Formula Sheets

Distance Formulas for K-12 Students

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Distance formulas are essential mathematical tools that help us measure the space between points, lines, and shapes. These formulas are foundational for geometry, physics, and many real-world applications.

Distance Formulas for K-12 Students

Distance formulas are essential mathematical tools that help us measure the space between points, lines, and shapes. These formulas are foundational for geometry, physics, and many real-world applications.

Below are the key distance formulas that K-12 students should know, organized by category with examples.

Distance Between Points

Distance Between Two Points (2D)
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

Example:

Find the distance between points (2, 3) and (5, 7)

\[ d = \sqrt{(5 - 2)^2 + (7 - 3)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Distance Between Two Points (3D)
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

Where \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) are the coordinates of the two points in 3D space.

Example:

Find the distance between points (1, 2, 3) and (4, 6, 8)

\[ d = \sqrt{(4 - 1)^2 + (6 - 2)^2 + (8 - 3)^2} = \sqrt{9 + 16 + 25} = \sqrt{50} = 5\sqrt{2} \approx 7.07 \]
Midpoint Formula (2D)
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]

Where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

Example:

Find the midpoint between (2, 3) and (8, 11)

\[ M = \left(\frac{2 + 8}{2}, \frac{3 + 11}{2}\right) = \left(\frac{10}{2}, \frac{14}{2}\right) = (5, 7) \]
Midpoint Formula (3D)
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right) \]

Where \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) are the coordinates of the two points in 3D space.

Example:

Find the midpoint between (2, 4, 6) and (8, 10, 12)

\[ M = \left(\frac{2 + 8}{2}, \frac{4 + 10}{2}, \frac{6 + 12}{2}\right) = (5, 7, 9) \]

Distance from Point to Line

Distance from Point to Line (Standard Form)
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

Where the line is \(Ax + By + C = 0\) and the point is \((x_0, y_0)\).

Example:

Find the distance from point (3, 4) to line \(3x - 4y + 8 = 0\)

\[ d = \frac{|3(3) - 4(4) + 8|}{\sqrt{3^2 + (-4)^2}} = \frac{|9 - 16 + 8|}{\sqrt{9 + 16}} = \frac{|1|}{5} = \frac{1}{5} \]
Distance from Point to Line (Slope-Intercept Form)
\[ d = \frac{|y_0 - mx_0 - b|}{\sqrt{1 + m^2}} \]

Where the line is \(y = mx + b\) and the point is \((x_0, y_0)\).

Example:

Find the distance from point (2, 3) to line \(y = 2x + 1\)

\[ d = \frac{|3 - 2(2) - 1|}{\sqrt{1 + 2^2}} = \frac{|3 - 4 - 1|}{\sqrt{5}} = \frac{|-2|}{\sqrt{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5} \]

Distance Between Lines and Planes

Distance Between Parallel Lines
\[ d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \]

Where the lines are \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\).

Example:

Find the distance between parallel lines \(3x + 4y - 10 = 0\) and \(3x + 4y + 5 = 0\)

\[ d = \frac{|-10 - 5|}{\sqrt{3^2 + 4^2}} = \frac{|-15|}{\sqrt{9 + 16}} = \frac{15}{5} = 3 \]
Distance from Point to Plane
\[ d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}} \]

Where the plane is \(Ax + By + Cz + D = 0\) and the point is \((x_0, y_0, z_0)\).

Example:

Find the distance from point (1, -2, 3) to plane \(2x + 3y - z + 4 = 0\)

\[ d = \frac{|2(1) + 3(-2) - (3) + 4|}{\sqrt{2^2 + 3^2 + (-1)^2}} = \frac{|2 - 6 - 3 + 4|}{\sqrt{4 + 9 + 1}} = \frac{|-3|}{\sqrt{14}} = \frac{3}{\sqrt{14}} \]

Special Distance Formulas

Manhattan Distance (Taxicab Geometry)
\[ d = |x_2 - x_1| + |y_2 - y_1| \]

The sum of the absolute differences of their Cartesian coordinates.

Example:

Find the Manhattan distance between points (1, 2) and (4, 6)

\[ d = |4 - 1| + |6 - 2| = 3 + 4 = 7 \]
Pythagorean Theorem
\[ c^2 = a^2 + b^2 \]

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Example:

Find the length of the hypotenuse in a right triangle with sides 3 and 4.

\[ c^2 = 3^2 + 4^2 = 9 + 16 = 25 \] \[ c = \sqrt{25} = 5 \]
Distance-Rate-Time Formula
\[ d = rt \]

Where \(d\) is the distance, \(r\) is the rate (speed), and \(t\) is the time.

Example:

Find the distance traveled by a car moving at 60 mph for 2.5 hours.

\[ d = 60 \text{ mph} \times 2.5 \text{ h} = 150 \text{ miles} \]
Great Circle Distance
\[ d = r \times \arccos(\sin(\phi_1) \sin(\phi_2) + \cos(\phi_1) \cos(\phi_2) \cos(\lambda_2 - \lambda_1)) \]

Where \(r\) is the Earth's radius, and \((\phi_1, \lambda_1)\) and \((\phi_2, \lambda_2)\) are the latitude/longitude coordinates in radians.

Example:

The great circle distance is useful for calculating flight distances or nautical routes on Earth's surface.

Note: Distance formulas are fundamental to many disciplines, from mathematics and physics to geography and computer science. Understanding these concepts helps students measure, analyze, and solve problems in the physical world and abstract mathematical spaces.
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