In mathematics, the "distance between two points" refers to the length of the line segment connecting them. Using coordinate geometry, we can calculate this distance by measuring the length of the line between the two points. This calculation applies to points in both two- and three-dimensional spaces, making it a versatile tool across various applications.
In any situation, only one line segment directly connects two points. By measuring this segment’s length, we determine the precise distance between the points.
The formula for calculating the distance between two points is derived from the Pythagorean Theorem. Imagine two points on a graph, represented by coordinates (x₁, y₁) and (x₂, y₂). To calculate the distance between these points, you can create a right-angled triangle, with the two points forming the triangle's corners.
To determine the lengths of the triangle’s sides, simply subtract the x- and y-values of the points. Once you know the lengths of these sides, the Pythagorean Theorem helps find the hypotenuse, which represents the distance between the points.
The formula for the distance (d) is as follows:
$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$
This formula works reliably, allowing you to calculate the distance between any two points on a coordinate plane.
To use this formula accurately, ensure that your x and y values are correctly paired. Here are a few essential steps to follow:
This method provides a precise solution for finding the distance between two points on a 2D plane. Always ensure you include the square root symbol in your calculations to avoid errors.
To illustrate, let’s calculate the distance between two points with coordinates (3, 2) and (7, 8).
The distance between (3, 2) and (7, 8) is approximately 7.21 units.
For points on Earth’s surface, distances are often calculated differently due to the planet’s curvature. Two common formulas for this are the Haversine Formula and Lambert’s Formula.
The Haversine formula calculates the great-circle distance between two points specified by latitude and longitude on a spherical surface. This method gives a close approximation of the shortest distance between two points on Earth, assuming it is a perfect sphere.
Great-Circle Distance: This is the shortest distance between any two points on a sphere's surface and is represented by the largest circle that can be drawn on the sphere.
Lambert's formula refines distance calculation on an ellipsoidal surface, such as Earth, by taking into account the flattening effect at the poles. It is more accurate than the Haversine formula, particularly for long distances, with an error margin of about 10 metres over thousands of kilometres.
Lambert’s formula uses the Earth's equatorial radius, central angle between points, and a flattening factor. However, neither Lambert’s nor the Haversine formula can fully account for Earth’s surface irregularities, so minor inaccuracies remain.
This calculator has numerous practical applications, including:
