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 * 7.1 Surface Area and Volume***

Surface area and Volume Formulas:

surface area= S volume=V length=l width=w height=h**
 * Key:

Volume: **The amount of space or measured in cubic units** Surface Area: **The extent of a 2-dimensional surface enclosed with a boundary**

Rectangular Prisim :**__S=2lw + 2lh__ or __V= lwh__** Example: S=2(5)(5)+2(5)(5) S=50+50 S=100**
 * If length is 5 and width is 5

Cube:**__S= 6s^2__ or __V=s^3__ **


 * 7.2 Surface Area and Volume of Prisms***

Altitude of a prism: **A perpendicular segment from one base to the plane of the other**

Height of a prism: **The distance between the two bases of a prism**

Surface Area of a Right Prism Surface area=S Right prism with lateral area=L Base area=B Perimeter=p Height=h
 * Key:

S=L+2B or S=hp+2B**

Example: Use-S=L+2B S=3+2(4) S=3+8 S=12**
 * Lateral area of a right prism is 3 and the base is 4.

Cavakueru's Principle: includes early work on logarithms and geometry, including the rule known today as Cavalieri's principle. [|**CLICK HERE to Learn more about Cavalieri's Principle**]**
 * //Interesting Fact:// Bonaventura Cavalieri was an Italian mathematician whose

Volume of a Prism: Volume=V prism with height=h base area=B V=Bh**
 * Key:

Example: V=Bh V=(10)(9) V=90**
 * If the base a prism is 10 and the height of the prism is 9.


 * 7.3 Surface Areas and Volume of Pyramids***



Pyramid: **A solid having a polygonal base, and triangular sides that meet in a point** Base: **The line or surface forming the part of a figure that is most nearly horizontal or which it is supposed to stand** Lateral faces: **The face or surface of a solid on its sides. That is, any face or surface that is not a base** Vertex of the pyramid: **The fixed point at the intersectino of all the faces of the pyramid** Base edge: **Common edges within lateral faces** Lateral edge: **where two lateral faces meet** Altitude: **The perpendicular distance from the vertex of a figure to the side oppsite the vertex or the line through the vertex of a figure perpendicular to the base** Height: **Extent or distance upward to a fixed point** Regular Pyramid: **At its base** Slant Height: **Altitude length of lateral faces**

Surface Area of a Regular Pyramid: lateral area=L base area=B perimeter of the base= p slant height= l
 * surface area=S

S=L+B or S=1/2 lp+B**

Example: S=1/2(5)(2)+10 S=5+10 S=15**
 * If the lateral area is 5, perimeter of the base is 2 and base area is 10

Volume of a Pyramid pyramid with height=h base area=B
 * volume=V

C=1/3Bh

[|CLICK HERE to find the volume of your pyramid]**

Example: C=1/3(5)(3) C=4.95 **
 * If the height was 3 and the base was 5

**
 * 7.4 Surface Area and Volume of Cylinders*****

Cylinder: **A surface or solid surrouned by two parallel planes and proceed by a straight line moving parallel to the given planes and tracing a curve bounded by the planes and lying in a plane perpendicular or oblique to the given planes ex. Picture a soup can ** Lateral surface: **The face or surface of a solid on its sides. That is, any face or surface that is not a base** Bases: **Bottem and top circular part of a cylinder** Altitude: **The line through the vertex of a figure perpendicular to the base** Height: **Distance upward from a given level to a fixed point** Axis: **A line about which a 3-dimensional body** Right cylinder: **A cylinder which has bases aligned one directly above the other

**

Oblique cylinder: **A cylinder with bases that are not aligned one directly above the other

** Surface Area of a Right Cylinder: Suface area= S Right cylinder with lateral area= L Base area= B Radius= r Height= h
 * __Key:__

S=L+2B or S=2(3.14)(h)+2(3.14)(r)^2**

Example: 2(3.14)(5)(9)+ 2(3.14)(5)^2 =19964989.9684**
 * If the radius is 5 and height 9

Volume of a Cylinder volume= V cylinder with radius= r height= h base area= B
 * key:

V=Bh or V= 3.14r^2h**

Example: V=3.14(2^2)(1)= V=12.56**
 * If the radius is 2 and the height is 1

Net of a Cylinder-




 * 7.5 Surface Area and Volume of Cones***

Cone- **Is a 3 dimensional with one base** Base- **Is on the bottem of the cone** Lateral surface- **Runs up and down both right and left sides** Vertex- **the point where everything meets** Altitude**- Connecting the vertex to the base with a line running down the middle** Height- **The altitude** Right cone- **A cone that has its apex aligned (or the vertex at the tip of a cone) directly about the center of its base. The base need not be a circle **

