A point is an exact location in space. It has no length, no width and no thickness, so it has no size at all. You draw a point as a dot and name it with a capital letter, such as point A. On a coordinate plane, an ordered pair such as (3, 2) names a point.
A dot only shows where a point is; the point itself has no size. On a grid, the ordered pair gives the address, and swapping the numbers moves you to a different point.
Formal definition and notation
In Euclidean geometry, point is an undefined term: it is described rather than defined, together with line and plane. A point marks a position and has zero dimensions; Euclid described it as that which has no part. Every other figure, such as a segment, a ray, an angle or a triangle, is a set of points. In the coordinate plane, each point matches exactly one ordered pair(x, y), and in space each point matches an ordered triple (x, y, z).
Symbol
Read it as
Meaning
A
point A
a point is named with one capital letter, written next to its dot
P(3, 2)
point P at 3, 2
point P with x-coordinate 3 and y-coordinate 2
AB
segment A B
all the points between A and B, including the endpoints
Key facts
A point has zero dimensions. A line has one (length), a plane has two (length and width), and space has three.
Through any two different points there is exactly one line.
Points on the same line are collinear. Points in the same flat surface are coplanar.
Order matters in coordinates: (4, 2) and (2, 4) are different points.
Some points have special names: a vertex is a corner, an endpoint ends a segment or ray, and a midpoint cuts a segment into two equal halves.
Try it: move the points
Interactive
Drag P or Q, or select one and use the arrow keys. The readout gives each point’s ordered pair and its quadrant. Try putting a point on an axis or at the origin.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Point examples, step by step
Example 1: Name a point from directions
Warm-up
Start at the origin, move 5 units left and 3 units up. What point are you at?
A line is a straight path of points that goes on forever in both directions. A point is one single location. Two points are enough to fix a line.
Pointvs.Decimal point
A decimal point is the dot in a number such as 3.5 that separates whole units from tenths. It is a symbol for writing numbers, not a location in geometry.
Quick check: 3 questions
Question 1 of 3
How many dimensions does a point have?
Show answer and explanation
Answer: A. A point has no length, width or height, so it has 0 dimensions. A line has 1, a plane has 2, and space has 3.
Question 2 of 3
Which point lies on the y-axis?
Show answer and explanation
Answer: B. Points on the y-axis have x-coordinate 0, so (0, −4) is the one. (5, 0) is on the x-axis, and the other two are inside quadrants.
Question 3 of 3
Points A(−3, 1) and B(4, 1) lie on a horizontal line. How far apart are they?
Show answer and explanation
Answer: C. Same y-coordinate, so subtract the x-coordinates: 4 − (−3) = 7. 1 adds 4 and −3, 0 subtracts the equal y-values, and a distance is never negative.
Point: frequently asked questions
Does a point have a size?
No. A point has no length, width or thickness, so its size is zero. The dot you draw is only a mark that shows where the point is, and a dot drawn bigger is still the same point. This idea lets a segment have a length even though it is made of points: length belongs to the segment, not to any point in it.
Why is a point called an undefined term?
Every definition uses other words, so a few basic words have to come first. In geometry those are point, line and plane. They are described rather than defined: a point is a location with no size. All the other terms, such as segment, ray, angle and polygon, are then defined using these three.
How do you name a point?
Use one capital letter written next to the dot, such as point A. On a coordinate plane, you can also give its coordinates as an ordered pair, with x first: P(3, −2) means point P is 3 units right of the origin and 2 units down. Lowercase letters are usually saved for lines, such as line m.
