FULL YEAR of High School GEOMETRY
GEOMETRY FORMULA &
REFERENCE PACKET7 pages of formulas, theorems, visuals and tricks! Must-have for students preparing for
end of course exams!
Created by: KoltyMath
GEOMETRY REFERENCE SHEET
Coordinate GeometryDistance Formula/LengthD = x2 − x1
2 + (y2 − y1)2
Midpoint Formula
Partitioning formulaPx = x1 + k(x2 – x1)
Py = y1 + k(y2 – y1)
Ratio a:b
= (x1+x2
2,
y1+y2
2)xm, ym
Finding the endpointPythagorean theorem
can replace the
distance formula.
a2 + b2 = c2
(–4, 2)
(1,–5)
(6,–12)
+5 –7
+5 –7
Visual Method
𝟒
𝟏𝟎=
𝟔
𝟏𝟓=
𝟐
𝟓=
𝐀𝐏
𝐀𝐁
Part
Whole
k
Rati
o AP
:PB
= 2:
3
Slope Formula
?
m =y2 − y1
x2 − x1=
rise
run
m = 2
Types of Slope
Linear Equations
y = mx + b
y − y1 = m(x − x1)
Slope-Intercept Form
Point-Slope Form
Mr.
Slo
pe G
uy
Negative – Positive + Undefined Zer0x = –2 y = 2
b
y = – ½ x + 2
KoltyMath
Angle RelationshipsParallel lines cut by a transversal
acute = acute
obtuse = obtuse
acute + obtuse = 180°
Alternate Interior
Angles Theorem
Consecutive Interior Angles Theorem
“Same Side Interior” Angles Theorem
Angle Types
Complementary = 90°
Supplementary = 180°
TrianglesTriangle Sum Theorem
a + b + c = 180
Exterior Angle Theorema + b = external c
Side-Angle RelationshipsIf a > b > c (sides)
Then A > B > C (angles)
Triangle Possible side lengths
Sum (+) of 2 shorter sides > 3rd side
Triangle Inequality Theorem
Range of Possible Values for a 3rd Side
Sum > x > Difference+ –
Ex: 7, 4, x
11 > x > 3
Triangle Types
60 60
60
Scalene Isosceles EquilateralEquiangular
Right
Angle Classification:
acute, obtuse, right, equiangular
Side Classification:
scalene, isosceles, equilateral
Isosceles Triangle Theorems Hinge Theoremaka SAS Inequality: the measure of the included angle between two pairs of
congruent sides dictates which triangle has the longer third side.
CD > AB
Altitude Theorem
o Bisects Vertex Angle
o Bisects Base Side
KoltyMath
CONGRUENCEASA
Angle-Side-Angle
AAS
Angle-Angle-SideHL
Hypotenuse-Leg
SSS
Side-Side-Side
SAS
Side-Angle-Side
Once the▲s are proven
congruent you can use CPCTC.
Similarity (Match it up!)Triangle similarity proofs
AA~ SAS~ SSS~
Angles must be Congruent and
Sides must be in Proportion
Similarity Tips
✓ Match! Match! Match!
✓ Proportions
Side Splitter TheoremIf a parallel segment intersects 2 sides of a triangle.
U1
U2=
L1
L2
U1
W1=
U2
W2
Similar Polygons
Sca
le F
acto
r Side Lengths a : b
Perimeter a : b
Area a2 : b2 (surface area too)Volume a3 : b3 (similar solids)
Using the appropriate scale factor
allows you to find missing
perimeters/areas/volumes with limited
information by setting up proportions.
a2
b2 =AreaA
AreaB
Ex:
Mean Proportional TheoremsUsed when an altitude is drawn from the
right angle to the hypotenuse of the▲.
Altitude Theorem Leg Theorem
short
altitude=
altitude
long
long
altitude=
altitude
short
Whole
Leg 1=
Leg 1
Part 1
Whole
Leg 2=
Leg 2
Part 2
KoltyMath
TransformationsTranslations: “shift”(x,y) (x + a, y + b)
+– +
–
Reflections: “FLIP”(x,y) ______
Ry-axis (–x,y)
Rx-axis (x,–y)
Ry=x (y,x)
Ry=-x (–y,–x)
You can always use a visual counting method instead of
using rules for reflections.
