two- and three-dimensional figures

Two- and Three-Dimensional Figures

Advanced GeometryInductive Reasoning

Lesson 3

Polygon

Named by: all vertices written in consecutive order

ExamplesNO HOLES

NO CURVES

SIDES CANNOT OVERLAP

Convex Concave

extend the sides“caves in”

any extension crosses inside the figure

all extensions lie outside the figure

extend the sides

# of sides and angles Name of Polygon

3456789

10n

Names of Polygons

trianglequadrilateral

pentagonhexagonheptagonoctagonnonagondecagonn - gon

Regular Polygon

convex polygon

all the sides are congruent

andall angles are congruent

Examples: Name each polygon by its number of sides. Then classify it as convex or concave and regular or irregular.

Pentagonconvexregular

Perimeter and Area of a RectanglePerimeter – the sum

of the lengths of the sides of a

polygon

Area – the number of square units

needed to cover a surfaceA lwADD ALL SIDES

Circumference and Area of a CircleArea

2A r

Circumference – the distance around a circle

2C r

Example:Mr. Smith has a circular fence that encloses an area with a diameter of 12 feet. Using the same fence, he wants to create a square fence. What is the maximum side length of the square?

Example: Find the perimeter of a square with an area of 30 square centimeters.

Example: Find the circumference of a circle with an area of 36 square units.

Example: Find the length of each side of the polygon below if its perimeter is 20 units.

Polyhedron Definition: a solid with all flat surfaces that enclose a

single region of space

• All flat surfaces are called faces.• The line segments where the faces intersect are edges. • The points where the edges intersect are vertices.

PrismsThe two bases are parallel AND congruent.

A regular prism has bases that are regular polygons.

Named by: the shape of the bases

Triangular Prism Rectangular Prism Pentagonal Prism

Pyramidsone base

Named for: the base

Triangular Pyramid

Rectangular Pyramid

Hexagonal Pyramid

Examples: Identify each solid. Name the bases, faces, edges, and vertices.

This picture is on page 62 in your

textbook.

Bases –

Faces –

Edges –

Vertices –

Examples: Identify each solid. Name the bases, faces, edges, and vertices.

This picture is on page 61 in your

textbook.

Bases –

Faces –

Edges –

Vertices –

CylindersThe two bases are congruent, parallel circles.

NOT polyhedra (polyhedrons)

Conesone circular base

Spheres

Examples: Identify each solid. Name the bases, faces, edges, and vertices.

This picture is on page 62 in your

textbook.

Bases –

Faces –

Edges –

Vertices –

Examples: Identify each solid. Name the bases, faces, edges, and vertices.

This picture is on page 61 in your

textbook.

Bases –

Faces –

Edges –

Vertices –

Height vs. Slant Height

Surface Area and Volume

Prisms Pyramids

2SA Ph B

V Bh

12

SA Pl B

13

V Bh

P Perimeter of thebaseB areaof the Basel slant height

Surface Area and Volume

Cylinders Cones22 2SA rh r

2V r h

2SA rl r

213

V r h

Examples: Find the surface area and volume of the square prism.

Examples: Find the surface area and volume of the square pyramid.

This picture is on page 62 in your

textbook.

Examples: Find the surface area and volume of the square pyramid.