UndefinedUndefined describes an expression that has no value, most often because it divides by zero. 70 is undefined, since no number times 0 gives 7. A vertical line has an undefined slope, and 3x − 2 is undefined at x = 2. Undefined does not mean zero: 07 = 0, but 70 has no value.Problem Solving · Grade 6, Grade 8, Algebra 1AngleAn angle is formed by two rays that share the same endpoint, called the vertex. It is measured in degrees by how far one ray is turned from the other. A right angle measures 90 degrees, a straight angle measures 180 degrees, and one full turn is 360 degrees.Geometry · Grade 4, Grade 5, Grade 6TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5Coordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7Ordered PairAn ordered pair is two numbers written to locate a point on the coordinate plane. It is written as (x, y): the x-value comes first, and the y-value comes second. From the origin, move x units across, then y units up or down. The order matters: (2, 5) and (5, 2) are different points.Geometry · Grade 5, Grade 6, Grade 7LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3X-AxisThe x-axis is the horizontal number line on the coordinate plane. It crosses the vertical y-axis at the origin, (0, 0). The x-coordinate of a point tells how far to move along the x-axis, and every point on the x-axis has a y-coordinate of 0, such as (5, 0).Geometry · Grade 5, Grade 6, Grade 7Y-AxisThe y-axis is the up-and-down number line on the coordinate plane, with positive numbers above the origin and negative numbers below it. It meets the x-axis at the origin, (0, 0). A point’s y-coordinate is read along the y-axis, and every point on the y-axis has x-coordinate 0, like (0, 4).Geometry · Grade 5, Grade 6, Grade 7OriginThe origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.Geometry · Grade 5, Grade 6, Grade 8VertexA vertex is a corner point where two or more lines, rays, or sides meet. It is the point of an angle, a corner of a polygon such as a triangle, a corner of a solid, and the turning point of a parabola. The plural of vertex is vertices: a square has 4 vertices.Geometry · Grade 4, Grade 5, Grade 6MidpointA midpoint is the point exactly halfway between the two endpoints of a line segment, so it splits the segment into two equal parts. On the coordinate plane, find it by averaging the coordinates: the midpoint of (x1, y1) and (x2, y2) is (x1 + x22, y1 + y22).Geometry · Grade 8, Algebra 1, GeometryQuadrantA quadrant is one of the four regions that the x-axis and y-axis split the coordinate plane into. They are numbered I, II, III and IV counterclockwise, starting at the top right. The signs of a point’s coordinates tell its quadrant: (+, +) is I, (−, +) is II, (−, −) is III and (+, −) is IV.Geometry · Grade 5, Grade 6, Grade 7SlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1AverageAn average is one number that stands for the typical size of the values in a data set. In everyday math, average means the mean: add all the values, then divide by how many values there are. In statistics, the median and the mode are also kinds of average.Statistics · Grade 5, Grade 6, Grade 7NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2LineA line is a perfectly straight path of points that goes on forever in both directions. It has no thickness and no endpoints, so it has no length you can measure. Through any two points there is exactly one line, which is named by those two points, as in line AB, or by one lowercase letter.Geometry · Grade 4, Grade 8, GeometryDecimal PointA decimal point is the dot that separates the whole-number part of a number from its fractional part. Digits to its left are ones, tens and hundreds; digits to its right are tenths, hundredths and thousandths. In 46.25 the point sits between 6 ones and 2 tenths, and you read it aloud as “and”.Number System · Grade 4, Grade 5, Grade 6TenthsTenths are the parts you get when one whole is split into 10 equal parts; each part is one tenth, written 110 or 0.1. The tenths place is the first digit to the right of the decimal point, so in 4.7 the 7 means 7 tenths, or 0.7.Number System · Grade 3, Grade 4, Grade 5PolygonA polygon is a flat, closed shape made of straight line segments, called sides, that meet only at their endpoints. Each corner is a vertex. Polygons are named by their number of sides: a triangle has 3, a quadrilateral 4, a pentagon 5, a hexagon 6 and an octagon 8.Geometry · Grade 2, Grade 3, Grade 5Number LineA number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.Number System · Grade 2, Grade 3, Grade 6Line SegmentA line segment is a straight part of a line with two endpoints. It starts at one point, stops at another, and includes every point between them. A segment with endpoints A and B is written AB. Unlike a line, which goes on forever, a segment has a definite length that you can measure.Geometry · Grade 4, Grade 5, Grade 6RayA ray is part of a line that starts at one endpoint and goes on forever in one direction, like a beam of light from a flashlight. It is named by its endpoint first and then any other point on it, so ray AB starts at A and passes through B. Two rays sharing an endpoint form an angle.Geometry · Grade 4, GeometryPlaneA plane is a perfectly flat surface that has no thickness and extends forever in every direction, like a tabletop that never ends. It is one of the undefined terms of geometry, along with point and line. Any three points that are not on one line lie in exactly one plane.Geometry · Grade 7, Grade 8, Geometry