3
3
Orie
ntat
ion
Chan
ges
afte
r a
Refle
ctio
n
Rotations: “turn”(x,y) ______ (centered at origin)
Clockwise Counterclockwise
– +
R90/-270 (–y,x)
R180/-180 (–x,–y)
R270/-90 (y,–x)
When you connect the pre-image and image to the
center of rotation, you form the degree of rotation.
Remember: you can also perform rotations
centered at the origin by turning your paper.
Dilations: “Grow/Shrink”(x,y) (kx, ky) (centered at origin)
Only Non-Rigid Transformation
Scale Factor (k) =new
oldOnly non-rigid transformation. Dilations enlarge or
shrink your object and form SIMILAR FIGURES.
Image, Pre-image and Center of Dilation are collinear.
To perform dilations not centered at the origin, you can
plot the center and pre-image points and use rise and
run to find the image points. Option 2, use the formula:
x’ = a + k(x – a) y’ = b + k(y – b)
k = 2
Right Triangles
a2 + b2 = c2
Pythagorean Theorem
Converse of the Pythagorean theoremTo determine if the triangle is acute, right or obtuse
from the side lengths only. Center of Dilation: (a,b)Pre-image Point: (x,y)Image Point: (x’,y’)
Notice: when you connect
the pre-image and image
points with a line, the point
of intersection for all three
lines is the center of
dilation (0,-2).
KoltyMath
trigonometrySimplifying radicals
Perfect Squares
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
112 = 121
122 = 144
132 = 169
20
4 5
2 5
Grea
test
Per
fect
Squ
are
= 12 25 2
4 5 ● 3 10
= 12 50
= 12●5 2
= 60 2
Special right trianglesAll 30-60-90 and 45-45-90 are similar and their sides can be found using the side relationships below.
30-60-90 45-45-90Isosceles Right Triangle
Useful with equilateral triangles when an altitude is drawn in.
Useful with squares when a diagonal is drawn in.
TrigonometrySOH CAH TOA
𝐭𝐚𝐧 𝟒𝟎 =𝟏𝟎
𝐱
x =10
tan 40
x = 11.92
𝐬𝐢𝐧Θ =𝟑
𝟓= . 𝟔
Θ = sin−1(.6)
Θ = 36.87°
If you’re solving for the angle (Θ)
Use the INVERSE TRIG Functions-1
Θ = 𝐬𝐢𝐧−𝟏𝐨
𝐡
Θ = 𝐜𝐨𝐬−𝟏(𝐚
𝐡)
Θ = 𝐭𝐚𝐧−𝟏(𝐨
𝐚)
Complementary angles
Sin A = Cos B
Sin B = Cos A
Ex 1: Sin T = Cos (90 – T)
Ex 2: Sin 30 = Cos 60
Special segments & CentersCircumcenter
formed by
perpendicular bisectors
result: AM=BM=CM
Incenter
formed by
angle bisectors
result: XM=YM=ZM
Centroid
formed by Medians
divides median into a 2:1BM = 2(MZ)
Orthocenter
formed by Altitudes2
(x1+x2+x3
3,
y1+y2+y3
3)
KoltyMath
Spatial reasoningArea Formulas (2d Shapes)Circle A = Πr2
C = 2Πr
Triangle A = ½ bh
Parallelogram A = bh
Rectangle A = bh
Rhombus/Kite/Square A = ½ d1 d2
Square A = s2
Trapezoid A = ½ h(b1+b2)
Any Regular Polygon A = ½ aP
a = apothemP = perimeter
n = number of sides
may
nee
d tr
ig
to
sol
ve fo
r bo
th
central angle = 𝟑𝟔𝟎
𝐧
Volume & Surface area formulas
Prism/Cylinder 𝐕 = 𝐁𝐡
Pyramid/Cone 𝐕 =𝟏
𝟑𝐁𝐡
Sphere 𝐕 =𝟒
𝟑Π𝐫𝟑
VOLUME FORMULAS
SURFACE AREA FORMULAS
Prism 𝐋𝐀 = 𝐏𝐡𝐒𝐀 = 𝐏𝐡 + 𝟐𝐁
Cylinder 𝐋𝐀 = 𝟐Π𝐫𝐡𝐒𝐀 = 𝟐Π𝐫𝐡 + 𝟐Π𝐫𝟐
Pyramid 𝐋𝐀 =𝟏
𝟐𝐏𝐥
𝐒𝐀 =𝟏
𝟐𝐏𝐥 + 𝐁
Cone 𝐋𝐀 = Π𝐫𝐥𝐒𝐀 = Π𝐫𝐥 + Π𝐫𝟐
Sphere 𝐒𝐀 = 𝟒Π𝐫𝟐
SA = LA + 2B
l = slant length
h = height
r = radius
B = Area of Base
P = Perimeter
12 in = h
13 in = l
5 in = a
circlesAngles in a circle
Central ∠ Inscribed ∠ Inscribed Right ∠
Intercepted Arc Tangent Radius Supplementary Opposite ∠s
ALTERNATE OPTION
Break the solid into its net
and find the area of each
face separately then add
them all together.
