Entry Task 1. How many vertices, edges, and faces are in the polyhedron below? List them using the proper notation. 2. Use your answers to part 1 to verify Euler’s Theorem for a polyhedron.

Surface Area 11-2 and 11-3

What is surface area? Surface area of a 3-D object is the sum of the areas of all of its flat surfaces (called faces). Polyhedrons are 3-D shapes where each of the flat surfaces are polygons. The lines where the polygons intersect are called edges.

Prisms are polyhedrons with 2 bases that are both congruent and parallel.

Lateral Faces The lateral faces are the faces that connect the bases. In this figure the lateral faces would be rectangles.

So what’s the formula? Regardless of the type of prism (rectangular, triangular, pentagonal, etc.) just find the area of each of the flat surfaces using the area formulas you have already learned. Then add them together to get the total surface area of the prism.

Example 1 Find the surface area of this rectangular prism. 6 cm 3 cm 4 cm

Example 2 Find the surface area of this triangular prism. 3 cm 4 cm 5 cm 8 cm

Cylinders A cylinder is not a polyhedron. Why not? A cylinder is similar to a prism because it has 2 bases that are congruent and parallel. The bases are the circles. What shape is the lateral part of a cylinder if you were to cut it and lay it flat?

Deriving the Formula A cylinder is made up of the following parts: 2 circles and a rectangle Area of each circle = ∏r 2 Area of the rectangle = 2∏rh h r h base = circumference = 2∏r

Surface Area Formula for Cylinders SA = 2∏r 2 + 2∏rh h r

Example 3 Find the surface area of this cylinder. 10 ft 5 ft

Homework page 704 (8-20 evens, 22, 23, 24, 26, 29, 30, 34, 37)

Pyramids Pyramids have only one base, which can be any type of polygon, not just squares. The other faces all intersect at one point, called the vertex. –What shape are all the lateral faces? The surface area is the sum of the area of the base and the area of the triangular faces.

Deriving the Formula SA = area of the base + lateral area SA= B + ½sℓn = B + ½Pℓ *B = area of the base *s = the base length of the triangle (which is also the side length of the polygon that is the base. *ℓ = slant height *n = the number of sides of the base *P = the perimeter of the base For a regular pyramid:

Example 4 Find the area of this square pyramid.

Cones What shape is the base? What shape is the lateral area if we were to “unroll” it?

Surface Area Formula for Cones SA = πr 2 +πrℓ

Example 5 Find the surface area of the cone. 26 mm 20 mm

Final Thoughts Surface area is the sum of the areas of all flat surfaces of a 3-D shape. The surface area of any prism, cylinder, pyramid, or cone is its lateral surface area plus the area(s) of its base(s).

Homework Page 713 (9-31 all)