A Naturally Occurring Function

Slides:

Advertisements

Similar presentations

The flat surfaces of three-dimensional figures are called faces.

Advertisements

Solid Geometry.

Problem of the Day If the figure shown is folded into a cube so that 6 is on the top, what number would be on the bottom? 2.

3D Figures and Nets.

Three-Dimensional Figure A three-dimensional figure is a shape whose points do not all lie in the same plane.

Polyhedron A polyhedron is simply a three-dimensional solid which consists of a collection of polygons, joined at their edges. A polyhedron is said to.

 Glasses not required!.  A polyhedron is a 3-dimensional, closed object whose surface is made up of polygons.  Common examples: cubes and pyramids.

Surface Area and Volume

Surface Area: Add the area of every side. = ( ½ 10 12) + 2( ½ 18 8) + ( ½ 9 8) = (60) + 2(72) + (36) = = 240 u 2 12 SA = ( ) 18.

Three-Dimensional Figures. Find each missing measure. 1. A = 56 cm 2 2. C = ft 3. A = 72 in 2 r 8 cm x cm x in 15 in 6 in.

Prisms Lesson 11-2.

SOLID FIGURES SPI

Geometric Perspectives. Everything has a name… Face Corner (Vertex) Edges.

Surface Area of Prisms SECTION After completing this lesson, you will be able to say: I can represent three-dimensional figures using nets made.

Holt CA Course Three-Dimensional Figures Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview.

Look at page 193 in your explorations book. Ignore the letters--they are not used for this. Each figure is made up of 5 squares that may or may not be.

Holt CA Course Three-Dimensional Figures Preparation for MG1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area.

Identifying 3-D Figures Lesson 12 – 7. Vocabulary Three Dimensional (3 – D) Figure: Shapes that have a length, width, and depth/height Face – a flat surface.

Week 24 - Vocabulary 3-Dimensional Figures.

Faces, Edges and Vertices- 3D shapes Faces, Edges and Vertices Three dimensional (3D) shapes are defined by the number of faces, edges and vertices.

Fill in the area formulas for the following: Circle ____________________ Rectangle ________________ Triangle __________________.

Learn to identify various three-dimensional figures.

12.1 – Explore Solids.

3-Dimensional Figures. Prisms – Two parallel bases – Named after the shape of its base – All other faces are rectangles Rectangular Prism Triangular Prism.

Space Figures & Nets, Surface Areas of Prisms & Cylinders Unit 5, Lesson 1 chapter%20ten.ppt.

What are these shapes? squarecircletrianglerectangle How many sides do each have? How many points do each have?

Holt CA Course Three-Dimensional Figures Warm Up Warm Up Lesson Presentation California Standards Preview.

Space Figures & Cross-Sections

Nets Nets A net is a pattern that you cut out and fold to form a 3 - dimensional figure.

Solid Figures Vocabulary.

Three Dimensional Figures

Solids: Three –Dimensional figures

Solids: Three – Dimensional figures EQ: How do you identify various three-dimensional figures?

Day 2. Prism and Pyramid In what ways are these shapes alike? In what ways are these shapes different? Distribute set.

Chapter 10 Measurement Section 10.5 Surface Area.

Exploring Solids and Shapes. Basic Definitions Face: A flat surface on a solid figure. Edge: A line segment where two faces meet Vertex: A point where.

PREPARING FOR SURFACE AREA AND VOLUME DRAWINGS, CROSS SECTIONS AND NETS.

Unit 4D:2-3 Dimensional Shapes LT5: I can identify three-dimensional figures. LT6: I can calculate the volume of a cube. LT7: I can calculate the surface.

Solids: Three – Dimensional figures EQ: How do you identify various three-dimensional figures? How do you distinguish between prisms and pyramids? 6.G.4.

12.1 Exploring Solids Hubarth Geometry. The three-dimensional shapes on this page are examples of solid figures, or solids. When a solid is formed by.

The difference between prisms & pyramids.

Three-Dimensional Figures Identify and classify pyramids and prisms by the number of edges, faces, or vertices Identify and classify pyramids and prisms.

P RACTICE AND R EVIEW $8.50 × 4 $36.00 $

Faces, Edges and Vertices

Geometric Solids POLYHEDRONS NON-POLYHEDRONS.

May look at figures in box to give you some ideas. Geometric Solid:

Geometric Solids.

Preview Warm Up California Standards Lesson Presentation.

What do they all have in common?

3-D Shapes Lesson 1Solid Geometry Holt Geometry Texas ©2007

12-1 Properties of Polyhedra

Three Dimensional Figures

Lesson 10.3 Three-Dimensional Figures

Three –Dimensional Figures

What in the World are Nets?

Solid Geometry.

