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May 30, 2018

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Volumes & Surfaces Areas

Shape Volume Surface Area

Sphere V=4r / 3

RectangularPrism

SA = 2(lw) + (2l + 2w)h

Cylinder

3

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Shape Volume Surface Area

PyramidSA = B + n(1/2sl)

Cube SA = 2(s ) +(4s)s = 6s

Cone

2

2

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Shape Area

Any RegularPolygon

Prism V = Bh

Trapezoid h/2 (b1 + b2)

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Surface Area of Any Prism

(b is the shape of the ends)

Surface Area = Lateral area + Area of twoends

(Lateral area) = (perimeter of shape b) * L

Surface Area = (perimeter of shape b) *L+ 2*(Area of shape b)

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RHS Congruence

- !RHS Right Angle Hypotenuse Side When the right angle and the

hypotenuse and the given side are

equal for a right angle trianglethen we say that the given 2.triangles are congruent

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Example proving RHS congruence

E

DA

FCB

< =< =B E 90 degrees = ( )AC DF hypotenuse = ( )BC EF given side

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Example proving AAA congruence

A

B

P

O

Q

In this figure QA andPB are perpendiculars

.to AB If AO is equal to

, , &10cm BO equal to 6cm,PB equal to 9cm Find

.AQ

Let us consider the

triangles OAQ and OBPcongruent.

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In a parallelogram if one angle A is equal to

110 degrees find the remaining angles?

All sides of a parallelogram have to equal.360 degrees So if Angle A is 110 degrees

then = + + +360 110 B C D- =-110 110 = + +250 B C D =D also s 110

- =360 220 140 & = .So B C 70

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If the diagonals of a parallelogram are

,equal then show it is a rectangle? .h e o r em 1 1 1 I f AB C D i s a p ar al le l o g r a m

h e n i ts no nc on se c u t i v e s i d e s a nd i ts.o n c o n se c ut iv e an g l e s a r e eq ua l

= , =ro o f W e n ee d to p r o v e t h a t A B C D B C.D

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SASSide/Angle/Side

SAS- If 2 sides and the included angleare congruent to 2 sides and the

included angle of a 2nd

triangle, the 2triangles are congruent. And includedangle is an angle created by 2 sides of

a triangle.

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SSSSide/Side/Side

It is a rule that is used in geometry toprove triangles congruent. The rule

states that if 3 sides on 1 triangle arecongruent to 3 sides of a 2nd triangle,the 2 triangles are congruent.

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AAAAngle/Angle/Angle

If in 2 triangles, corresponding anglesare equal, then their corresponding

sides are in the same ratio and hencethe 2 triangles are similar.

b

a

fc

d

e

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ASAAngle/Side/Angle

ASA is a rule used in geometry to provetriangles are congruent. The rule

states that if 2 angles and theincluded side of one triangle arecongruent to 2 angles and the included

side of another triangle, the trianglesare congruent.

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AASAngle/Angle/Side

AAS is used in geometry to provetriangles are congruent. The rules

state that if 2 angles and a non-included side of 1 triangle arecongruent to 2 angles and the

corresponding non-included side ofanother triangle the 2 triangles arecongruent.

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Corresponding Parts of Congruent

Triangle Are Congruent/Equal When 2 triangles are congruent, all 6pairs of corresponding parts {angles &

sides} are congruent. This statementis usually simplified as correspondingparts of congruent triangles are

congruent.

If

then the following conditions are true:

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E q u i v a l e n c er e l a t i o n s

Reflexivity: a~ a*Every triangle is congruent to itself

Symmetry: if a~ b then b~ a

Transitive: if a~ band b~ cthen a~ c.

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123 4

5678

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Supplementary Angles

A pair of angles are supplementary iftheir respective measures sum to

180. If the two supplementary angles are

adjacent (i.e. have a common vertex

and share a side, but do not have anyinterior points in common) their non-shared sides form a line.

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Complementary Angles

A pair of angles are complementary ifthe sum of their angles is 90.

If the two complementary angles areadjacent (i.e. have a common vertexand share a side, but do not have any

interior points in common) their non-shared sides form a right angle.

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Area of Circle= r

Arc length= circumference-2r* Area/360

Arc length=Circumferencemultiplied by r divided by 2r

Area of a sector= A= mAB/360

* r

(

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What =

http://en.wikipedia.org/wiki/Image:Pi-unrolled-720.gif
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SHAPE AREA PERIMETER

Square 2 4s

Rectangle lw +2l 2w

Triangle /bh 2 Add all thesides

Trapezoid( + ) /a b h 2 Ad

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