Oblique cone- **A cone with an apex ( or the the vertex at the tip of a cone) that is not aligned about the center of the base

** Slant height**- The radius is the sector of the cone**

Surface Area of a Right Cone:

Surface area=S Right cone with lateral area=L Base or area= B Radius= r Slant height= l
 * Key-

S=L+B or S=3.14l+3.14r^2**

Example: S=L+B S=6+6 S=12 ** Volume of a Cone Volume: V Cone with radius: r Height: h base area: B
 * If the lateral area is 6 and the base or area is 6
 * Key:

V=1/3Bh or V=1/33.14r^2h**

Example: V=1/3(12)(4) V=15.84**
 * If the base area is 12 and height is 4

Net of a Cone-


 * Surface Area and Volume of Spheres***

Sphere: **All spheres have the same distance and radius around the circle and there is one center point in the middle **

Annulus: **The region lying between two concentric circles.** Volume of a Sphere Volume=V Sphere with radius= r
 * Key-

V=4/3(3.14)r^3 [|CLICK HERE for help on finding the volume and area of a sphere]**

Surface Area of a Sphere Radius= r S=4(3.14)r^2**
 * Surface area=S

Many 3-dimensional symmetry figure may be reflected across a plane, just as a 2-dimensional figure can be reflected across a line More Help: [|More Help with Area] [|Help/Games with Volume]
 * 7.7 Three- Dimensional Symmetry***

volume= 1/2*length*width*height volume=1/2*2*2*2 volume=1/2*8 volume=4
 * 1. Volume of triangular prism-**

Surface area = bh + (S1+ S2 + S3)H SA= 5*3+ (3+3+3)2 SA= 15+(9)2 SA=24*2 SA=48
 * 2. Surface area of triangular prism-**

V=1/3Bh
 * 3. Volume of pyramid-**

S= L+B or S=1/2lp=B
 * 4. Surface Area of pyramid-**

V= Bh or V=π r2 h
 * 5. Volume Cylinder-**

S=L+ 2B or S= 2 π r h+ 2 π r2
 * 6. Surface area of cylinder-**

V= 1/3Bh or V=1/3 π r2 h
 * 7. Volume of cone-**

S=L+B or S= π r l+ π r2
 * 8. Surface Area of cone-**

V= 4/3 π r3
 * 9. Volume of a sphere-**

S= 4 π r2
 * 10. Surface Area of a sphere-**

__***8.1 DILATIONS AND SCALE FACTORS*** Dilation- The process of expanding

__ _ Similar Figures- If one figure is congruent to another by dilation Polygon Similarity Postulate- The only way two polygons are similar is setting up a correspondence between each side/angle with the following: - Congruent angles are corresponding to each pair -Proportional sides correspond to each pair Properties of Proportions- -Cross-Multiplication Property- a/b=c/d and ad=bc -Reciprocal Property- a/b=c/d and a, b, c, then b/a=d/c -Exchange Property- a/b=c/d and a, b, c, then a/c=b/d -"Add- One" Property-a/b=c/d then a+b/b = c+d/d _ AA (Angle-Angle) Similarity Postulate-If two angles of a triangle are congruent to two angles of another triangle, then the two triangles are similar. SSS( Side-Side-Side) Similarity Theorem-If all three pairs of corresponding sides of two triangles are proportional, then the two triangles are similar. SAS (Side-Angle-Side) Similarity Theorem-If an angle of one triangle is congruent to an angle of another triangle and the corresponding sides that include these angles are proportional, then the triangles are similar. _ Side- Splitting Theorem- A triangle that is divided equally into two sides by a parallel line Two-Transversal Proportionality Corollary-Three or more parallel lines divide two intersecting transversals proportionally.
 * 8.2 SIMILAR POLYGONS***
 * 8.3 TRIANGLE SIMILARITY***
 * 8.4 The Side- Splitting Theorem***

_ Proportional Altitudes Theorem- Two triangles are similar if there corresponding altitudes have the same ratio as their corresponding sides. Proportional Median Theorem- Two triangles are similar if their corresponding median have the same ratio as their corresponding sides. Proportional Angle Bisectors Theorem- Two triangles are similar if corresponding angle bisectors have the same ratio as the corresponding sides Proportional Segments Theorem- An angle bisector of a triangle divides the opposite side into two segments that have same ratio as the other two sides
 * 8.5 INDIRECT MEASUREMENT AND ADDITIONAL SIMILARITY THEOREMS***