Faces, Edges and Vertices

Three-Dimensional Figures

Three-Dimensional Figures

Solid Geometry.

Objective - To identify solid figures.

I have 4 faces. I have 6 edges. I have 4 vertices.

Solid Geometry.

Faces, Edges and Vertices

– I can find the surface areas of prisms, pyramids, and cylinders

Have Out: Bellwork: 7/11/12 Homework, red pen, pencil, gradesheet

Presentation transcript:

A Naturally Occurring Function Euler’s Formula A Naturally Occurring Function

Leonhard Euler was a brilliant Swiss mathematician Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

Euler discovered an interesting relationship between the number of faces, vertices, and edges for any polyhedron.

Poly-what?

A polyhedron is a 3 dimensional shape with flat sides. A polyhedron is a 3 dimensional shape with flat sides.

All prisms and pyramids are examples of polyhedra (plural for polyhedron). PRISMS PYRAMIDS

Any polyhedron has faces, vertices, and edges. VERTEX

A face is a flat side.

This rectangular prism has 6 faces.

This rectangular prism has 6 faces. FRONT

This rectangular prism has 6 faces. BACK FRONT

This rectangular prism has 6 faces. TOP BACK FRONT

This rectangular prism has 6 faces. TOP BACK FRONT BOTTOM

This rectangular prism has 6 faces. TOP BACK FRONT LEFT BOTTOM

This rectangular prism has 6 faces. TOP BACK FRONT RIGHT LEFT BOTTOM

This rectangular prism has 6 faces. TOP BACK FRONT RIGHT LEFT BOTTOM

This rectangular prism has 6 faces. TOP BACK FRONT RIGHT LEFT BOTTOM

This rectangular prism has 6 faces. TOP LEFT BACK FRONT RIGHT BOTTOM

This rectangular prism has 6 faces. TOP LEFT BACK FRONT RIGHT BOTTOM

This rectangular prism has 6 faces. TOP RIGHT LEFT BACK FRONT BOTTOM

This square pyramid has 5 faces. The faces consist of 4 triangles and a square.

The faces consist of 4 triangles and a square.

A triangular pyramid has 4 faces.

A triangular pyramid has 4 faces.

A triangular pyramid has 4 faces.

A triangular pyramid has 4 faces.

A triangular pyramid has 4 faces.

A triangular pyramid has 4 faces.

An edge is a line segment where two faces meet.

A rectangular prism has 12 edges.

A triangular pyramid has 6 edges.

A vertex is a corner. It is a point that connects 2 or more edges.

A vertex is a fancy word for “corner.” B Every triangle has 3 vertices (corners). Points A, B, and C are vertices.

A rectangular prism has 8 vertices.

A rectangular prism has 8 vertices.

A triangular pyramid has 4 vertices.

A triangular pyramid has 4 vertices.

Euler studied the faces, vertices, and edges of different polyhedra.

Like most great mathematicians and scientists, he organized his data in a chart. Polyhedron # of Faces # of Vertices # of Edges Cube 6 8 12 Sq. Pyramid 5 Tri. Prism 9

Euler looked for a relationship between these numbers. Polyhedron # of Faces # of Vertices # of Edges Cube 6 8 12 Sq. Pyramid 5 Tri. Prism 9

Can you determine Euler’s formula that relates the # of Faces and # of Vertices to the # of Edges? Polyhedron # of Faces # of Vertices # of Edges Cube 6 8 12 Sq. Pyramid 5 Tri. Prism 9

Faces + Vertices –2 = Edges Polyhedron # of Faces # of Vertices # of Edges Cube 6 8 12 Sq. Pyramid 5 Tri. Prism 9 - 2 = + - 2 = + - 2 = +

Use Euler’s Formula to determine the number of edges in a pentagonal prism. Polyhedron # of Faces # of Vertices # of Edges Cube 6 8 12 Sq. Pyramid 5 Tri. Prism 9 Pent. Prism 7 10 - 2 = + - 2 = + - 2 = +

Use Euler’s Formula to determine the number of edges in a pentagonal prism. Polyhedron # of Faces # of Vertices # of Edges Cube 6 8 12 Sq. Pyramid 5 Tri. Prism 9 Pent. Prism 7 10 - 2 = + - 2 = + - 2 = + - 2 = +

Use Euler’s Formula to determine the number of edges in a pentagonal prism. Polyhedron # of Faces # of Vertices # of Edges Cube 6 8 12 Sq. Pyramid 5 Tri. Prism 9 Pent. Prism 7 10 15 - 2 = + - 2 = + - 2 = + - 2 = +

Euler’s Formula works for any polyhedron. SUMMARY: Euler’s Formula says that if you add the number of faces and vertices, then subtract by 2, the result is the number of edges. Euler’s Formula works for any polyhedron.

THE END!