Geometry · Units 2, 11–12 · Quick reference

GeometryPART 1

Grades 8–124 pages

Formulas for perimeter, area, volume and surface area, the Pythagorean theorem, and the terms the rest of geometry stands on: points, lines, planes, space, shapes, segments, rays and angles.

1Flat figures2Solids3Pythagoras4Terms & shapes5Lines & planes6Segments & rays7Angles

1Perimeter & Area: Flat FiguresP = distance around · A = square units inside

Three measures, three kinds of units

Perimeter (P): the length of the boundary. Units: in, cm, ft.
Circumference (C): the perimeter of a circle.
Area (A): how many unit squares cover the inside. Units: in², cm², ft².
Always match units first: 2 ft × 18 in → 24 in × 18 in.

Square

P = 4s

A = s²

EXs = 6 cm → P = 24 cm; A = 36 cm²

Rectangle

P = 2l + 2w

A = lw

EXl = 9, w = 4 → P = 26; A = 36 units²

Triangle

P = a + b + c

A = 12bh

EXb = 10, h = 7 → A = ½ · 10 · 7 = 35

Height is always ⊥ to the base

The height can fall outside an obtuse triangle; extend the base with a dashed line and measure straight up to the top vertex. A slanted side is not the height.

Parallelogram

P = 2a + 2b

A = bh

EXb = 12, h = 5, slant side 6 → A = 60; P = 36

Trapezoid

A = 12(b₁ + b₂)h

P = sum of 4 sides

EXb₁ = 6, b₂ = 10, h = 4 → A = ½ · 16 · 4 = 32

Rhombus · kite

A = 12d₁d₂

P (rhombus) = 4s

EXd₁ = 8, d₂ = 6 → A = 24; each side = √(4² + 3²) = 5, so P = 20

Circle

d = 2r

C = 2πr = πd

A = πr²

EXr = 5 → C = 10π ≈ 31.4; A = 25π ≈ 78.5

Equilateral triangle

h = √32s

A = √34s²

EXs = 6 → h = 3√3 ≈ 5.2; A = 9√3 ≈ 15.6

Arcs & sectors (a slice of the circle)

arc = θ360 · 2πr

sector = θ360 · πr²

EXr = 6, θ = 60° → arc = 2π ≈ 6.28; sector = 6π ≈ 18.8

Radius: center to the circle
Diameter: chord through the center (2r)
Chord: segment joining two points of the circle
Arc: a piece of the circle; a semicircle is 180°
Sector: region between two radii and their arc

Regular polygon (n equal sides, apothem a)

P = ns

A = 12aP

EXhexagon, s = 4, a = 2√3 ≈ 3.46 → P = 24; A = ½ · 3.46 · 24 ≈ 41.6

Fence or carpet?

Perimeter is the fence around the yard (a length). Area is the carpet that covers the floor (square units). Ask “am I walking around it or covering it?”

Common slips

Using the diameter where the formula wants the radius (halve it first).
Taking a slanted side as the height.
Mixing units (feet with inches) or forgetting the ² on area.
Rounding π too early: keep π to the end, then round once.

Pie for circles

“Cherry pie’s delicious” → C = πd

“Apple pies r 2” → A = πr²

Area answers always wear a square hat (units²).

Triangle by three sides (Heron)

s = a + b + c2, A = √(s(s − a)(s − b)(s − c))

EX13, 14, 15 → s = 21; A = √(21 · 8 · 7 · 6) = √7056 = 84

Composite figures: add or subtract Worked

10 × 6 rectangle; a 2 × 2 square cut from one corner
A = 60 − 4 = 56 units²subtract the piece
P = 32 units (unchanged)the two new edges replace two lost ones
Circle of r = 3 removed from a 10 × 10 square: A = 100 − 9π ≈ 71.7

GeometryGeometry · Units 2, 11–12 · continued

2Volume & Surface Area: SolidsV = cubes inside · SA = total area of every face

Two measures for solids

Volume (V): unit cubes that fill it. Units: in³, cm³, ft³.
Surface area (SA): add the areas of all faces (the “net”). Units²
Lateral area (LA): sides only, no bases.
B = area of a base; h = height ⊥ to the base; ℓ = slant height.

