Solid GeometryAn exploration of 3-dimentional figures

Modeling the World

The purpose of this exploration is to move beyond my basic knowledge of 3-dimentional shapes, which consists of the classification of solids; the formulas for the calculation of surface area; and some strategies for drawing them on a plane; to gaining a deeper understanding:

• What are the properties and relationships of solids that make them especially interesting not only for problem solving but also for modeling and analyzing real world issues specially in this age of high technology?

• Are there specific properties of solids as useful and powerful as the Pythagoras theorem is to triangles?

• What characteristics of special solids have significant impact in, for instance, the modeling of virtual worlds?

• What are the underlying patterns that facilitate estimation and visual brainstorming?

Big IdeaSolid Geometry concerns itself with the

study of 3-dimentional or space figures. Areas in interest include:

• Classification• Relationship to 2-dimentional shapes• Geometrical Properties• Location (co-ordinate systems)• Visualization, perspective• Drawing 3-d shapes• Calculating area and volume• Problem Solving

The Anatomy of Space Figures

• 3-dimentional figures, or solids have:

length

heig

ht

width

• a flat surface of a solid is called a face• an edge is where two faces meet• a vertex is where three or more edges meet• the face used to classify a solid is the base edge

face

base

vertex

(Bennett et al., 2004)

Polyhedrons• a polyhedron is a three-dimensional figure in

which all the faces are polygons.

• A prism is a polyhedron that has two congruent parallel bases. The bases can be any polygon, the other faces are all parallelograms.

• A pyramid is a polyhedron that has only one base. The base can be any polygon, all other faces are triangles.

(Bennett et al., 2004)

Regular Polyhedrons• A solid is a regular polyhedron (also known as platonic solids) if all its faces

are congruent regular polygons, and the same number of faces meet at each vertex. (Bass et al., 2004)

• The regular polyhedrons are :

• Tetrahedron

• Hexahedron

• Octahedron

• Dodecahedron

• icosahedron

(pictures form wikipedia.org )

Regular Polyhedrons• Regular polyhedrons have a number of characteristics that make them

specially interesting, in particular in terms of how they are manifested in nature and other real world connections:

• All their vertices lie within a sphere, there is a sphere tangent to all faces and one tangent to all edges

• All solid angles are congruent

• Regular polyhedrons are the most symmetrical solids

• Each of the regular polyhedrons occurs in nature, for example in mineral crystals , biology, and chemistry.

Cylinders and cones• a cylinder is typically defined as a solid with two

congruent parallel bases which are circles. Other definitions however, consider ALL solid with two congruent parallel faces a cylinder and classify prisms as special types of cylinders. This more inclusive definition of cylinder is useful in deriving formulas that apply to all types of cylinders and prisms alike. (Van De Valle et al., 2010)

• a cone is typically defined as a solid with one circular base. Other definitions however, consider all prisms as a subset of cones. This more inclusive definition of cone is useful in deriving formulas that apply to all types of cylinders and prisms alike. (Van De Valle et al., 2010)

Sphere

• “A sphere is the set of all point in space equidistant from a given point called the center.• A radius is a segment that has one endpoint at

the center and the other on the sphere• One a plane intersects a sphere at more than

one point, the intersection is a circle. If the center of the circle is also the center of the sphere, the circle is called the great circle of the sphere. A great circle divides the sphere into two hemispheres.” (Bass et al., 2010)

Drawing solidsKnowing how to draw 3-D figures is important to help with visualization,

developing formulas for area and volume calculation, decomposition of more complex shapes and real-world problem solving.

• Nets are used to represent solids in two dimensions

• Euler’s formula is useful in determining the number of faces (F), vertices (V), and edges (E) in a polygon: F+V=E+2

• EXAMPLE: I want to create a play structure with five rectangular sides and two sides that are pentagons. I would like to put a lamp on each vertex, how many lamps do I need?

Drawing solidsIsometric, Orthographic, and foundation drawings are three ways to representthree dimensional figures.

• Isometric: shows three sides of a figure from a corner view

• Orthographic: shows the figure for one side view (top, front, right)

• Foundation: shows the base of a structure and the height

Drawing solids – Online ResourcesThere are many good on-line resources to aid in the creation and manipulation of

3-d images, here are a few:

• http://www.learner.org/interactives/geometry/3d.html

• http://illuminations.nctm.org/activitydetail.aspx?id=70

• http://illuminations.nctm.org/activitydetail.aspx?id=70

Application Examples Virtual worlds, Google Earth

The rapid advance of computer and internet technology has enabled the creation of 3-d representations of our world. Examples of this are the Massive Multi-player Online Role Playing Games (MMORPG) like WarCraft.

Another prominent example is the 3-d views in Google Earth.

Users can create their own 3-d world representation objects for Google Earth or other applications using Google SketchUp

Drawing solids – Cross sections• Cross sections are ‘slices’ of a solid created by the

intersection of a solid with a plane. Because each face of a polyhedron is a plane and the intersection of two planes is a line. Cross sections of polyhedrons can be drawn by making a straight line on each face of the polyhedron according to the geometrical properties of the polyhedron and the angle of the intersecting plane.