x = ½ (arc 1 + arc 2)
a●b = x●yIntersecting Chords
ext.(whole) = ext.(whole)Secant-Secant Rule
ext.(whole) = tangent2
Secant-Tangent Rule
Segment lengths in a circle
x = ½ (far arc – near arc)
Sector Area & arc length
Sector Area
Πr2 =Central Angle
360
Arc Length
2Πr=
Central Angle
360
Area of Circle
Circumference
Θ
Θ
Θ
Θ
Lateral Area (LA)is the
area of all the surfaces
EXCEPT for the BASE.
KoltyMath
Angles of a polygon
Density 𝐃 =𝐌𝐚𝐬𝐬
𝐕𝐨𝐥𝐮𝐦𝐞
𝐏𝐨𝐩𝐮𝐥𝐚𝐭𝐢𝐨𝐧 𝐃𝐞𝐧𝐬𝐢𝐭𝐲 =𝐏𝐨𝐩𝐮𝐥𝐚𝐭𝐢𝐨𝐧
𝐋𝐚𝐧𝐝 𝐀𝐫𝐞𝐚
Degree & radian conversion
𝐑●180
Π
Radians to Degrees
𝐃●Π
180
Degrees to Radians
D =
Degr
ees
R =
Radi
ans
Equation of a circle(𝐱 − 𝐡)𝟐+(𝐲 − 𝐤)𝟐= 𝐫𝟐
(𝐱 − 𝟑)𝟐+(𝐲 + 𝟓)𝟐= 𝟏𝟒𝟒Ex:
Center: (3,–5) r = 12
Quadrilateral properties
Quadrilateral
360°
1. Opposite sides are parallel (II)
2. Opposite sides are congruent (≌)
3. Opposite angles are congruent (≌)
4. Consecutive angles are supplementary
5. Diagonals bisect each other (same midpoint)
Parallelogram
Rectangle
1. 4 right angles
2. Diagonals are congruent (≌)
Rhombus
1. 4 congruent (≌) sides
2. Diagonals are perpendicular ()
3. Diagonals bisect the angles
Square
Absorbs all properties from the
parallelogram, rectangle, and rhombus
Trapezoid
1. Only 1 pair of parallel (II) sides
2. Same Side Interior Angles are Supplementary
3. Median = ½ (Base 1 + Base 2)
Isosceles Trapezoid
1. Legs are congruent (≌)
2. Base angles are congruent (≌)
3. Diagonals are congruent (≌)
1. 2 Pairs of Consecutive Sides are congruent
2. Diagonal BD bisects ∠B and ∠D
3. Diagonals are perpendicular
4. ≌ opposite angles formed at ∠C and ∠A
KiteB D
A
C
KoltyMath
A = ½ d1d2
A = ½ d1d2
A = ½ d1d2
A = ½ d1d2Area = ½ diagonal 1 * diagonal 2