Cube

V = s³

SA = 6s²

EXs = 4 → V = 64; SA = 6 · 16 = 96

Rectangular prism

V = lwh

SA = 2(lw + lh + wh)

EX5 × 3 × 2 → V = 30; SA = 2(15 + 10 + 6) = 62

Scale factor k

Double every length (k = 2): perimeter ×2, area ×4, volume ×8. In general: lengths ×k, area ×k², volume ×k³.

Euler: F + V − E = 2

Cube: 6 + 8 − 12 = 2 (any convex polyhedron)

Any prism

V = Bh

SA = 2B + Ph

(P = perimeter of base)

EXbase = right △ 3-4-5, length 10 → B = 6; V = 60; SA = 12 + 12 · 10 = 132

Cylinder

V = πr²h

SA = 2πr² + 2πrh

EXr = 3, h = 5 → V = 45π ≈ 141.4; SA = 18π + 30π = 48π ≈ 150.8

Cone

V = 13πr²h

SA = πr² + πrℓ

ℓ = √(r² + h²)

EXr = 3, h = 4 → ℓ = 5; V = 12π ≈ 37.7; SA = 9π + 15π = 24π ≈ 75.4

Composite solid Worked

cylinder r = 3, h = 10, topped by a hemisphere
V = π · 9 · 10 + 23π · 27 = 90π + 18πhalf of 43πr³
= 108π ≈ 339.3

Pyramid

V = 13Bh

SA = B + 12Pℓ

EXsquare base 6, h = 4 → ℓ = 5; V = ⅓ · 36 · 4 = 48; SA = 36 + ½ · 24 · 5 = 96

Sphere

V = 43πr³

SA = 4πr²

EXr = 3 → V = 36π ≈ 113.1; SA = 36π ≈ 113.1 (equal only at r = 3)

Pointy solids pay one-third

A cone or pyramid holds exactly ⅓ of the cylinder or prism with the same base and height. Sharpen it to a point → divide by 3.

Prism and cylinder: V = Bh (“Base times height”) for both.

Converting square & cubic units

Square the length factor for area, cube it for volume.
1 ft = 12 in → 1 ft² = 144 in²; 1 ft³ = 1,728 in³
1 yd = 3 ft → 1 yd² = 9 ft²; 1 yd³ = 27 ft³
1 m = 100 cm → 1 m² = 10,000 cm²; 1 m³ = 1,000,000 cm³

3The Pythagorean Theoremright triangles only

The square on the hypotenuse equals the two leg squares together: 9 + 16 = 25.

In any right triangle

a, b = legs (form the right angle) · c = hypotenuse (longest side, across from the right angle)

a² + b² = c²

Distance = Pythagoras

d = √((x₂ − x₁)² + (y₂ − y₁)²); (1, 2) to (4, 6): 5

Find a missing side Worked

Hypotenuse: legs 6 and 8
c² = 36 + 64 = 100 → c = 10
Leg: c = 13, a = 5
b² = 169 − 25 = 144 → b = 12subtract for a leg
Ladder 10 ft, base 6 ft from wall: top at √(100 − 36) = 8 ft

Converse: classify by sides (c longest)

c² = a² + b² → right
c² < a² + b² → acute
c² > a² + b² → obtuse
EX 6, 7, 9: 81 < 36 + 49 = 85 → acute

Diagonal of a box Worked

d = √(l² + w² + h²)

3 × 4 × 12 box: √(9 + 16 + 144) = √169 = 13

Common triples (and multiples)
a	b	c
3	4	5
5	12	13
8	15	17
7	24	25
6	8	10

“3-4-5, stay alive”

Learn 3-4-5 and 5-12-13; every multiple works too (6-8-10, 9-12-15, 10-24-26). Builders square a corner by measuring 3 ft, 4 ft and checking for 5 ft.

