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
(
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
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