Math
Distance from Point to Plane Calculator
Enter a point and a plane in either ax + by + cz = d or ax + by + cz + d = 0 form. mathcheck substitutes into the distance formula, shows the numerator and the radical denominator, rationalizes the answer (for example 7√6/6), and gives the foot of the perpendicular and which side of the plane the point is on.
Switch to the parallel-planes mode to measure the gap between two planes, with a check that they really are parallel.
Right side for = d, or the constant on the left for + d = 0.
Try an example
Result
Distance
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- Distance (decimal)
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- Plane (reduced)
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- Foot of the perpendicular
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- Position relative to the plane
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- Normal vector
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
The point-to-plane distance formula
The shortest path from a point to a plane runs along the plane's normal A quantity with both a size and a direction, often drawn as an arrow.. Projecting the vector from any point of the plane to P onto the unit normal gives the distance, and that projection simplifies to a formula that uses only the plane's The number multiplying a variable, like the 3 in 3x. and the point's coordinates.
From (1, 2, 5) to 2x - y + 2z = 6
Numerator: |2(1) - 2 + 2(5) - 6| = |4| = 4. Denominator: sqrt(4 + 1 + 4) = 3. Distance = 4/3. The foot of the perpendicular is (1, 2, 5) - (4/9)⟨2, -1, 2⟩ = (1/9, 22/9, 37/9).
How to use this calculator
- Choose point-to-plane or parallel-planes mode.
- Enter the plane coefficients, choosing whether your equation has d on the right (= d) or on the left (+ d = 0).
- Enter the point, or the second plane.
- Read the exact distance (rationalized A root expression written with the √ symbol, such as a square root.) and its decimal, plus the foot of the perpendicular and the side of the plane.
- Open Show the work to see the substitution, the radical, and the reduction of the plane equation.
How to read the answer
A distance of 0 means the point lies on the plane. The sign of a·x0 + b·y0 + c·z0 - d before the How far a number sits from zero, so the answer is never negative. tells you which side of the plane the point is on: positive means the normal vector points from the plane toward the point. The foot of the perpendicular is the closest point on the plane. In parallel-planes mode, non-parallel planes are rejected because they meet along a line and have no gap.
Common mistakes and edge cases
- Forgetting the absolute value; the numerator can come out negative but a distance cannot.
- Mixing the two plane forms: in ax + by + cz + D = 0 the constant is added, in ax + by + cz = d it is subtracted.
- Using sqrt(a + b + c) or a^2 + b^2 + c^2 without the square root in the The bottom number of a fraction; it says how many equal parts make one whole..
- Computing the gap between parallel planes before scaling them to the same normal: 2x + 2y + 2z = 4 and x + y + z = 1 are 1/√3 apart, not 3/√12.
- Trying to find the distance between planes that are not parallel. They intersect, so the distance is 0 everywhere along the line.
Worked examples
Textbook example
(1, 2, 5) to 2x - y + 2z = 6
Distance
4/3 (1.3333)
Irrational distance, rationalized
(1, 1, 1) to x + 2y - z + 4 = 0
Distance
√6 (2.4495)
Point on the plane
(1, 0, 0) to x + y + z = 1
Distance
0
Distance from the origin
(0, 0, 0) to 3x - 4y + 12z = 26
Distance
2
Decimal coefficients
(0.5, -1.5, 2) to 0.5x + 0.5y + 0.5z = 1
Distance
√3/3 (0.5774)
Two parallel planes
2x - y + 2z = 6 and 4x - 2y + 4z = 3
Distance
3/2 (1.5)
Same plane twice
x + y + z = 1 and 2x + 2y + 2z = 2
Distance
0
Planes that are not parallel
Distance
—
Not a plane
Distance
—
Frequently asked questions
What is the formula for the distance from a point to a plane?+
For the plane ax + by + cz = d and the point (x0, y0, z0): distance = |a·x0 + b·y0 + c·z0 - d| / sqrt(a^2 + b^2 + c^2). It is the length of the projection of any plane-to-point vector onto the unit normal.
How do I find the closest point on the plane?+
Move from P along the normal: foot = P - t·⟨a, b, c⟩ where t = (a·x0 + b·y0 + c·z0 - d)/(a^2 + b^2 + c^2). The calculator reports this foot of the perpendicular.
How do I find the distance between two parallel planes?+
Write both with the same normal vector, ax + by + cz = d1 and ax + by + cz = d2, then distance = |d1 - d2| / sqrt(a^2 + b^2 + c^2). Equivalently, pick any point on one plane and use the point-to-plane formula with the other.
What does a negative value before the absolute value mean?+
It tells you which side of the plane the point is on. Positive means the normal ⟨a, b, c⟩ points from the plane toward the point; negative means the point is on the other side. The distance itself is never negative.
Why is the answer written as 7√6/6?+
The denominator sqrt(a^2 + b^2 + c^2) is often irrational. Multiplying top and bottom by the radical rationalizes the denominator, which is the form textbooks expect; the decimal is shown beside it.
Does the formula work for a point on the plane?+
Yes: the numerator becomes 0, so the distance is 0. That is also a quick way to check whether a point satisfies the plane equation.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026