A crash course in: Three Dimensional Shapes Surface Area and Volume Formulas Platonic Solids

Parts of a polyhedra: faces edges vertices Remember from last time Today, we’re going to talk about specific polyhedra called: Prisms and Pyramids

Bases Lateral Faces FACES can be either

PRISMS PYRAMIDS *have 1 base. *have 2 bases: they are || and  *Lateral faces are ALWAYS triangles *have 2 bases: they are || and  *Lateral faces are ALWAYS rectangles or parallelograms

Naming Prisms and Pyramids They 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

Volume Surface Area The number of cubic units inside a shape. units3 The number of square units on the surface of a shape. units2 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. Pentagon Square Hexagon Triangle Heptagon Dodecagon Octagon Nonagon

Areas of Regular Polygons What part of A=1/2bh is the perpendicular bisector? Can you find the area of a triangle? 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 apothem p is the perimeter of the polygon

Let’s see how this works… 10 A = 1/2ap A = ½(6.88)(50) A = 172 sq.units 6.88 PAINLESS!! Let’s kick it up a notch…

Let’s find the area of this one 12 units …and since we LOVE triangles, let’s start there. 60° 60° 60° How many degrees would the central angle of each Δ have? 60° 60° 60° 30° 60° Think of the center as a circle (360°) and divide 30° 60° 60° Since the Δs are isosceles, what are the measures of the base angles? 60° 30° 60° Since the apothem is an angle bisector, then what is the measure of the small top angle? The short side = 6 The apothem = 6√3 A = 1/2ap A = ½(6√3)(72) = 216√3 (exact) A = 374.12 (approx.) 6 units

The second one is always easier… Find the area of this regular pentagon: Find the central angle Chop it in half Find the base angles Find the apothem Find the area: A = ½ ap 72° 8 units 36° 72° 54° 36° 36° a 5.5 36° 5.5 54° 54° a A = ½ (5.5)(8)(5) A = 110 sq. units 5.5 54° 4 units

Assignment *Shape Identification Activity * Area & Perimeter Wksht #1