Concave and convex

Concave means curving inwards, like the inside of a bowl or the mouth of a cave. Convex means curving outwards, like the outside of a ball. The same idea is used in three places: a convex polygon has no corners pointing inwards, while a concave one has at least one; a convex lens is thicker in the middle and brings light together, while a concave lens is thinner in the middle and spreads it out; and in A-level maths a curve is convex where it bends upwards and concave where it bends downwards.

Concave vs convex: an easy way to remember

Concave has a "cave" in it: it goes in. Convex bulges out. Look at a spoon: the inside, where the soup sits, is concave; the back of the spoon is convex. Whether a surface is concave or convex depends on which side you look from.

Concave and convex polygons

A convex polygon has every interior angle less than 180°, so all its corners point outwards. The regular pentagon has five angles of 108°. A concave polygon has at least one interior angle greater than 180°, a reflex angle, so one corner is pushed in. The arrowhead has a reflex angle of 250°.

Two quick tests for a convex polygon: any straight line between two points inside it stays inside it, and every diagonal lies inside the shape. In the arrowhead, a line between its two lower points passes outside. Every triangle is convex; a polygon needs at least four sides to be concave. See what is a polygon for more.

Convex and concave lenses

In physics a lens forms an image by refracting light. Its shape decides what it does to parallel rays of light.

Convex lens: converging

A convex lens is thicker in the middle than at the edges. AQA GCSE Physics says that in a convex lens, parallel rays of light are brought to a focus at the principal focus, and the distance from the lens to the principal focus is the focal length. The image can be real (it can be shown on a screen) or virtual, depending on how far away the object is. Magnifying glasses, cameras and the lens in your eye are convex.

Concave lens: diverging

A concave lens is thinner in the middle. It spreads parallel rays out so they seem to come from a principal focus on the same side as the light came from, shown by the dashed lines. AQA GCSE Physics says the image produced by a concave lens is always virtual. Glasses for short-sighted people use concave lenses.

For both kinds of lens, magnification = image height ÷ object height, with both heights in the same unit. Trace rays through both kinds of lens in the Ray Optics simulator.

Concave and convex mirrors

Mirrors work the other way round. A concave mirror, curved inwards like the inside of a spoon, brings light to a focus: it is used in torches and makeup mirrors. A convex mirror, curved outwards, spreads light out and shows a wider view: it is used for car wing mirrors and on blind corners.

Convex and concave curves (A-level maths)

In A-level maths, convex and concave describe how a graph bends. A curve is convex on a section where it bends upwards like a bowl: its gradient is increasing, so f″(x) > 0. It is concave where it bends downwards like an arch: its gradient is decreasing, so f″(x) < 0. In the US these are called concave up and concave down.

Example: where is y = x³ − 6x² convex?

The curve is convex where 6x − 12 > 0, so x > 2, and concave where x < 2. At x = 2 it changes from concave to convex, which makes (2, −16) a point of inflection. More on this in point of inflection.

Check your understanding.

Common mistakes

• Mixing the two words up. Concave caves in; convex bulges out.
• Calling a shape concave because it looks unusual. Check for an interior angle over 180°.
• Mixing up lenses and mirrors. A convex lens converges light, but a convex mirror spreads it out.
• Saying a concave lens can make a real image. Its image is always virtual.
• Using the US words in a UK A-level answer. AQA uses convex and concave; concave up means convex.

Practise with a short timed paper

Curves come up in A-level Maths and lenses in GCSE Physics. The daily A-level Maths: Algebra & Functions and GCSE Physics Higher papers are short timed papers of 8 questions, with a new one every day.

Read next

Maths · GeometryWhat is a polygon? Definition, types, names and anglesA polygon is a flat, closed shape with straight sides. See polygon names from 3 to 20 sides, the types, interior and exterior angles, and examples.