Enter four 3D points and test coplanarity accurately. See determinant, distance, volume, and plane equation. Download neat exports and learn each calculation step clearly.
| A | B | C | D | Expected Result |
|---|---|---|---|---|
| (0, 0, 0) | (2, 0, 0) | (0, 3, 0) | (1, 1, 0) | Coplanar |
| (0, 0, 0) | (1, 0, 0) | (0, 1, 0) | (1, 1, 2) | Not coplanar |
| (1, 2, 3) | (2, 4, 6) | (3, 6, 9) | (0, 1, 5) | Coplanar because A, B, and C are collinear |
Let A, B, C, and D be four points in 3D space.
Step 1: Build vectors u = B - A, v = C - A, and w = D - A.
Step 2: Compute the scalar triple product T = (u × v) · w.
Step 3: The points are coplanar when |T| ≤ tolerance.
Step 4: The tetrahedron volume is |T| / 6.
Step 5: When u × v ≠ 0, the plane through A, B, and C has normal vector n = u × v.
Step 6: The distance of D from plane ABC is |T| / |n|.
Coplanarity tells you whether several points sit on one flat plane in three-dimensional space. This idea appears in geometry, graphics, surveying, design, robotics, and physics. A reliable coplanarity calculator helps students verify answers fast. It also helps professionals check spatial models before further calculations begin.
The standard method uses the scalar triple product. First, the calculator builds three vectors from the entered coordinates. Next, it finds the cross product of two vectors. Then it takes the dot product with the third vector. A zero result means the enclosed tetrahedron has zero volume. That confirms the four points are coplanar.
Real data often contains rounding noise. Coordinates from measurements or software may not produce a perfect zero. That is why this calculator includes a tolerance field. You can decide how close to zero the scalar triple product must be. This gives more practical answers for homework, modeling, and engineering checks.
This tool does more than classify the points. It shows the scalar triple product, the tetrahedron volume, and the distance of the fourth point from plane ABC. When the first three points define a unique plane, the calculator also returns the plane equation. These extra values make the result easier to understand and explain.
Use this page when studying analytic geometry, revising vector algebra, preparing exam solutions, or checking a 3D dataset. The worked vector outputs help you follow each stage clearly. The example table gives a quick reference. The CSV and PDF exports also make it easy to save results for notes, reports, and classroom practice.
Coplanar points all lie on the same geometric plane. In 3D space, four points are coplanar when one flat surface can pass through all of them.
The scalar triple product measures signed volume. When that volume is zero, the tetrahedron collapses into a plane. That is why it is a standard coplanarity test.
Tolerance handles decimal noise and rounding error. Instead of demanding an exact zero, the calculator checks whether the scalar triple product is close enough to zero.
If A, B, and C are collinear, they do not define one unique plane. Still, the four points remain coplanar because a line and another point can lie on at least one plane.
The tetrahedron volume gives geometric meaning to the determinant result. A zero volume means the four points do not form a solid figure and instead lie in one plane.
Yes. The input fields accept integers and decimals. This makes the calculator useful for classroom examples, measured data, and coordinate values from software tools.
The plane equation helps describe the exact flat surface passing through A, B, and C. It is useful for later distance checks, modeling, and analytic geometry work.
Exports help you keep a clean record of the coordinates and outputs. They are useful for homework files, reports, revision notes, or project documentation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.