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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 =
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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 Add all the
sidesParallelogram bh ( + )2 a b
Circle r2 - -no perimeter
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-l length-l length
-b base-b base
-h height-h height
- W width- W width
-a just a side-a just a side
-s side-s side