1. BELLRINGER Complete this assignment: You have 20 minutes.
Choir students: Get the formulas
from your group.

2. 11.1 Space Figures and Cross Sections
A polyhedron is a space figure, or three-dimensional figure, whose surfaces are polygons.
Each polygon is a face of the polyhedron.
An edge is a segment that is formed by the intersection of two faces.
A vertex is a point where three or more edges intersect.

3. Identifying Vertices, Edges, and Faces
How many vertices, edges, and faces are in the polyhedron? List them.
Vertices: D, E, F, G, H
Edges:
Faces: Quadrilateral DEFG,

4. Euler’s Formula
Leonhard Euler – a Swiss Mathematician – discovered relationship among the number of faces, vertices, and edges of any polyhedron.
Known as Euler’s Formula
The sum of the number of faces (F) and the vertices (V) of a polyhedron is two more than the number of its edges (E).
F + V = E + 2

5. Using Euler’s Formula
How many vertices, edges, and faces does the polyhedron have? Use your results to verify Euler’s Formula.
F = 8
V = 12
E = 18
F + V = E + 2 =
20 = 20

6. Cross Section
A cross section is the intersection of a solid and a plane.
Can think of a cross section as a very thin slice of the solid.

7. Describing a Cross Section
What is the cross section formed by the plane and the solid?
The cross section is a rectangle.

8. 11.2 Surface Areas of Prisms and Cylinders
A prism is a polyhedron with two congruent, parallel faces, called bases.
The other faces are lateral faces.
You name prisms based on using the shapes of the bases.

9. Prisms
An altitude of a prism is a perpendicular segment that joins the planes of the bases.
The height, h, of a prism is the length of an altitude.
A prism may either be right or oblique.
Right prism – the lateral faces are rectangles and a lateral edge is an altitude.
Oblique prism – some or all of the lateral faces are nonrectangular.

10. Prisms
Lateral Area (L.A.) of a prism is the sum of the areas of the lateral faces.
Surface Area (S.A.) of a prism is the sum of the lateral area and the area of the two bases.

11. Find the Surface Area of a Prism
What is the surface area of the prism?

12. Lateral and Surface Areas of a Prism
The lateral area of a right prism is the product of the perimeter of the base and the height of the prism.
L.A. = Ph
The surface area of a right prism is the sum of the lateral area and the areas of the two bases.
S.A. = L.A. + 2B

13. Using Formulas to Find Surface Area of a Prism
What is the surface area of the prism?

14. Cylinders
A cylinder is a solid that has two congruent parallel bases that are circles.
An altitude of a cylinder is a perpendicular segment that joins the planes of the bases.
The height, h, of a cylinder is the length of an altitude.
In a right cylinder, the segment joining the centers of the bases is an altitude.
In an oblique cylinder, the segment joining the centers is not perpendicular to the planes containing the bases.

16. Lateral and Surface Areas of a Cylinder
The lateral area of a right cylinder is the product of the circumference of the base and the height of the cylinder.
L.A. = 2πrh or L.A. = πdh
The surface area of a right cylinder is the sum of the lateral area and the areas of the two bases.
S.A. = L.A. + 2B or S.A. = 2πrh + 2πr²

17. Finding Surface Area of a Cylinder
The radius of the base of a cylinder is 4 in. and its height is 6 in. What is the surface area of the cylinder in terms of π?

18. Finding Lateral Area of a Cylinder
You are using a cylindrical stencil roller to paint patterns on your floor. What area does the roller cover in one full turn if it has a diameter of 2.5 in. and a height of 6 in.?

19. 11.3 Surface Areas of Pyramids and Cones
A pyramid is a polyhedron in which one face (the base) can be any polygon and the other faces (the lateral faces) are triangles that meet at a common vertex (called the vertex of the pyramid).

20. Pyramids
The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base.
The length of the altitude is the height of the pyramid.
A regular pyramid is a pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangles.
The slant height is the length of the altitude of a lateral face of the pyramid.

21. Lateral and Surface Areas of a Pyramid

22. Cones
A cone is a solid that has one base and a vertex that is not in the same plane as the base.
The base of a cone is a circle.
In a right cone, the altitude is the perpendicular segment from the vertex to the center of the base.
The height, h, is the length of the altitude.
The slant height, ℓ, is the distance from the vertex to a point on the edge of the base.

23. Surface Area of a Cone