Distance & Midpoint Calculator
Calculate the exact straight-line Euclidean distance and geometric midpoint between two points in 2D (Cartesian plane) or 3D space.
Point 1 Coordinates (P₁)
Point 2 Coordinates (P₂)
Solved Euclidean Distance (d)
10.0000
Exact Radical Form: 10
Axis-Wise Coordinate Deltas
Step-by-Step Distance Solving
Find Coordinate Deltas:
Δx = 7 - 1 = 6.00
Δy = 10 - 2 = 8.00
Apply Euclidean Distance Equation:
d = √[ (6.00)² + (8.00)² ]
d = √[ 36.00 + 64.00 ]
d = √[ 100.00 ]
Simplify Root & Solve:
Exact: 10
Decimal Approximation: d ≈ 10.00000
Determine Segment Midpoint:
x_mid = (1 + 7) / 2 = 4.00
y_mid = (2 + 10) / 2 = 6.00
Understanding Coordinate Geometry
Calculating distances between points is one of the most fundamental operations in coordinate geometry and vector algebra. Euclidean distance represents the straight-line displacement between two coordinates as measured by a ruler.
The Pythagorean Theorem in Multi-Dimensions
The distance formula is a direct extension of the **Pythagorean Theorem** ($a^2 + b^2 = c^2$). In a 2D plane:
d = √[ (x₂ - x₁)² + (y₂ - y₁)² ]
This creates a right triangle where the horizontal change represents leg $a$ ($\Delta x$), and the vertical change represents leg $b$ ($\Delta y$). The hypotenuse $c$ represents the straight-line distance. In a 3D grid, the formula expands by adding a third dimension ($\Delta z^2$):
d = √[ (x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)² ]
Locating the Geometric Midpoint
The **midpoint** represents the exact middle coordinate of a line segment connecting two points. It is solved by calculating the simple arithmetic average of each coordinate pair. For example, the horizontal midpoint coordinate is $(x_1 + x_2) / 2$.
Frequently Asked Questions
What is the formula for the distance between two points?
The distance formula in 2D is d = sqrt((x2 - x1)^2 + (y2 - y1)^2). In 3D space, it expands to include the z-axis: d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2).
How is the midpoint of a line segment calculated?
The midpoint is computed by taking the average of the coordinates. The formula is M = ((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2).
How do you calculate the distance between two points in a 3D system?
3D distance is calculated by summing the squares of the coordinate differences across all three axes (x, y, z) and then taking the square root of the total sum.
Can this calculate 3D distances?
Yes. Solve Euclidean distances between 2D or 3D coordinate points instantly.