Surface Area Formulas

Surface Area Formulas
In general, the surface area is the sum of all the areas of all the shapes that cover the surface of the object.

Note: "ab" means "a" multiplied by "b". "a²" means "a squared", which is the same as "a" times "a".

Be careful!! Units count. Use the same units for all measurements.

 

Surface Area of a Cube = 6 a 2

(a is the length of the side of each edge of the cube)
In words, the surface area of a cube is the area of the six squares that cover it. The area of one of them is a*a, or a ² . Since these are all the same, you can multiply one of them by six, so the surface area of a cube is 6 times one of the sides squared.

Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac

(a, b, and c are the lengths of the 3 sides)
In words, the surface area of a rectangular prism is the area of the six rectangles that cover it. But we don't have to figure out all six because we know that the top and bottom are the same, the front and back are the same, and the left and right sides are the same.

The area of the top and bottom (side lengths a and c) = a*c. Since there are two of them, you get 2ac. The front and back have side lengths of b and c. The area of one of them is b*c, and there are two of them, so the surface area of those two is 2bc. The left and right side have side lengths of a and b, so the surface area of one of them is a*b. Again, there are two of them, so their combined surface area is 2ab.
 

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)
 

Surface Area of a Sphere = 4 pi r 2

(r is radius of circle)
 

Surface Area of a Cylinder = 2 pi r 2 + 2 pi r h

(h is the height of the cylinder, r is the radius of the top)
Surface Area = Areas of top and bottom +Area of the side
Surface Area = 2(Area of top) + (perimeter of top)* height
Surface Area = 2(pi r ²) + (2 pi r)* h

In words, the easiest way is to think of a can. The surface area is the areas of all the parts needed to cover the can. That's the top, the bottom, and the paper label that wraps around the middle. 

You can find the area of the top (or the bottom). That's the formula for area of a circle (pi r²). Since there is both a top and a bottom, that gets multiplied by two.
The side is like the label of the can. If you peel it off and lay it flat it will be a rectangle. The area of a rectangle is the product of the two sides. One side is the height of the can, the other side is the perimeter of the circle, since the label wraps once around the can. So the area of the rectangle is (2 pi r)* h. 

Add those two parts together and you have the formula for the surface area of a cylinder.
Surface Area = 2(pi r ²) + (2 pi r)* h
 

formulae on Volumes

cube = a ³
rectangular prism = a b c
irregular prism = b h
cylinder = b h = pi r ² h
pyramid = (1/3) b h
cone = (1/3) b h = 1/3 pi r ² h
sphere = (4/3) pi r ³
ellipsoid = (4/3) pi r₁ r₂ r₃
 
Units
Volume is measured in "cubic" units. The volume of a figure is the number of cubes required to fill it completely, like blocks in a box.
Volume of a cube = side times side times side. Since each side of a square is the same, it can simply be the length of one side cubed.
If a square has one side of 4 inches, the volume would be 4 inches times 4 inches times 4 inches, or 64 cubic inches. (Cubic inches can also be written in³.)

Area Formulas

Area Formulas

Note: "ab" means "a" multiplied by "b". "a²" means "a squared", which is the same as "a" times "a".

Be careful!! Units count. Use the same units for all measurements. Examples

square = a ²
rectangle = ab
parallelogram = bh 

trapezoid = h/2 (b₁ + b₂)
circle = pi r ²
ellipse = pi r₁ r₂
 

triangle =1/2(bh

one half times the base length times the height of the triangle

  

equilateral triangle =

 
triangle given SAS (two sides and the opposite angle)
= (1/2) a b sin C
triangle given a,b,c = [s(s-a)(s-b)(s-c)] when s = (a+b+c)/2 (Heron's formula)
regular polygon = (1/2) n sin(360°/n) S²
   when n = # of sides and S = length from center to a corner

Polygons

What is a Polygon?
A closed plane figure made up of several line segments that are joined together. The sides do not cross each other. Exactly two sides meet at every vertex.

Types of Polygons
Regular - all angles are equal and all sides are the same length. Regular polygons are both equiangular and equilateral.
Equiangular - all angles are equal.
Equilateral - all sides are the same length.

	Convex - a straight line drawn through a convex polygon crosses at most two sides. Every interior angle is less than 180°.
	Concave - you can draw at least one straight line through a concave polygon that crosses more than two sides. At least one interior angle is more than 180°.

