GeometryFormula

Coordinate Geometry Formulas

Coordinate geometry uses a grid of (x, y) points to measure shapes with algebra. The three essential formulas — distance, midpoint, and slope — all come from two points, and the single interactive plane below computes every one of them at once as you drag.

One plane, three formulas

M(0, 1.5) A (-3, -1) B (3, 4)

Drag a point, or click it and use the arrow keys. Coordinates snap to the grid.

Distance 7.81
Midpoint (0, 1.5)
Slope 0.83

Step-by-step substitution

  • d = √((x₂−x₁)² + (y₂−y₁)²) = √((3−-3)² + (4−-1)²) = √(36 + 25) = 7.81
  • M = ((x₁+x₂)/2, (y₁+y₂)/2) = ((-3+3)/2, (-1+4)/2) = (0, 1.5)
  • m = (y₂−y₁)/(x₂−x₁) = (4−-1)/(3−-3) = 5/6 = 0.83

The core formulas

Distance

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Full lesson + explorer →

Midpoint

M=(x1+x22, y1+y22)M = \left(\dfrac{x_1 + x_2}{2},\ \dfrac{y_1 + y_2}{2}\right)

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Slope

m=y2−y1x2−x1m = \dfrac{y_2 - y_1}{x_2 - x_1}

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Section formula (ratio m : n)

P=(mx2+nx1m+n, my2+ny1m+n)P = \left(\dfrac{m x_2 + n x_1}{m + n},\ \dfrac{m y_2 + n y_1}{m + n}\right)

The midpoint is the case m = n = 1.

Area of a triangle (shoelace)

A=12 ∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣A = \tfrac{1}{2}\,\bigl|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\bigr|

Area from three vertices’ coordinates.

Worked example: area from coordinates

For the triangle with vertices (0, 0), (4, 0), and (0, 3), the shoelace formula gives A = ½|0(0 − 3) + 4(3 − 0) + 0(0 − 0)| = ½|12| = 6 square units — matching ½ · base · height = ½ · 4 · 3.

Frequently asked questions

What is coordinate geometry?

Coordinate geometry (analytic geometry) studies shapes using a coordinate plane and algebra. Points are (x, y) pairs, and formulas compute distance, midpoint, slope, and area directly from coordinates.

What are the main coordinate geometry formulas?

The core formulas are distance d = √((x₂−x₁)² + (y₂−y₁)²), midpoint M = ((x₁+x₂)/2, (y₁+y₂)/2), and slope m = (y₂−y₁)/(x₂−x₁). The section formula and the shoelace area formula extend them.

What is the section formula?

The section formula finds the point dividing a segment in ratio m : n: ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)). The midpoint is the special case m = n = 1.