Three Dimensional Shapes

Surface Area and Volume Formulas

Platonic Solids

A crash course in:

Parts of a polyhedra:facesedges

vertices

Remember from last time

Today, we’re going to talk about specific polyhedra called:

Prisms

and

Pyramids

FACES can be eitherBases

Lateral Faces

PRISMS

*have 2 bases: they are || and

*Lateral faces are ALWAYS rectangles or parallelograms

PYRAMIDS*have 1 base.

*Lateral faces are ALWAYS triangles

Naming Prisms and PyramidsThey have 3 names – just like most of you

First name:

RIGHT – straight up and down – all lateral sides are rectangles

or

OBLIQUE – at least one lateral side is a parallelogram (slanted)

Middle name:

Names the shape of the base:

“triangular”

“rectangular”

“octagonal”

“trapezoidal”

“hexagonal”

Last name:

Names the family:

Prism

Or

Pyramid

Surface Area

The number of

square units on the surface of a shape.

units2

VolumeThe number of

cubic units inside a shape.

units3

You should have a paper that lists all the formulas for surface area and volume for various shapes.

A regular polygon is one where all the sides have the same length and all the angles are the same measure.

TriangleSquare

Heptagon Octagon Nonagon

Pentagon Hexagon

Dodecagon

Areas of Regular Polygons

The perpendicular bisector of a triangle in a polygon is called an APOTHEM.

The formula for the area of a regular polygon is:

A = ½ap

a is the length of the apothemp is the perimeter of the polygon

Can you find the area of a triangle?

What part of A=1/2bh is the perpendicular bisector?

Let’s see how this works…

A = 1/2ap

A = ½(6.88)(50)

A = 172 sq.units

10

6.88

PAINLESS!!

Let’s kick it up a notch…

Let’s find the area of this one…and since we LOVE triangles,

let’s start there.

How many degrees would the central angle of each Δ have?

Since the Δs are isosceles, what are the measures of the base angles?

Since the apothem is an angle bisector, then what is the measure of the small top angle?

60°

60°60°

30°

30°

60°

The short side = 6The apothem = 6√3A = 1/2ap A = ½(6√3)(72) = 216√3 (exact)A = 374.12 (approx.)

Think of the center as a circle (360°) and divide

60°60°60°

60°60°

60°

60°60°

30° 30°

12 units

6 units

The second one is always easier…

Find the area of this regular pentagon:

8 units

1.Find the central angle

2.Chop it in half

3.Find the base angles

4.Find the apothem

5.Find the area: A = ½ ap

72°

54°

36°

72°

54°54°

36°

36° 36°

54°

a

a

4 units

5.5

A = ½ (5.5)(8)(5) A = 110 sq. units

5.5

5.5

Assignment

*Shape Identification Activity

* Area & Perimeter Wksht #1