Three-Dimensional Figures
Tenth Grade Geometry – Complete Guide
1. Parts of Three-Dimensional Figures
Key Components
- Face: A flat or curved surface of a 3D shape
- Edge: The line segment where two faces meet
- Vertex (Vertices): A point where three or more edges meet (corner)
Euler’s Formula
[F + V = E + 2]
Where: F = number of faces, V = number of vertices, E = number of edges
Note: Applies to polyhedra (solids with flat faces), not curved surfaces
3D Shape | Faces (F) | Vertices (V) | Edges (E) |
---|---|---|---|
Cube | 6 | 8 | 12 |
Cuboid | 6 | 8 | 12 |
Triangular Prism | 5 | 6 | 9 |
Square Pyramid | 5 | 5 | 8 |
Triangular Pyramid | 4 | 4 | 6 |
Pentagonal Prism | 7 | 10 | 15 |
Hexagonal Prism | 8 | 12 | 18 |
Cone | 2 | 1 | 1 |
Cylinder | 3 | 0 | 2 |
Sphere | 1 | 0 | 0 |
2. Three-Dimensional Figure Vocabulary
Polyhedron
A solid figure where every surface is a polygon (flat faces). Examples: prisms, pyramids, cubes.
Prism
A polyhedron with two parallel, congruent bases connected by rectangular lateral faces. The cross-section parallel to the base is congruent throughout.
- Types: Triangular, Rectangular (Cuboid), Pentagonal, Hexagonal, etc.
- Base: The polygon that defines the prism
- Lateral Faces: Rectangular faces connecting the bases
Pyramid
A polyhedron with a polygon base and triangular lateral faces that meet at a common point called the apex.
- Types: Triangular, Square, Pentagonal, Hexagonal, etc.
- Apex: The top vertex where all triangular faces meet
- Height: Perpendicular distance from apex to base
Curved Surface Solids
- Cylinder: Two parallel circular bases connected by a curved surface
- Cone: One circular base with a curved surface tapering to an apex
- Sphere: All points equidistant from a center point (perfectly round)
- Hemisphere: Half of a sphere
Platonic Solids
Regular polyhedra where all faces are congruent regular polygons. There are exactly 5 Platonic solids:
- Tetrahedron: 4 equilateral triangular faces
- Cube (Hexahedron): 6 square faces
- Octahedron: 8 equilateral triangular faces
- Dodecahedron: 12 pentagonal faces
- Icosahedron: 20 equilateral triangular faces
3. Nets and Drawings of 3D Figures
What is a Net?
A net is a two-dimensional pattern that can be folded to form a three-dimensional figure. It shows all faces of the 3D shape laid out flat.
Using Nets to Find Surface Area
Surface Area = Sum of areas of all faces in the net
Surface Area Formulas
Cube (side length = (a))
Surface Area: [SA = 6a^2]
Volume: [V = a^3]
Cuboid (length = (l), breadth = (b), height = (h))
Total Surface Area: [TSA = 2(lb + bh + hl)]
Lateral Surface Area: [LSA = 2h(l + b)]
Volume: [V = l times b times h]
Cylinder (radius = (r), height = (h))
Curved Surface Area: [CSA = 2pi rh]
Total Surface Area: [TSA = 2pi r(r + h)]
Volume: [V = pi r^2 h]
Cone (radius = (r), height = (h), slant height = (l))
Slant Height: [l = sqrt{r^2 + h^2}]
Curved Surface Area: [CSA = pi rl]
Total Surface Area: [TSA = pi r(l + r)]
Volume: [V = frac{1}{3}pi r^2 h]
Sphere (radius = (r))
Surface Area: [SA = 4pi r^2]
Volume: [V = frac{4}{3}pi r^3]
Hemisphere (radius = (r))
Curved Surface Area: [CSA = 2pi r^2]
Total Surface Area: [TSA = 3pi r^2]
Volume: [V = frac{2}{3}pi r^3]
Frustum of Cone (radii = (r_1, r_2), height = (h), slant height = (l))
Slant Height: [l = sqrt{h^2 + (r_1 – r_2)^2}]
Curved Surface Area: [CSA = pi l(r_1 + r_2)]
Volume: [V = frac{1}{3}pi h(r_1^2 + r_2^2 + r_1 r_2)]
4. Cross Sections of 3D Figures
Definition
A cross-section is the two-dimensional shape formed when a plane intersects (cuts through) a three-dimensional solid. Think of it as a “slice” of the object.
Types of Cross Sections
- Parallel to Base: The cutting plane is in the same direction as the base
- Perpendicular to Base: The cutting plane forms a 90° angle with the base
- Diagonal: The cutting plane is at an angle to the base
Common Cross Sections
3D Solid | Type of Cut | Cross Section Shape |
---|---|---|
Sphere | Any direction | Circle |
Cylinder | Parallel to base | Circle |
Cylinder | Perpendicular to base | Rectangle |
Cone | Parallel to base | Circle |
Cone | Perpendicular through apex | Triangle (Isosceles) |
Cone | Diagonal (not through apex) | Ellipse or Parabola |
Cube | Parallel to face | Square |
Cube | Diagonal | Rectangle or Hexagon |
Pyramid | Parallel to base | Same shape as base (smaller) |
Pyramid | Perpendicular through apex | Triangle |
Key Insight: Cross sections help us understand 3D shapes by examining their 2D slices, which is crucial in calculus, engineering, and medical imaging (CT scans, MRI).
5. Solids of Revolution
Definition
A solid of revolution is a three-dimensional object obtained by rotating a two-dimensional curve or region around an axis (usually the x-axis or y-axis) through 360°.
Common Solids of Revolution
Sphere
Formed by rotating a semicircle about its diameter
Cone
Formed by rotating a right triangle about one of its legs (perpendicular sides)
Cylinder
Formed by rotating a rectangle about one of its sides
Torus (Donut Shape)
Formed by rotating a circle about an axis that doesn’t intersect the circle
Volume of Solid of Revolution
When rotating a function (y = f(x)) from (x = a) to (x = b) about the x-axis:
Disk Method
[V = int_a^b pi y^2 , dx = int_a^b pi [f(x)]^2 , dx]
Washer Method (hollow solid)
When rotating the region between two curves (y = f(x)) and (y = g(x)) where (f(x) geq g(x)):
[V = int_a^b pi left([f(x)]^2 – [g(x)]^2right) , dx]
Rotating about the y-axis
For a function (x = g(y)) from (y = c) to (y = d):
[V = int_c^d pi x^2 , dy = int_c^d pi [g(y)]^2 , dy]
Applications: Solids of revolution are used in calculus to find volumes of complex shapes, and in engineering to design objects like vases, bottles, bells, and other symmetrical objects.
Quick Reference Summary
Euler’s Formula
(F + V = E + 2)
Polyhedron Types
Prisms, Pyramids, Platonic Solids
Net Purpose
2D pattern to find Surface Area
Cross Section
2D slice of a 3D solid
Revolution Volume
(V = int_a^b pi y^2 , dx)
Sphere Volume
(V = frac{4}{3}pi r^3)
📐 Master these concepts for success in Geometry! Practice identifying 3D shapes, drawing nets, and visualizing cross sections. 📐