays never intersect because they are parallel on the other side. In this case, the image produced exists at an infinite distance away.

### Concave lenses

Any concave lens will have a thicker diameter, and a thinner center since a concave lens is curved inward. The shape of concave lenses allows light to spread out once it makes contact with the lens, which allows virtual objects to appear smaller. The concave lens’s ability to spread out light makes it an ideal lens for tools such as flashlights, where a central source of light can be used over a wider surface area.

Concave lenses produce images with three principals that are similar to convex lenses:

Principle 1: Any ray of light traveling above the object and parallel to the principal axis will refract upon contact with the lens and appear as though it’s coming from the direction of the nearest focal point.

Principle 2: Any ray of light passing through the center of the concave lens will pass through without refraction.

Principle 3: Any ray of light directed toward a focal point on the other side of the concave lens will refract and become parallel to the principal axis.

While the size of the object’s image will vary depending on the object’s distance from a concave lens, the image produced is always virtual, upright, smaller, and appears on the same side of the lens as the object.

## Concave vs. convex in writing

The words concave and convex are opposite of each other, and they essentially describe the shape of objects or surfaces in the same way we contrast words like:

- Small vs. big
- Tall vs. short
- Slim vs. wide
- Round vs. flat

Because concave and convex are technical words, using them descriptively for prose can produce a more metaphorical or abstract interpretation for casual audiences. What this means is that if writers decide to use concave or convex outside the realm of their typical use, the writer must decide how clear they intend their writing to be.

### Consider the adjective

A prominent example of using concave or convex in literature is to consider the writings of American author David Foster Wallace (or DFW for short). DFW is notorious for his complex and technical writing style–– and yes, he used convex and concave to describe nouns several times. Wallace uses concave and convex in the following examples:

“My handeye was okay, but I was neither large nor quick, had a near concave chest and wrists so thin I could bracelet them with a thumb and pinkie…”

– “Tennis, Trigonometry, Tornados: A Midwestern Boyhood,” Harper’s Magazine.

“…doll’s eyes that open with the pull of a heartstring, concave where I am convex.”

– “Order and Flux in Northampton,” Conjunctions.

The point of using DFW examples isn’t merely to show how writers have used convex or concave outside of mathematical context, but rather, for writers to consider whether using technical or abstract terms is helpful for clearly communicating to their audience.

In the first DFW example, the use of concave makes sense because he is describing the structure of something. But, using concave and convex to describe non-objective nouns, such as feelings or thoughts, tends to obscure the meaning of what we’re trying to explain. The second DFW example walks a fine line while using convex and concave because their use is both literal and metaphorical.

## FAQ: Related terms

What is a function?

A function is the equation of a line on a graph. Functions are different for every line on a graph, but they take the general form of,

f(x) = x + 1

With any line function, the dependent variable is typically f(x), while any unknown variable within the function is called an independent variable.

What is the domain?

The domain is a set of independent variables on a graph that corresponds to the function of a line. Domains are important to understand because they locate specific points on the chart that corresponds to any real output of an equation.

What is the range?

The range of any function represents any possible value along the x or y-axis on a graph that produces a valid value for the dependent variable. All minimum and maximum integers are identified by substituting variables for the x or y-axis.

What is an interval?

An interval is a set of numbers that represent the integers of a line, aka the domain. A closed interval includes two finite points of a line, where the line no longer exists outside of an integer’s range. An open interval is a set of integers that do not represent the absolute end of a line’s domain.

## Test Yourself!

See how well you understand the difference between convex and concave with the following multiple choice questions:

- While looking at a shiny spoon, you notice your reflection is upside down. The surface of the spoon is a _________ surface.

a. Convex

b. Mirror

c. Concave

d. Metal - In terms of graphing functions, a convex function is sometimes called:

a. Concave upwards

b. Convex downwards

c. Concave downwards

d. A and B - Which of the following does not describe a concave surface:

a. Hollow half sphere

b. Eye lense

c. Cereal bowl

d. Hemisphere - Which of the following does not describe a convex surface:

a. The character Stewie from Family Guy

b. Sphere

c. Parking garage mirror

d. Football - Which of the following shapes cannot exist as a concave polygon:

a. Pentagon

b. Hexagon

c. Triangle

d. Octagon

Answers

- C: Concave
- D: A and B
- D: Hemisphere
- B: Sphere
- C: Triangle

#### Sources:

- Bourne, M. “Domain and Range of a Function.” Interactive Mathematics, Jan. 4, 2019.
- “Concave.” Merriam-Webster Dictionary, 2019.
- “Concave.” The Oxford Dictionary of Word Histories, Ed. Chantrell, 2002, p. 110.
- “Concave and Convex Mirrors.” Manocha Academy, YouTube, 2019.
- “Concave Lens: Definition & Uses.” Study.com, 2019.
- “Convex.” Merriam-Webster Dictionary, 2019.
- “Convex and Concave Lenses.” Manocha Academy, YouTube, 2019.
- “Mathematics / Understanding Polygons.” Learnhive Inc., 2018.
- “Reflection from a concave mirror.” UNSW Physics, YouTube, 2017.
- Wallace, D.F. “Tennis, Trigonometry, Tornadoes: A Midwestern Boyhood.” Harper’s Magazine, 1991.
- Wallace, D.F. “Order and Flux in Northampton.” Conjunctions, 1991.
- Weisstein, Eric W. “Concave Function.” MathWorld–A Wolfram Web Resource, n.d.
- Weisstein, Eric W. “Convex Function.” MathWorld–A Wolfram Web Resource, n.d.

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