Unit 09 notes - Ed W. Clark High School



Geometry Unit 9 – Notes

Surface Area and Volume

Review topics: 1)polygon 2)ratio 3)area formulas 4)scale factor

SURFACE AREA:

Polyhedron – a solid that is bounded by polygons, called faces, that enclose a single region of space. Plural is polyhedra, or polyhedrons.

Syllabus Objective: 10.3 - The student will sketch solid figures.

10.4 – The student will explore relationships among the parts of solid

figures.

10.5 – The student will develop strategies for finding the surface area

of basic solid figures.

Faces – the polygonal regions on a solid bounded by edges.

Edges – a line segment formed by the intersection of two faces of a polyhedron.

Vertices – a point where three or more edges of a polyhedron meet.

Base(s) – one face (or two congruent faces) of a solid that determine the category of a specific solid.

Cross Section – the intersection of a plane and a solid.

The circle on the right is a cross section of the cylinder on the left.

Platonic Solids – Five regular polyhedra, named after the Greek mathematician Plato, including a regular tetrahedron, a cube (regular hexahedron), a regular octahedron, a regular dodecahedron, and a regular icosahedron.

Euler’s Theorem: The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula [pic].

Examples:

a) A polyhedron has 10 vertices and 7 faces. Find the number of edges.

[pic]→ [pic]→ [pic]→ [pic](a pentagonal prism).

b) Is it possible for a polyhedron to have 6 faces, 6 vertices and 6 edges?

[pic]→ [pic]→ [pic] → NO.

Syllabus Objective: 9.2 - The student will solve surface area and volume problems of various geometric solids.

Review concept: Special right triangles.

Many common household items can be used to demonstrate these concepts. Try to collect a few as you go through the year so that you don’t have to depend on the students to bring them in.

Prism – a polyhedron with two congruent faces, called bases, that lie in parallel planes. The other faces, called lateral faces, are parallelograms formed by connecting corresponding vertices of the bases. The segments connecting the vertices are lateral edges. The altitude, or height, of a prism is the perpendicular distance between the bases.

|Cube |Rectangular Prism |Regular Square Prism |Triangular Prism |Regular Pentagonal |Regular Hexagonal |

| | | | |Prism |Prism |

Surface Area – the sum of the areas of the faces and bases. {S}

Net – the two-dimensional representation of all of the faces of a polyhedron. As if the edges were opened and it was laid out flat.

Net of a Regular Pentagonal Prism:

Volume of a solid – the number of cubic units contained in its interior.

Surface Area of a Rectangular Prism: The surface area S of a rectangular prism can be found using the formula S = 2lw+2lh+2wh, where l and w and h are the length, width and height of the prism.

Surface Area of a Right Triangle Prism: S= 2 (area of the triangle) +3 areas of rectangles

How do I find the Surface Area? Rectangular Prism:

1) Identify the length (l), width (w), and height (h).

2) Find the area of each face.

3) Add the area together.

Examples:

a) Find the lateral area and the total surface area of the right triangular prism. Break it into smaller tasks:

1) Identify all of the shapes in the figure.

2) Write the formula for the area of each face.

3) Fill in all values.

4) Use the order of operations for combining values.

*Find the value of x: [pic] → [pic].

Perimeter of base: [pic] Height of prism: [pic]

Rectangle area: (6)(12) =72,

(8)(12)= 96,

(10)(12) = 120.

Area of the (triangular)base: [pic] → [pic] → 24.

Total surface area: [pic]

** notice that the area of the triangle was doubled to account for the top and bottom of the fgure.

b) Find the total surface area of the rectangular prism.

[pic]

c) Find the width of a rectangular solid with length 15 cm, height 8 cm, and lateral area 400 cm2.

LA = Ph and P = 2l + 2w, so

LA = (2l + 2w)h.

400 = [2(15) + 2w]8

50 = 30 + 2w

20 = 2w → w =10. The width is 10 cm.

