Key Formulas(MENSURA Formulas(MENSURATION) TION) Length of a Diagonal of a Cube 2
d ¥3 s
= s¥3, where s is the length of one edge of the cube.
Length of a Diagonal of a Rectangular Solid d =
, wherel is the length, w is the width, and h is the height of the rectangular r ectangular solid.
Surface Area of a Cube
urface Area =6 s2, where s is the length of one edge.
S
Surface Area of a Rectangular Solid
urface Area = 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height.
S
Surface Area of a Cylinder
urface Area = 2r 2 + 2r h, where r is the radius and h is the height.
S
Surface Area of a Sphere
urface Area = 4r 2, where r is the radius.
S
Lateral Surface Area of a Cone
ateral Surface Area = r l l, where r is the radius of the base and l is the cone¶s lateral height.
L
Total Surface Area of a Cone
urface Area = r 2 + r l l, where r is the radius of the base, and l is the cone¶s lateral height.
S
Volume
of a Cube
olume = s3, where s is the length of one edge.
V
Volume
of a Rectangular Solid
olume =lwh, where l is the length, w is the width, and h is the height.
V
Volume
of a Prism
olume = Bh, where B is the area of the base, and h is the height.
V
Volume
of a Cylinder
olume = r 2h, where r is the radius of the circular base and h is the cylinder¶s height.
V
Volume
of a Cone
olume = 1/3 r 2h, where r is the radius and h is the height.
V
Volume
of a Pyramid
olume =1/3 Bh, where B is the area of the base and h is the height.
V
Volume
of a Sphere
olume =4/3 r 3, where r is the radius.
V
Key Formulas Pythagorean Theorem a2 + b2 = c2, where a and b are the lengths of the legs of a right triangle, and c
hypotenuse.
is the length of the
Area of a Triangle
Area = 1/2 bh, where b is the length of the base and h is height. Sum of the Interior Angles of a Polygon
he sum of the interior angles of a polygon is (n ± 2)180°, where n is the number of sides in the polygon. T
Area of a Trapezoid
Area = s1 + s2 / 2 , h h, where s1 and s2 are the lengths of the bases of the trapezoid, and h is the height. Area of a Parallelogram, Rectangle, and Rhombus
Area = bh, where b is the length of the base, and h is the height. Area of a Square
Area = s2, where s is the length of a side of the square. Circumference of a Circle C
ircumference = 2r , where r is the radius of the circle.
Arc Length
Arc Length = n/ 360o × 2r , where n is the measure of the degree of the arc, and r is the radius of the circle. Area of a Circle
Area = r 2, where r is the radius of the circle. Area of a Sector
Area of Sector = n/ 360o × r 2, where n is the measure of the central angle which forms the boundary of the sector, and r is the radius of the circle.
Key Formulas (CO-ORDINATE GEOMETRY) Distance in the Coordinate Plane
istance = y2).
if you¶re measuring the distance between the points ( x1, y1) and ( x2,
Distance in the Coordinate Space Distance
istance = y1, z1) and ( x2, y2, z2).
if you¶re measuring the distance between the points ( x1,
Midpoint between Two Points M
idpoint = (
,
where the endpoints of a line segment are ( x1, y1) and ( x2, y2).
Point-Slope Form of the Equation of a Line y
± y1 = m( x ± x1), where m is the slope of the line, and ( x1, y1) is a point on the line.
Slope-Intercept Form of the Equation of a Line
= mx + b, where m is the slope of the line, and b is the y-intercept of the line. Both m and b are
y
constants.
Slope of a Line
S
lope =
, where two points on the line are ( x1, y1) and ( x2, y2).
Standard Form of the Equation of a Circle
( x ± h)2 + ( y ± k )2 = r 2, where (h, k ) is the center of the circle, and r is the radius. When centered at the origin, the equation simplifies to x2 + y2 = r 2. Standard Form of the Equation of a Parabola y
2
= a( x ± h) + k , where a, h, and k are constants.
Pythagorean Identities
sin(2x) = 2 sin x cos x cos(2x) = cos^2(x) - sin^2(x) = 2 cos^2(x) - 1 = 1 - 2 sin^2(x) tan(2x) = 2 tan(x) / (1 - tan^2(x)) sin^2(x) = 1/2 - 1/2 cos(2x) cos^2(x) = 1/2 + 1/2 cos(2x) sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 ) cos x - cos y = -2 sin( (x - y)/2 ) sin( (x + y)/2 ) LAW OF COSINES
c^2 = a^2 + b^2 - 2ab cos(C) b^2 = a^2 + c^2 - 2ac cos(B) a^2 = b^2 + c^2 - 2bc cos(A) L
AW OF TANGENT
(a - b)/(a + b) = tan [(A-B)/2] / tan [(A+B)/2]
P
olygon Formulas
(N = # of sides and S = length from center to a corner) Area of a regular polygon = (1/2) N sin(360°/N) S 2 Sum of the interior angles of a polygon = (N - 2) x 180°
he 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) T
square
= a2
rectangle = ab
parallelogram = bh
trapezoid = h/2 (b1 + b2)
circle = pi r 2
triangle
one half times the base length times the height of the triangle
=
equilateral triangle
=
triangle given SAS (two sides and the oppo site 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)
POLYNOMIAL IDENTITIES
(a+b) 2 = a 2 + 2ab + b 2 (a+b)(c+d) = ac + ad + bc + bd a 2 - b 2 = (a+b)(a-b) (Difference of squares) a 3 b 3 = (a b)(a 2 ab + b 2) (Sum and x 2 + (a+b)x + AB if ax 2 + bx + c
=
=
0 then x
=
( -b
(b 2 - 4ac) ) / 2a (Quadratic Formula)
x a x b = x (a + b) x a y a = (xy) a =
x (ab)
x (a/b) = bth root of (x a)
=
( bth (x) ) a
x (-a) = 1 / x a x (a - b) = x a / x b
Logarithms
y = logb(x) if and only if x =b y logb(1) = 0 logb(b) = 1 logb(x*y)
=
of Cube s)
(x + a)(x + b)
Power s
(x a) b
Difference
logb(x) + log b(y)
logb(x/y) logb(x n)
= =
logb(x) - log b(y) n logb(x)
logb(x) = logb(c) * log c(x) = logc(x) / log c(b)
