About

Errors in coordinate geometry propagate. A misread sign on one vertex coordinate shifts an entire polygon's computed area. Manual plotting on graph paper introduces parallax and rounding that compound across multi-step constructions. This calculator renders an interactive Cartesian plane where you place points by click or direct numeric entry, then computes d (Euclidean distance), M (midpoint), m (slope), and polygon area via the Shoelace formula with full decimal precision. All coordinates snap to the resolution you set. The tool assumes a flat Euclidean R² plane with no curvature correction.

Computed line equations are returned in both slope-intercept (y = mx + b) and standard (Ax + By + C = 0) forms. Vertical lines are handled as x = k. Area calculations require at least 3 non-collinear points and use vertex order as entered. Pro tip: for convex polygons, enter vertices sequentially (clockwise or counter-clockwise) to avoid self-intersecting edge artifacts that halve the reported area.

Formulas

The primary distance metric between two points P₁(x₁, y₁) and P₂(x₂, y₂) on the Cartesian plane uses the Euclidean norm derived from the Pythagorean theorem:

d = √(x₂ − x₁)² + (y₂ − y₁)²

The slope m of the line through those two points:

m = y₂ − y₁x₂ − x₁

The area of a simple polygon with n vertices ordered sequentially is computed via the Shoelace (Gauss) formula:

A = 12 |n∑i=1(x_iy_i+1 − x_i+1y_i)|

Where d = Euclidean distance between two points. m = slope of the line (rise over run). A = signed area of the polygon. x_i, y_i = coordinates of the i-th vertex, with indices wrapping so that vertex n + 1 = vertex 1.

Reference Data

Metric	Formula	Minimum Points	Notes
Euclidean Distance	√(x₂ − x₁)² + (y₂ − y₁)²	2	Always ≥ 0
Midpoint	(x₁ + x₂2, y₁ + y₂2)	2	Arithmetic mean of coordinates
Slope	y₂ − y₁x₂ − x₁	2	Undefined if x₁ = x₂
Line Equation (slope-intercept)	y = mx + b	2	Not applicable for vertical lines
Line Equation (standard)	Ax + By + C = 0	2	Integer coefficients when possible
Polygon Area (Shoelace)	12 \|n∑i=1(x_iy_i+1 − x_i+1y_i)\|	3	Vertices must be ordered
Perimeter	n∑i=1 d(P_i, P_i+1)	3	Closed polygon assumed
Centroid	(1nn∑i=1x_i, 1nn∑i=1y_i)	1	Geometric center of point set
Angle Between Segments	arccos(A ⋅ B\|A\| ⋅ \|B\|)	3	Returns 0° - 180°
Manhattan Distance	\|x₂ − x₁\| + \|y₂ − y₁\|	2	L₁ norm (taxicab metric)
Quadrant	Sign of (x, y)	1	I - IV, or on axis
Distance to Origin	√x² + y²	1	Radius in polar form
Polar Coordinates	(r, θ) where r = √x² + y²	1	θ in degrees and radians

Frequently Asked Questions

The Shoelace formula yields zero when all vertices are collinear (they lie on a single line). Three or more points must be non-collinear to enclose a positive area. Check that your points are not all on the same line by verifying that at least one pair of consecutive segments has a different slope.

The Shoelace formula computes a signed area. Counter-clockwise ordering produces a positive value; clockwise ordering produces a negative value. This calculator takes the absolute value, so the reported area is always positive. However, if you enter vertices in a non-sequential order (e.g., jumping across the polygon), the formula treats the path as self-intersecting, which typically underestimates the true area.

The slope formula divides by (x₂ − x₁). When this difference is zero, the slope is undefined. The calculator detects this case and reports the line equation as x = k (a vertical line) instead of the slope-intercept form y = mx + b. Distance and midpoint calculations remain valid regardless.

No. This tool operates strictly in the two-dimensional Euclidean plane (R²). All formulas assume flat geometry with no curvature. For spherical coordinates (e.g., GPS), you need the Haversine formula. For 3D Cartesian space, the distance formula extends to include a z-component: √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²).

Results are displayed to 4 decimal places. JavaScript uses IEEE 754 double-precision floating-point, which provides approximately 15-17 significant digits. For typical coordinate values (−1000 to 1000), rounding errors remain below 10⁻¹⁰. Precision concerns arise only with extremely large coordinates (beyond ±10⁹) or when subtracting nearly equal large numbers.

The angle between two segments sharing a common vertex is computed using the dot product formula, which returns a value between 0° and 180° (the interior angle). It does not distinguish clockwise from counter-clockwise direction. For signed angles, you would need the cross product (atan2), which this tool does not currently report.