• Because cross-sections can be complicated, in middle and high-school math only special cases of cross-sections are considered (cross section parallel or perpendicular to the bases of right cylinders, prisms, cones, pyramids and spheres) (http://online.math.uh.edu/MiddleSchool/Modules/Module_3_Measurement/Content/Solids,Nets,andCrossSections-TEXT.pdf)

(picture from iconmath.com)

Application Example - GPSIn addition to cross sections, intersections between solids have many real-world problem-

solving connections on of my favorites is the use of the intersection of spheres for trilateration in the implementation of Global Positioning Systems.

(picture from iconmath.com)

A GPS receiver calculates the distanceto 4 different satellites. For each satellite, there is a sphere representing all the points in space where the receiver can be the given distance from the satellite. The intersection of the spheres for each of the satellites as well as the sphere representing earth can pin-point the exact location of the GPS receiver.

volume and area of solids The most common application of solid geometry is the calculation of volume

and surface area. Here are the most commonly used formulas for volume:• Volume of cylinders and prisms (note that prisms are sometimes

considered a subset of cylinders and the reason is self-evident in the formulas):• Cylinder: V= pr2h (note that pr2 is simply the cylinder)• Prism: V=Bh

• Volume of pyramids, cones and spheres (note that prisms are sometimes considered a subset of cylinders and the reason is self-evident in the formulas):• Cone: V= 1/3 pr2h • Pyramid: V=1/3 Bh• Sphere: V=4/3 pr2

volume and area of solidsAdditional observations

• Many three- dimensional figures are made of a combination of cylinders, cones, etc. the volume of these solids can be found by calculating the volumes of the composing shapes.

• It can be demonstrated that the volume of a cone or pyramid is one third the volume of a cylinder or prism of the same height and a congruent base.

(Bennett et al., 2004)

Surface area of solids Surface and lateral area of solids can be calculated through standard formulas

for well known shapes or by adding the areas of all the faces and bases.

• Surface area of rectangular prisms: S=2lwx2wh

• Surface area of cylinder: S= 2pr2 +2prh

• Surface area of sphere: S= 4pr2

volume and area of solids Surface and lateral area of solids can be calculated through standard formulas

for well known shapes or by adding the areas of all the faces and bases.

• The lateral area of solids is equal to the sum of the areas of all faces.• The surface area of solids is equal to the sum of the areas of the faces

plus the area of the bases.Example: Assuming that with one gallon of paint I can cover 200 square

feet, how many gallons do I need to paint the outside of this building, (lateral surface), what if want to paint the roof as well?

Building height is 180 ft.

60 ft.

30 ft. 60

ft.

35

ft.

Real World Connections Surface and lateral area of solids can be calculated through standard formulas

for well known shapes or by adding the areas of all the faces and bases.

• The lateral area of solids is equal to the sum of the areas of all faces.• The surface area of solids is equal to the sum of the areas of the faces

plus the area of the bases.Example: Assuming that with one gallon of paint I can cover 200 square

feet, how many gallons do I need to paint the outside of this building, (lateral surface), what if want to paint the roof as well?

Curricular connections

Solid Geometry

Plane Geometry

AlgebraProportional

Reasoning

Numbers and computation

Cross-curricular connections

Solid Geometry

Science• Area and

Volume• Measurement

s• GPS

Arts• Drawing• Perspecti

ve drawing

Social studies• Surface area• Perimeter

• Maps, navigation

Reflection

• This journey of discovery into the world of Solid Geometry has taken me from a basic knowledge of classification of solids together with area/volume formulas to a deeper appreciation and understanding of the meaning and relevance of 3-d geometry.

• One of my goals was to discover ways in which space figures help model the world. I was fascinated to discover, or example that all of the regular polyhedrons appear in nature (my interesting in discovering patterns was also stimulated by this discovery). In addition to natural occurrences of solids, my research led me to several applications that have been enabled by current technological advances and that make extensive use of solid geometry concepts, for example Google Earth, GPS, and virtual on-line worlds.

• From a theoretical perspective, this study increased the depth of my knowledge regarding relationships between 2-D and 3-D shapes, between properties of different solids (e.g. cones and pyramids) and a more though understanding of graphical representation of three dimensional figures.

References

• Bass, L. E., Charles, R. I., Johnson, A., & Kennedy, D. (2004). Geometry. Needham, Ma: Pearson.

• Bennett , J.M., Chard, D. J., Jackson, A., Milgram, J., Scheer, J.K., & Waits, B.K. (2004). Middle School Math. Holt, Reinhart and Winston.

• Van De Valle, J.A., Karp, K. S., & Bay-Williams, J.M. (2010. Elementary & Middle School Mathematics. Pearson Education.

• Indirect references through Wikipedia:– Haeckel, E. (1904). Kunstformen der Natur. Available as Haeckel, E. Art forms in nature, Prestel

USA (1998), ISBN 3-7913-1990-6, or online at http://caliban.mpiz-koeln.mpg.de/~stueber/haeckel/kunstformen/natur.html

– Smith, J. V. (1982). Geometrical And Structural Crystallography. John Wiley and Sons.