Difficulty: Introductory | Prerequisites: familiarity with 2D coordinate geometry, Pythagorean theorem, basic algebra.
This is the opening topic of Calculus III (multivariable calculus). It extends everything you know about 2D coordinate geometry into three dimensions, adding a z-axis. You need this foundation before you can talk about vectors, surfaces, or any of the later material in the course. If you are comfortable plotting points in the xy-plane and using the 2D distance formula, you have everything you need to start here.
Three-dimensional space uses three mutually perpendicular axes (x, y, z). The distance formula in 3D is a natural extension of the 2D version, and the equation of a sphere is simply the set of all points at a fixed distance (the radius) from a centre point. Most problems in this section come down to computing distances and recognising what geometric object an equation describes.
Three-dimensional coordinate system (R³)
A system of three mutually perpendicular axes (x, y, z) meeting at an origin. Every point in space is described by an ordered triple (x, y, z).
In simple terms, it is the familiar xy-plane with an extra axis poking straight out of the page.
Distance formula in R³
The distance between points P₁ = (x₁, y₁, z₁) and P₂ = (x₂, y₂, z₂) is:
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}Think of it as the Pythagorean theorem applied twice: once in a plane, then once more to account for the third dimension.
Sphere
The set of all points in R³ at a fixed distance r (the radius) from a centre point (a, b, c). Its standard equation is:
(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2In simple terms, a sphere is the 3D version of a circle.
Completing the square
An algebraic technique used to convert a general quadratic expression into standard form. Essential for rewriting expanded sphere equations back into centre-radius form.
Think of it as reverse-engineering an equation so you can read off the centre and radius directly.
Coordinate planes
The three planes formed by pairs of axes: the xy-plane (z = 0), the xz-plane (y = 0), and the yz-plane (x = 0). These divide R³ into eight octants.
Tangent (touching)
Two surfaces are tangent if they meet at exactly one point. For a sphere tangent to a plane, the radius equals the perpendicular distance from the centre to that plane.
A point (x, y, z) is located by moving x units along the x-axis, y units along the y-axis, and z units along the z-axis.
The right-hand rule gives the standard orientation: curl the fingers of your right hand from the positive x-axis toward the positive y-axis, and your thumb points in the positive z-direction.
The coordinate planes (xy, xz, yz) divide space into eight octants. The first octant is where x > 0, y > 0, z > 0.
The distance between two points in R³ follows directly from applying the Pythagorean theorem in three dimensions.
To find the distance from P₁ = (x₁, y₁, z₁) to P₂ = (x₂, y₂, z₂): square the differences in each coordinate, sum them, and take the square root.
Special case: the distance from a point (a, b, c) to a coordinate plane is simply the absolute value of the coordinate perpendicular to that plane. For example, the distance from (a, b, c) to the yz-plane is |a|.
Standard form: (x - a)² + (y - b)² + (z - c)² = r², where (a, b, c) is the centre and r is the radius.
General (expanded) form: x² + y² + z² + Dx + Ey + Fz + G = 0. To extract the centre and radius, complete the square for each variable.
Group the x, y, and z terms separately.
For each group, add and subtract the square of half the linear coefficient.
Rewrite in standard form and read off centre and radius.
Sphere tangent to a plane: the radius equals the perpendicular distance from the centre to the plane.
Tangent to the yz-plane: r = |a| (the absolute value of the x-coordinate of the centre).
Tangent to a plane x = k: r = |a - k|.
Two spheres tangent to each other: the distance between their centres equals the sum of their radii (external tangency) or the absolute difference of their radii (internal tangency).
The set of all points equidistant from two given points lies on a plane. To find its equation, set the distance from the unknown point to each given point equal and simplify.
