Three-Dimensional

Geometry
Kalam Khan
A plane is a flat surface with no
thickness that extends forever.
Our world has three dimensions, but a plane
has only two dimensions:
1. length and width make a plane, or
2. horizontal and vertical make a plane, or
3. x and y also make a plane

In Geometry we can have different dimensions:

A Point has no dimensions, only position
A Line is one-dimensional
A Plane is two dimensional (2D)
A Solid is three-dimensional (3D)
We live in a 3D world, but we often work
with 2D spaces, like triangles, circles,
squares, etc
In three dimensions, an area which extends forever in every direction is called a space
 Understanding Skew Lines, Planes, and 3D Figures
1. Lines in Space
Parallel Lines: Same direction, never intersect.
Skew Lines: Different directions, never
2. Planes in 3D
A plane is an infinite 2D flat surface (like a sheet of paper)
Line-Plane Interaction:
1. Lies in the plane.
2. Intersects at one point.
3. Parallel (no intersection).
3. Defining a Plane
•3 non-collinear points determine a plane (e.g., a triangle).
•4+ points may not be coplanar (e.g., triangle + point lifted above it).
4. Surface Area
•Total Surface Area (TSA): Cloth covering all faces (e.g., sum of a box’s sides).
•Lateral Surface Area (LSA): Excludes top/bottom (e.g., sides of a cylinder).

Shape TSA Formula Volume Formula

Cube 6a2 a3

Cylinder 2πr(r+h) πr2h

Sphere 4πr2 4/3πr3

Volume: Space enclosed
 Spheres
A sphere is the 3D analog of a circle.
Set of all points in space at a fixed distance (radius)
from a center point.
Tangent Plane: Intersects at 1 point (e.g., a finger touching a ball). Real-World Examples
Great Circle: Plane through the center → largest possible circle •Great Circles: Longitude lines on Earth.
(e.g., Earth’s equator). •Surface Area: Calculating paint needed for a ball.
•Volume: Measuring liquid in a spherical tank.
Cubes and Boxes
Basic Properties
•A cube is a 3D shape with:
• 6 faces: All identical squares.
• 12 edges: All equal in length (ss).
• 8 vertices: Corners where edges meet. •Surface Area = Sum of all faces = 6s2.
•Adjacent faces are perpendicular to each other. •Volume = Base area × Height = s3
 parallelepiped
• Parallelepiped is a 3-D shape whose faces
are all parallelograms
• it is formed by six parallelogram sides to
result in a three-dimensional figure or a Prism
 Prisms and Cylinders •A cylinder is a prism with circular bases.
•Right Circular Cylinder: Lateral surface is
•A prism is a 3D shape with:
perpendicular to the bases (e.g., a soda can).
• Two parallel, congruent bases (any •Axis: The line connecting the centers of the
polygon). two circular bases.
• Lateral faces = parallelograms formed
by connecting corresponding vertices
of the bases.
Types:
Key Formulas:
•Regular Prism: Bases are regular polygons (e.g., regular hexagonal
•Volume = Base Area × Height = πr2h
prism).
•Lateral Surface Area = Unrolled rectangle = 2πrh.
•Right Prism: Lateral edges are perpendicular to the bases (e.g.,
•Total Surface Area = Lateral Area + 2(Base Area) = 2πrh+2πr2
cubes, rectangular boxes).
Key Formulas:
•Volume = Base Area × Height.
•Total Surface Area = Sum of all faces (bases + lateral faces).
•Lateral Surface Area = Sum of only the lateral faces (excludes bases)
•A pyramid is a 3D shape with:
•A cone is a pyramid-like shape with a circular
• One polygonal base (can be any polygon,
base and a single vertex.
including triangles).
• Triangular lateral faces meeting at a •Right Circular Cone: Vertex is directly above the
common vertex. center of the base.
• Altitude (h): Perpendicular distance from •Slant Height (l): Distance from vertex to edge of
the vertex to the base. the base.

Advanced Insight:
 Cone
A cone is a pyramid with a circular base.
•Right Circular Cone:
• Vertex (apex) is directly above the center of the base.
• Slant height (l): Distance from apex to any point on the base edge.
• Pythagorean Relation: h2+r2=l2
Using this "unrolling" approach,
we can prove that the lateral
surface area of a cone is πrl.
 Polyhedrons & Platonic Solids
Polyhedron Basics
•Definition: A 3D shape with flat polygonal faces, straight edges, and
sharp vertices.
• No curved surfaces (e.g., cubes , spheres ).
• Examples: Cubes, pyramids, prisms, dodecahedrons.

Regular Polyhedron (Platonic Solids)

•Requirements:
• Congruent Faces: All faces are identical regular polygons
(equilateral triangles, squares, etc.).
• Uniform Vertices: Same number of faces meet at every
vertex.
•Only 5 Exist due to geometric constraints (see table below).
Key Observations:
•Euler’s Formula: Holds for all polyhedrons (regular or not): V+F−E=2
•Dual Pairs:
• Cube Octahedron.
• Dodecahedron Icosahedron. Beyond Platonic Solids
• Tetrahedron is self-dual. •Archimedean Solids: Semi-regular polyhedrons with mixed
Why Only 5 Platonic Solids? regular polygons (e.g., soccer ball: hexagons + pentagons).
•Angle Sum Constraint: The sum of face angles at each vertex must be < 360°. •Johnson Solids: Convex polyhedrons with regular faces but non-
uniform vertices (e.g., square pyramid).
How to Solve 3D Problems 4. Leverage Perpendicular Lines
•Why?: Right triangles simplify calculations.
Break Down Complex Figures • Example: To find the height of a pyramid, use
•Divide and Conquer: Split the 3D object into simpler shapes (cubes, the altitude (perpendicular from apex to base) and base
pyramids, prisms, etc.) whose surface area or volume you know. dimensions.
• Example: An octahedron can be split into two square
pyramids. 5. Strategic Auxiliary Lines
• Formula: Total Volume = Sum of parts’ volumes. • Add Lines: Draw diagonals, midsegments, or perpendiculars to
create solvable triangles.
• Use Subtraction for Hollow/Composite Shapes
• Larger Object – Unwanted Parts: • Example: In a prism, draw a diagonal to split a trapezoidal
• Example: Find the volume of a drilled hole in a cylinder face into two triangles.
→ Volume of cylinder – Volume of the hole 6. Real-World Analogies
•Visualization: Imagine unfolding a 3D shape (e.g., a cylinder’s lateral
Reduce 3D Problems to 2D surface → rectangle).
•Key Insight: Most 3D length/angle/area problems are 2D problems • Example: A cone’s slant height problem → Unroll to a
in disguise. sector of a circle.
• Step 1: Identify the plane or triangle containing the unknown.
• Step 2: Apply 2D geometry (Pythagoras, trigonometry, area
formulas).
• Example: Find the diagonal of a cube → Focus on a right
triangle spanning the cube’s edge, face diagonal, and space
diagonal.