Basic Math

Three-dimensional figures | Tenth Grade


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 ShapeFaces (F)Vertices (V)Edges (E)
Cube6812
Cuboid6812
Triangular Prism569
Square Pyramid558
Triangular Pyramid446
Pentagonal Prism71015
Hexagonal Prism81218
Cone211
Cylinder302
Sphere100

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 SolidType of CutCross Section Shape
SphereAny directionCircle
CylinderParallel to baseCircle
CylinderPerpendicular to baseRectangle
ConeParallel to baseCircle
ConePerpendicular through apexTriangle (Isosceles)
ConeDiagonal (not through apex)Ellipse or Parabola
CubeParallel to faceSquare
CubeDiagonalRectangle or Hexagon
PyramidParallel to baseSame shape as base (smaller)
PyramidPerpendicular through apexTriangle

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. 📐

Shares: