Distance & Midpoint Formulas — Complete Reference Sheet A complete reference for distance and midpoint formulas in coordinate geometry. Includes the 2D distance and midpoint formulas, their 3D extensions, partition (section) formulas, distance to a line, and number-line distance.
Distance Formulas d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 3D Distance Formula
d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 + ( z 2 − z 1 ) 2 d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 + ( z 2 − z 1 ) 2 Distance on Number Line
d = ∣ x 2 − x 1 ∣ d = |x_2 - x_1| d = ∣ x 2 − x 1 ∣ Distance from Origin
d = x 2 + y 2 d = \sqrt{x^2 + y^2} d = x 2 + y 2 Distance from Origin (3D)
d = x 2 + y 2 + z 2 d = \sqrt{x^2 + y^2 + z^2} d = x 2 + y 2 + z 2 Midpoint Formulas M = ( x 1 + x 2 2 , y 1 + y 2 2 ) M = \left(\tfrac{x_1 + x_2}{2},\ \tfrac{y_1 + y_2}{2}\right) M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) 3D Midpoint
M = ( x 1 + x 2 2 , y 1 + y 2 2 , z 1 + z 2 2 ) M = \left(\tfrac{x_1 + x_2}{2},\ \tfrac{y_1 + y_2}{2},\ \tfrac{z_1 + z_2}{2}\right) M = ( 2 x 1 + x 2 , 2 y 1 + y 2 , 2 z 1 + z 2 ) Number-Line Midpoint
M = a + b 2 M = \tfrac{a + b}{2} M = 2 a + b Partition (Section) Formulas Internal Division (Ratio m:n)
P = ( m x 2 + n x 1 m + n , m y 2 + n y 1 m + n ) P = \left(\tfrac{m x_2 + n x_1}{m + n},\ \tfrac{m y_2 + n y_1}{m + n}\right) P = ( m + n m x 2 + n x 1 , m + n m y 2 + n y 1 ) External Division (Ratio m:n)
P = ( m x 2 − n x 1 m − n , m y 2 − n y 1 m − n ) P = \left(\tfrac{m x_2 - n x_1}{m - n},\ \tfrac{m y_2 - n y_1}{m - n}\right) P = ( m − n m x 2 − n x 1 , m − n m y 2 − n y 1 ) Centroid of Triangle
G = ( x 1 + x 2 + x 3 3 , y 1 + y 2 + y 3 3 ) G = \left(\tfrac{x_1 + x_2 + x_3}{3},\ \tfrac{y_1 + y_2 + y_3}{3}\right) G = ( 3 x 1 + x 2 + x 3 , 3 y 1 + y 2 + y 3 ) Distance to a Line Point to Line (general form)
d = ∣ A x 0 + B y 0 + C ∣ A 2 + B 2 d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} d = A 2 + B 2 ∣ A x 0 + B y 0 + C ∣ Point to Line (slope-intercept)
d = ∣ m x 0 − y 0 + b ∣ m 2 + 1 d = \frac{|m x_0 - y_0 + b|}{\sqrt{m^2 + 1}} d = m 2 + 1 ∣ m x 0 − y 0 + b ∣ Distance Between Parallel Lines
d = ∣ C 1 − C 2 ∣ A 2 + B 2 d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} d = A 2 + B 2 ∣ C 1 − C 2 ∣ Related Identities & Theorems Pythagorean Theorem (derivation)
a 2 + b 2 = c 2 a^2 + b^2 = c^2 a 2 + b 2 = c 2 Magnitude of a Vector
∣ v ⃗ ∣ = v x 2 + v y 2 |\vec{v}| = \sqrt{v_x^2 + v_y^2} ∣ v ∣ = v x 2 + v y 2 Magnitude of 3D Vector
∣ v ⃗ ∣ = v x 2 + v y 2 + v z 2 |\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2} ∣ v ∣ = v x 2 + v y 2 + v z 2 Circumradius (from sides)
R = a b c 4 ⋅ Area R = \frac{abc}{4 \cdot \text{Area}} R = 4 ⋅ Area ab c Common Applications Diameter of a Circle (from points on circle)
d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 Side Length of a Square (vertices given)
s = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 s = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} s = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 Perimeter of Triangle (3 vertices)
P = d A B + d B C + d C A P = d_{AB} + d_{BC} + d_{CA} P = d A B + d B C + d C A Find Center of a Circle (3 points)
Use perpendicular bisectors of two chords \text{Use perpendicular bisectors of two chords} Use perpendicular bisectors of two chords 