Recognising geometric regions from inequalities: if an expression can be rewritten as (x - a)² + (y - b)² + (z - c)² ≥ r², the region is the exterior of (or on) a sphere.
d(P_1, P_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2 \quad \text{(sphere, standard form)}\text{Distance from } (a,b,c) \text{ to the plane } x = k \text{ is } |a - k|GPS and 3D modelling software locate objects using three coordinates in exactly this way. Any time you place a pin on a map and add an altitude, you are working in R³. Sphere equations appear in physics (gravitational and electric fields around a point source), computer graphics (bounding spheres for collision detection), and medical imaging (tumour boundaries modelled as spheres).
Students often forget to square the radius when writing the standard equation of a sphere. The right-hand side is r², not r.
When completing the square, students frequently forget to add the same constant to both sides of the equation (or to account for a leading coefficient before completing the square).
The distance from a point to the yz-plane is |x|, not |y| or |z|. Students mix up which coordinate corresponds to which plane.
Two spheres that are tangent do not necessarily have equal radii. Tangency means the distance between centres equals the sum (external) or difference (internal) of radii.
⚠️ Completing the square to convert general-form sphere equations to standard form is a near-certain exam topic. Practise until it is automatic.
⚠️ Recognising what geometric object an equation or inequality describes (sphere, plane, exterior of a sphere) is frequently tested.
⚠️ Problems involving tangency between a sphere and a plane, or between two spheres, require you to compute the correct distance for the radius. Get the perpendicular distance formula down cold.
True or false: the distance from the point (3, -2, 5) to the xz-plane is 5.
Fill in the blank: the equation (x + 1)² + (y - 3)² + (z - 4)² = 9 describes a sphere with centre ______ and radius ______.
True or false: to find whether a sphere is tangent to the yz-plane, you check whether the radius equals the absolute value of the centre's x-coordinate.
Fill in the blank: the set of all points equidistant from two fixed points lies on a ______.
True or false: x² + y² + z² + 2x - 4y + 6z + 14 = 0 represents a sphere with radius 0.
Answers: 1. False (it is |y| = |-2| = 2). 2. (-1, 3, 4), radius 3. 3. True. 4. Plane. 5. True (completing the square gives (x+1)² + (y-2)² + (z+3)² = 0, a degenerate sphere, i.e. a single point).
Q: Find the equation of the sphere with centre (-1, 3, 4) that is tangent to the yz-plane.
A: The distance from (-1, 3, 4) to the yz-plane is |-1| = 1, so r = 1. The equation is (x + 1)² + (y - 3)² + (z - 4)² = 1.
Q: The equation x² + 4x + y² + z² + 6z + 9 = 0 describes a sphere. What are its centre and radius?
A: Complete the square. x² + 4x = (x + 2)² - 4. z² + 6z = (z + 3)² - 9. Substituting: (x + 2)² - 4 + y² + (z + 3)² - 9 + 9 = 0, so (x + 2)² + y² + (z + 3)² = 4. Centre: (-2, 0, -3), radius: 2.
Q: Find the equation for the set of all points equidistant from S = (-1, 2, -5) and P = (2, -3, 4).
A: Set the squared distances equal: (x + 1)² + (y - 2)² + (z + 5)² = (x - 2)² + (y + 3)² + (z - 4)². Expand and simplify to get 6x - 10y + 18z = 6, or equivalently 3x - 5y + 9z = 3. This is a plane (the perpendicular bisector plane of SP).
Q: Describe, in words, the region given by 4x² - 16x + 4y² + 4z² - 20 ≥ 0.
A: Factor out 4 from the x terms and complete the square: 4[(x - 2)² + y² + z²] ≥ 36, so (x - 2)² + y² + z² ≥ 9. This is the set of all points on or outside the sphere of radius 3 centred at (2, 0, 0).
This material feeds directly into §12.2 (Vectors), where distances become magnitudes and points become position vectors. Sphere equations reappear in §12.6 when you study quadric surfaces. The completing-the-square technique will also return when you work with conic sections and level curves of multivariable functions.
3D coordinate system, R³, three-dimensional space, distance formula 3D, equation of a sphere, standard form sphere, completing the square sphere, tangent sphere, perpendicular bisector plane, locus of equidistant points, coordinate planes, octants, Calculus III, MATH 215, multivariable calculus