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Use our angle between two planes calculator to find the angle between two plane equations. Get fast results, a simple formula, and clear steps.
| Term | Meaning |
|---|---|
| θ = arccos(|n₁·n₂| ÷ (|n₁||n₂|)) | Smaller angle between two planes, from their normal vectors |
| n₁ = (A₁,B₁,C₁), n₂ = (A₂,B₂,C₂) | Normal vectors taken directly from the plane equation coefficients |
| D₁, D₂ | Constant terms — do not affect the angle, only used to distinguish parallel vs. coincident planes |
Need to find the angle between two planes? Our angle between two planes calculator makes it easy. Enter the plane values and get your answer in seconds. You don't need to solve each step by hand.
This tool helps students, teachers, and engineers solve geometry problems with ease. It shows the angle and the steps used to find it. You can use it for homework, study, or basic design work.
The angle between two planes depends on their normal vectors. A normal vector points straight out from a plane. Think of it as an arrow that stands at a right angle to a flat floor.
Write the two plane equations in this form:
Plane 1: A₁x + B₁y + C₁z + D₁ = 0
Plane 2: A₂x + B₂y + C₂z + D₂ = 0
The angle between two planes formula is:
θ = cos⁻¹(|A₁A₂ + B₁B₂ + C₁C₂| / [√(A₁² + B₁² + C₁²) × √(A₂² + B₂² + C₂²)])
Here, θ is the smaller angle between the planes. The answer ranges from 0° to 90°. The constant values D₁ and D₂ don't affect the angle.
Follow these five simple steps to get your result.
The calculator does the math for you. It saves time and helps reduce common errors.
Let's find the angle between these two planes:
Plane 1: 2x + y − 2z − 5 = 0
Plane 2: 3x − 6y − 2z − 7 = 0
First, find their normal vectors:
n₁ = (2, 1, −2)
n₂ = (3, −6, −2)
Next, find the dot product:
(2 × 3) + (1 × −6) + (−2 × −2) = 4
Now, find the lengths of both vectors:
|n₁| = √(4 + 1 + 4) = 3
|n₂| = √(9 + 36 + 4) = 7
Put these values into the formula:
θ = cos⁻¹(|4| / (3 × 7))
θ = cos⁻¹(4/21)
Final answer: θ ≈ 79.02°
So, the smaller angle between these planes is about 79 degrees.
Our angle between two planes calculator makes 3D geometry easier. It gives you a quick answer and shows how the formula works. Use it to check your work or learn each step. For the best result, enter the correct plane values.
Find the normal vector of each plane. Then use the dot product formula and take the inverse cosine. Our calculator handles these steps for you.
For Ax + By + Cz + D = 0, the normal vector is (A, B, C). These three values set its direction.
Yes. You can use an online calculator on a phone, tablet, or computer. If the tool supports mobile screens, you don't need to install an app.
The angle between two lines uses their direction vectors. The angle between two planes uses their normal vectors. Both methods use the dot product.
Use the line's direction vector and the plane's normal vector. First, find the angle between the line and the normal. Then subtract that angle from 90° to get the smaller line-to-plane angle.
Find a direction vector by taking the cross product of the two normal vectors. Then solve both plane equations together to find a point on the line. If the planes are parallel and distinct, they have no line of intersection.
A normal plane calculator may help with plane normals or related geometry tasks. For the angle between two planes, use each plane's normal vector in the formula above.