Polygon Formulas
(N = # of sides and S = length from center to a corner)
Area of a regular polygon = (1/2) N sin(360°/N) S²
Sum of the interior angles of a polygon = (N - 2) x 180°
The number of diagonals in a polygon = 1/2 N(N-3)
The number of triangles (when you draw all the diagonals from one vertex) in a polygon = (N - 2)
Polygon Parts

Side - one of the line segments that make up the polygon. Vertex - point where two sides meet. Two or more of these points are called vertices. Diagonal - a line connecting two vertices that isn't a side. Interior Angle - Angle formed by two adjacent sides inside the polygon. Exterior Angle - Angle formed by two adjacent sides outside the polygon.

Special Polygons
Special Quadrilaterals - square, rhombus, parallelogram, rectangle, and the trapezoid.
Special Triangles - right, equilateral, isosceles, scalene, acute, obtuse.

Polygon Names
Generally accepted names

Sides	Name
n	N-gon
3	Triangle
4	Quadrilateral
5	Pentagon
6	Hexagon
7	Heptagon
8	Octagon
10	Decagon
12	Dodecagon

Names for other polygons have been proposed.

Sides	Name
9	Nonagon, Enneagon
11	Undecagon, Hendecagon
13	Tridecagon, Triskaidecagon
14	Tetradecagon, Tetrakaidecagon
15	Pentadecagon, Pentakaidecagon
16	Hexadecagon, Hexakaidecagon
17	Heptadecagon, Heptakaidecagon
18	Octadecagon, Octakaidecagon
19	Enneadecagon, Enneakaidecagon
20	Icosagon
30	Triacontagon
40	Tetracontagon
50	Pentacontagon
60	Hexacontagon
70	Heptacontagon
80	Octacontagon
90	Enneacontagon
100	Hectogon, Hecatontagon
1,000	Chiliagon
10,000	Myriagon

To construct a name, combine the prefix+suffix

Sides Prefix 20 Icosikai... 30 Triacontakai... 40 Tetracontakai... 50 Pentacontakai... 60 Hexacontakai... 70 Heptacontakai... 80 Octacontakai... 90 Enneacontakai...

+

Sides Suffix +1 ...henagon +2 ...digon +3 ...trigon +4 ...tetragon +5 ...pentagon +6 ...hexagon +7 ...heptagon +8 ...octagon +9 ...enneagon

Examples:
46 sided polygon - Tetracontakaihexagon
28 sided polygon - Icosikaioctagon

Conic Sections formulae

The Conic Sections. For any of the below with a center (j, k) instead of (0, 0), replace each x term with (x-j) and each y term with (y-k).

	Circle	Ellipse	Parabola	Hyperbola
Equation (horiz. vertex):	x2 + y2 = r2	x2 / a2 + y2 / b2 = 1	4px = y2	x2 / a2 – y2 / b2 = 1
Equations of Asymptotes:				y = ± (b/a)x
Equation (vert. vertex):	x2 + y2 = r2	y2 / a2 + x2 / b2 = 1	4py = x2	y2 / a2 – x2 / b2 = 1
Equations of Asymptotes:				x = ± (b/a)y
Variables:	r = circle radius	a = major radius (= 1/2 length major axis) b = minor radius (= 1/2 length minor axis) c = distance center to focus	p = distance from vertex to focus (or directrix)	a = 1/2 length major axis b = 1/2 length minor axis c = distance center to focus
Eccentricity:	0	c/a	1	c/a
Relation to Focus:	p = 0	a2 - b2 = c2	p = p	a2 + b2 = c2
Definition: is the locus of all points which meet the condition...	distance to the origin is constant	sum of distances to each focus is constant	distance to focus = distance to directrix	Difference between distances to each foci is constant

Closure Property

Closure Property of Addition

Sum (or difference) of 2 real numbers equals a real number

Additive Identity

a + 0 = a

Additive Inverse

a + (-a) = 0

Associative of Addition

(a + b) + c = a + (b + c)

Commutative of Addition

a + b = b + a

Definition of Subtraction

a - b = a + (-b)

Closure Property of Multiplication

Product (or quotient if denominator 0) of 2 reals equals a real number

Multiplicative Identity

a * 1 = a

Multiplicative Inverse

a * (1/a) = 1     (a 0)

(Multiplication times 0)

a * 0 = 0

Associative of Multiplication

(a * b) * c = a * (b * c)

Commutative of Multiplication

a * b = b * a

Distributive Law

a(b + c) = ab + ac

Definition of Division

a / b = a(1/b)