Special right triangles

45°-45°-90°: legs x, x; hypotenuse x√2
30°-60°-90°: short leg x; long leg x√3; hypotenuse 2x
EX 30-60-90 with hypotenuse 14 → short leg 7; long leg 7√3 ≈ 12.1

GeometryGeometry · Units 2, 11–12 · continued

4Building Blocks: Terms, Space & Shapesthe words every proof is built from

Undefined terms

Named, not defined: we describe them and build on them.

Point: an exact location; no length, width or thickness. Drawn as a dot, named by a capital letter: point A.
Line: straight, endless both ways, no thickness. Named by two of its points (BC) or a script letter (line ℓ).
Plane: a flat surface endless in every direction, no thickness. Named by a script capital (plane 𝒫) or three noncollinear points (plane ABC).

Dot · Dash · Floor · Room

0-D point = a dot · 1-D line = a dash that never ends · 2-D plane = an endless floor · 3-D space = the whole room. Each adds one direction.

Defined terms

Space: the set of all points; it has three dimensions (length, width, height).
Collinear: points on one line.
Coplanar: points or lines in one plane.
Intersection: the set of points two figures share.
Postulate: a statement accepted without proof (also called an axiom).
Theorem: a statement proved from definitions, postulates and earlier theorems.
Figure: any set of points; a shape is a figure with a boundary.

Conjecture & counterexample

One counterexample disproves a conjecture: “all primes are odd” fails at 2.

What makes a good definition

It names the class and the difference, and it reverses. A right angle is an angle (class) that measures 90° (difference). Reversed: an angle that measures 90° is a right angle. If the reverse fails, the definition is too loose.

Ruler & protractor postulates

Points on a line match real numbers; the distance between them is |b − a|.
Rays from one point match the degrees 0 to 180; m∠ = |difference of the readings|.

D, E, F collinear; G is not. Any 3 points are coplanar.

Point-line-plane postulates

Through any 2 points there is exactly one line.
Through any 3 noncollinear points there is exactly one plane.
If 2 points lie in a plane, the whole line through them lies in it.
Two lines meet in at most one point.
Two planes meet in exactly one line (if they meet at all).

Planes 𝒫 and 𝒬 intersect in line ℓ.

Symbols to know
Sym.	Means	Sym.	Means
∠	angle	≅	congruent
m∠	measure of	∼	similar
⊥	perpendicular	△	triangle
∥	parallel	≈	about
°	degrees	∈	is on / in

Polygons

A polygon is a closed plane figure made of segments (sides) that meet only at endpoints (vertices). Convex: no dent; every diagonal stays inside. Concave: at least one dent. Regular: all sides and all angles equal.

Sides	Name	Angle sum
3	triangle	180°
4	quadrilateral	360°
5	pentagon	540°
6	hexagon	720°
7	heptagon	900°
8	octagon	1080°
10	decagon	1440°
12	dodecagon	1800°
n	n-gon	(n − 2) · 180°

Each angle of a regular n-gon = (n − 2) · 180° ÷ n (hexagon: 120°).

Names that count

Pentagon: the 5-sided Pentagon building · Hexagon: 6-sided honeycomb cells · Octagon: an octopus has 8 arms · Decagon: a decade has 10 years.

Triangles · Quadrilaterals

By sides: scalene (none equal), isosceles (≥ 2), equilateral (3).
By angles: acute (all < 90°), right (one 90°), obtuse (one > 90°).
Square ⊂ rectangle ⊂ parallelogram; square ⊂ rhombus ⊂ parallelogram; trapezoid has exactly 1 pair of parallel sides (some texts: ≥ 1).

Regular hexagon Worked

interior 4 · 180°6 = 120°; exterior 360°6 = 60°

diagonals n(n − 3)2 = 6 · 32 = 9

5Lines & Planeshow lines and planes meet, or never meet

Matching arrowheads on two lines mark them parallel.

Skew lines skip

Skew lines skip past each other in 3-D: no meeting point, no common plane. Two lines in one plane must cross or be parallel.

