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