three dimensional shapes surface area and volume formulas platonic solids a crash course in:
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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