Endpoint Calculator

The Endpoint Calculator helps you find the missing endpoint of a line segment on a coordinate plane. You can use it when you already know the starting point and either the midpoint or the direction and length of the line. This calculator is useful in many fields such as geometry, computer graphics, engineering, navigation systems, architecture, and game design.

In school-level math problems, finding the second endpoint is a common task when working with midpoints. In real-world applications, developers and designers use this concept to construct geometric paths, simulate object movement, or draw symmetrical shapes. Whether you are solving a problem on paper or writing code for a visual system, this calculator gives you fast and accurate results.

This tool falls under the Mathematics and Coordinate Geometry Calculators category. It supports both 2D and 3D coordinate systems, making it highly flexible for modern uses.

Formula of Endpoint Calculator

The formula depends on the type of input you have.

1. Given Start Point and Midpoint

If you know the starting point (x₁, y₁) and the midpoint (xₘ, yₘ), use this formula to find the missing endpoint (x₂, y₂):

x₂ = 2 × xₘ − x₁
y₂ = 2 × yₘ − y₁

This formula works by mirroring the known point across the midpoint.

2. Given Start Point, Direction (Δx, Δy), and Distance (d)

When you know the direction and the total length of the line, use this formula:

x₂ = x₁ + Δx × (d / √(Δx² + Δy²))
y₂ = y₁ + Δy × (d / √(Δx² + Δy²))

Here, Δx and Δy describe the direction vector from the starting point. The distance d tells you how far the endpoint lies from the start.

This method is useful when working with direction vectors, such as in vector drawing or robotics.

3. For 3D Coordinates

If the problem includes 3D coordinates with an added z-dimension, and you know the midpoint (xₘ, yₘ, zₘ) and the start point (x₁, y₁, z₁), use:

x₂ = 2 × xₘ − x₁
y₂ = 2 × yₘ − y₁
z₂ = 2 × zₘ − z₁

This works exactly like the 2D midpoint formula but includes vertical positioning.

Reference Table for Common Use Cases

This table helps you quickly identify which formula to use depending on your known inputs:

Use this chart as a quick guide to save time during geometry assignments or development tasks.

Example of Endpoint Calculator

Let’s say you know a line starts at (3, 4) and has a midpoint at (5, 7). You want to find the second endpoint.

Apply the midpoint-based formula:

x₂ = 2 × 5 − 3 = 10 − 3 = 7
y₂ = 2 × 7 − 4 = 14 − 4 = 10

So, the other endpoint is (7, 10).

If you're working in 3D and the start point is (2, 3, 4) and midpoint is (5, 6, 7), the second endpoint would be:

x₂ = 2 × 5 − 2 = 8
y₂ = 2 × 6 − 3 = 9
z₂ = 2 × 7 − 4 = 10

So, the endpoint is (8, 9, 10).

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