Parallel (∥): coplanar lines that never meet. Same slope on a graph.
Perpendicular (⊥): lines that meet at 90°. Slopes multiply to −1.
Intersecting: lines with exactly one common point.
Skew: lines in different planes: never meet, yet not parallel (a box’s top front edge and a back side edge).

Parallel planes: never meet (floor and ceiling).
Line ∥ plane: the line never touches the plane.
Line ⊥ plane: ⊥ to every line in the plane through its foot.

Slopes tell you Worked

y = 2x + 1 vs y = 2x − 5: slopes 2, 2 → parallel

y = 2x + 1 vs y = −½x + 3: 2 · (−½) = −1 → perpendicular

GeometryGeometry · Units 2, 11–12 · continued

6Segments & Rayspieces of a line

Endpoints are dots; arrows mean “keeps going.”

Segment AB: points A and B and every point between them. Its length is AB (no bar).
Ray AB: starts at endpoint A, passes through B, goes on forever. Order matters: AB ≠ BA.
Congruent (≅): segments with equal lengths: AB ≅ CD means AB = CD.

Bar = thing, no bar = number

AB is the segment itself (a figure); AB is its length (a number). Figures are congruent (≅); numbers are equal (=).

Segment addition postulate

If B is between A and C, then AB + BC = AC.

Algebra with segments Worked

B between A, C; AB = 2x + 1, BC = 3x − 4, AC = 27
(2x + 1) + (3x − 4) = 27add the parts
5x − 3 = 27 → x = 6
AB = 13, BC = 14 (13 + 14 = 27 ✓)check

Midpoint & distance

Number line: M = (−2 + 8) ÷ 2 = 3; AB = |8 − (−2)| = 10.

Formulas

Number line: M = a + b2; length = |b − a|
Plane: M = (x₁ + x₂2, y₁ + y₂2)
d = √((x₂ − x₁)² + (y₂ − y₁)²)

d = √(6² + 8²) = 10; M = (4, 5).

Midpoint = average · distance = Pythagoras

Add and halve to find a midpoint. The distance formula is just a² + b² = c² on a grid: run, rise, then the slanted side.

A segment bisector passes through the midpoint; a ⊥ bisector also makes a right angle.

Rays: opposite, union, intersection

Opposite rays share an endpoint and point in opposite directions; together they form a line.
BA ∪ BC = AC when they are opposite.
AB ∩ BA = AB (the part they share).
Two rays with one endpoint that are not opposite form an angle.

7Anglestwo rays, one endpoint

Name it ∠ABC, ∠CBA, ∠B or ∠1; the vertex is always the middle letter.

Angle: two rays (sides) with a common endpoint (vertex).
m∠ABC: its measure in degrees (°); a full turn = 360°.
Congruent ∠s: equal measures; marked with matching arcs.
Adjacent ∠s: share a vertex and a side, with no interior points in common.

Angle addition postulate

If D is in the interior of ∠ABC, then m∠ABD + m∠DBC = m∠ABC.

Solve for x Worked

m∠ABD = 3x + 5, m∠DBC = 2x + 10, ∠ABC is right
5x + 15 = 90 → x = 15
m∠ABD = 50°, m∠DBC = 40°50 + 40 = 90 ✓

Complementary: two angles summing to 90°. Complement of 62° = 28°.
Supplementary: two angles summing to 180°. Supplement of 62° = 118°.
Linear pair: adjacent angles whose outer sides are opposite rays; always supplementary.
Vertical ∠s: opposite angles formed by two crossing lines; always congruent.
Around a point: angles fill 360°.

Corner & Straight

Complementary → Corner (90°). Supplementary → Straight (180°). C comes before S in the alphabet, and 90 comes before 180.

Vertical angles Worked

Vertical angles: 4x − 10 and 2x + 30
4x − 10 = 2x + 30 → x = 20set equal
each angle = 70°; its neighbors = 110°linear pair

BD bisects ∠ABC: it splits it into two ≅ angles.