VOLUMES & SURFACES AREAS

Shape Volume Surface Area

Sphere V=4πr / 3

Rectangular Prism

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

Cylinder

3

Shape Volume Surface Area

Pyramid SA = B + n(1/2sl)

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

Cone

2

2

Shape Area

Any Regular Polygon

Prism V = Bh

Trapezoid h/2 (b1 + b2)

Surface Area of Any Prism  (b is the shape of the ends) Surface Area = Lateral area + Area of two

ends (Lateral area) = (perimeter of shape b) * L Surface Area = (perimeter of shape b) * L+

2*(Area of shape b)

       

RHS CONGRUENCE

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

and the given side are equal for a right angle triangle then we say that the given 2 triangles are congruent.

EXAMPLE PROVING RHS CONGRUENCE

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

E

DA

FCB

EXAMPLE PROVING AAA CONGRUENCEA

B

P

O

Q

In this figure QA and PB 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 OBP congruent.<A=<B A<AOQ=<BOP (vertically opposite angles) A<P=<Q (corresponding) A

10 AQ 6 = 990=6AQ

AQ=15

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=- 110250= B + C + D D also =s 110360-220= 140So B & C = 70.

IF THE DIAGONALS OF A PARALLELOGRAM ARE EQUAL, THEN SHOW IT IS A RECTANGLE?

Theorem 11.1 If ABCD is a parallelogram then its nonconsecutive sides and its nonconsecutive angles are equal.

Proof We need to prove that AB = CD, BC = AD.

SASSIDE/ANGLE/SIDE SAS- If 2 sides and the included angle are

congruent to 2 sides and the included angle of a 2nd triangle, the 2 triangles are congruent. And included angle is an angle created by 2 sides of a triangle.

SSSSIDE/SIDE/SIDE It is a rule that is used in geometry to prove

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

AAAANGLE/ANGLE/ANGLE If in 2 triangles, corresponding angles are

equal, then their corresponding sides are in the same ratio and hence the 2 triangles are similar.

b

a

fc

d

e

<a=<d<b=<e<c=<f

ASAANGLE/SIDE/ANGLE ASA is a rule used in geometry to prove

triangles are congruent. The rule states that if 2 angles and the included side of one triangle are congruent to 2 angles and the included side of another triangle, the triangles are congruent.

AASANGLE/ANGLE/SIDE AAS is used in geometry to prove triangles

are congruent. The rules state that if 2 angles and a non-included side of 1 triangle are congruent to 2 angles and the corresponding non-included side of another triangle the 2 triangles are congruent.

CPCTCCORRESPONDING PARTS OF CONGRUENT TRIANGLE ARE CONGRUENT/EQUAL

When 2 triangles are congruent, all 6 pairs of corresponding parts {angles & sides} are congruent. This statement is usually simplified as corresponding parts of congruent triangles are congruent.

If

                         then the following conditions are true:

                                                                                                                

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 ~ b and b ~ c then a ~ c.

1

23 4

567 8

<1, <5

<2,<6

<3,<7

<4,<8

Corresponding angles

<3, <5

<4, <6

Alternate Interior Angles

<1, <7

<2, <8

Alternate Exterior Angles

In geometry, adjacent angles are angles that have a common ray coming out of the vertex going between two other rays.

Ex. Of adjacent Angles

SUPPLEMENTARY ANGLES

A pair of angles are supplementary if their 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 any interior points in common) their non-shared sides form a line.

COMPLEMENTARY ANGLES

A pair of angles are complementary if the sum of their angles is 90°.

If the two complementary angles are adjacent (i.e. have a common vertex and share a side, but do not have any interior points in common) their non-shared sides form a right angle.

Area of Circle= πr²

Arc length= circumference-2πr * Area/360

Arc length=Circumference multiplied by πr² divided by 2πr

Area of a sector= A= mAB/360 * πr²

(

WHAT Π =

SHAPE AREA PERIMETER

Square s 2 4s

Rectangle lw 2l + 2w

Triangle bh/2 Add all the sides

Trapezoid (a+b)h/2 Add all the sides

Parallelogram

bh 2(a+b)

Circle πr2 -no perimeter-

l- length

b- base

h- height

W- width

a- just a side

s- side