11-1 Space Figures and Cross Sections

Quick Review

A polyhedron is a three-dimensional figure whose surfaces are polygons. The polygons are faces of the polyhedron.

An edge is a segment that is the intersection of two faces.

A vertex is a point where three or more edges intersect. A cross section is the intersection of a solid and a plane.

Example

How many faces and edges does the polyhedron have?

The polyhedron has 2 triangular bases and 3 rectangular faces for a total of 5 faces.

The 2 triangles have a total of 6 edges. The 3 rectangles have a total of 12 edges. The total number of edges in the polyhedron is one half the total of 18 edges, or 9.

Exercises

Draw a net for each three-dimensional figure.

Use Euler's Formula to find the missing number.

6. F = 5 , f equals 5 comma   V = 5 , v equals 5 comma   E = □ e equals white square
7. F = 6 , f equals 6 comma   V = □ , E = 12 v equals white square comma . e equals 12
8. How many vertices are there in a solid with 4 triangular faces and 1 square base?

9. Describe the cross section in the figure below.

   
10. Sketch a cube with an equilateral triangle cross section.

11-2 Surface Areas of Prisms and Cylinders

Quick Review

The lateral area of a right prism is the product of the perimeter of the base and the height. The lateral area of a right cylinder is the product of the circumference of the base and the height of the cylinder. The surface area of each solid is the sum of the lateral area and the areas of the bases.

Example

What is the surface area of a cylinder with radius 3 m and height 6 m? Leave your answer in terms of π.

S . A . = L . A . + 2 B Use the formula for surface area of a cylinder.   = 2 π r h + 2 ( π r 2 ) Substitute formulas for lateral area and area of a circle.   = 2 π ( 3 ) ( 6 ) + 2 π ( 3 ) 2 Substitute  3  for  r  and  6  for  h .   = 36 π + 18 π Simplify.   = 54 π table with 5 rows and 4 columns , row1 column 1 , cap s . cap a . , column 2 equals , column 3 cap l . cap a . , plus 2 b , column 4 table with 2 rows and 1 column , row1 column 1 , cap use the formula for surface area , row2 column 1 , of a cylinder. , end table , row2 column 1 , , column 2 equals , column 3 2 pi r h , plus 2 , open pi , r squared , close , column 4 table with 2 rows and 1 column , row1 column 1 , cap substitute formulas for lateral , row2 column 1 , area and area of a circle. , end table , row3 column 1 , , column 2 equals , column 3 2 pi open 3 close open 6 close , plus 2 , pi . open 3 , close squared , column 4 cap substitute . 3 , for , r , and , 6 , for , h . , row4 column 1 , , column 2 equals , column 3 36 pi , plus 18 , pi , column 4 cap simplify. , row5 column 1 , , column 2 equals , column 3 54 pi , end table

The surface area of the cylinder is 54 π  m 2 . 54 pi , m squared , .

Exercises

Find the surface area of each figure. Leave your answers in terms of π where applicable.

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15. A cylinder has radius 2.5 cm and lateral area 20 π  cm 2 . 20 pi . cm squared , .  What is the surface area of the cylinder in terms of π?

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