Cylinder – A solid with congruent circular bases that lie in parallel planes. The altitude, or height, of a cylinder is the perpendicular distance between its bases. The radius of the base is also called the radius of the cylinder.

Surface Area of a Right Cylinder Theorem: The surface area S of a right cylinder is

S = [pic], where r is the radius of a base, and h is the height. {[pic]

Although it looks a little intimidating… this formula only requires 2 pieces of information, the radius and the height.

How do you find the Surface Area of a Cylinder?

1) Identify the radius and height.

2) Fill the values into the formula.

3) Simplify using the order of operations.

Example:

The total surface area of a cylinder is [pic]cm2. If r = h, find r and LA.

[pic] [pic][pic]

Pyramid – A polyhedron in which the base is a polygon and the lateral faces are triangles with a common vertex. The intersection of two lateral faces is a lateral edge. The intersection of the base and a lateral face is a base edge. The altitude, or height, is the perpendicular distance between the base and the vertex.

Slant height – the height of the lateral side of a pyramid or cone. Symbol: l.

It’s nice to have a set of solids, preferably the plastic ones, to demonstrate the different parts of each one.

Plastic is nice because it can written on with overhead or whiteboard markers and erased afterwards.

It’s also nice to have fillable ones to demonstrate the concept of three full pyramids fitting into a prism with the same base & height or three full cones fitting into a cylinder with the same base & height.

Beans or rice are better than sand or water!!!

Sand and water make a MESS!

Surface Area of a Regular Pyramid Theorem: The surface area S of a regular pyramid is the area of the base (whatever shape) added to the area of the number of triangle as faces.

Surface Area Formulas:

Rectangular Pyramid: S= Area of the rectangle base + area of 4 triangles

Square Pyramid: S=Area of square base + area of 4 triangles

Triangular Pyramid: S= Area of triangular base + area of 3 triangles

Trapezoidal Pyramid: S= Area of trapezoidal base + area of 4 triangles.

How do I find the surface area of all Pyramids?

1) Identify the shape of the base. How many triangle faces?

2) Find the area of the base and each triangular face.

3) Simplify using order of operations.

4) Add them all together.

Example: Find the lateral area and total surface area of the regular triangular pyramid.

To find the lateral area:

Perimeter of the base = 36. Which makes it equilateral

since it’s regular. In the base ∆ if 12 is a side then

since it’s equilateral it can be divided into 2 30-60-90

triangle with height of 6√3.

The area of the base is : .5(6√3) (12) = 36√3

In right [pic], [pic], so [pic]

Since each face would be congruent their areas would be the same. Area of each face : [pic]

[pic]

B= 48

To find total surface area:

Area of base + area of 3 triangles

36√3 + 48+48+48

[pic]

**Does the order matter?

Use 30°-60°90° triangle relationship to find the height of the base triangle in the previous example. Short leg is half of 12, or 6, so long leg is [pic]

Cone – A solid with a circular base and a vertex that is not in the same plane as the base. The lateral surface consists of all segments that connect the vertex with points on the edge of the base. The altitude, or height, is the perpendicular distance between the vertex and the plane that contains the base.

Surface Area of a Right Cone Theorem: The surface area S of a right cone is the area of the base and the lateral area. [pic], where r is the radius of the base and l is the slant height. {[pic] and [pic]}

How do I find the Surface area of a Cone?

Remember the cone is made up of a circular base and a lateral face…the cone

1) Identify the radius ( r) and the slant height (l).

2) Fill the values into the formula.

3) Simplify using order of operations.

Examples:

a) Find the lateral area and total surface area of the cone.

Use the Pythagorean Theorem (or a Pythagorean triple) to find r.

[pic]

[pic]

b) A cone has lateral area [pic]and total surface area [pic]. Find its radius.

Since [pic] and [pic],

[pic]

-----------------------

A method for finding the Surface Area of a Right Rectangular Prism:

[pic] (length, width, and height)

8

4

6

r

12

13

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