Volume, Area
Curves, lines   Imaginary math   Trigonometry   Calculus   Math

Circle A = πr²
length of chord
L = 2√(r²-d²)
    r = radius
    d = perpendicular distance from chord to center
length of chord = L, r = radius
L = r crd θ
where crd θ = 2 sin (θ/2)
L = 2r sin (θ/2)
solve for θ
sin (θ/2) = (L/2r)
θ = 2 arcsin (L/2r)

Area of a segment
    A = (r²/2)[(π/180)θ – sinθ]
equation of circle with center at a,b and radius r
    (x–a)² + (y–b)² = r²
center at the origin
    x² + y² = r²

Ellipse A = πab
where a and b are one-half of the ellipse's major and minor axes,
  ie, the lengths of the semi-major and semi-minor axes.

Ellipse whose axis correspond to the x and y axis has the equation
  (x/a)² + (y/b)² = 1
a & b are distance from origin to curve along x or y axis.

Ellipse centered at x₁ and y₁, the equation is
  ((x – x₁)/a)² + ((y – y₁)/b)² = 1
a & b are distance from center to curve along x or y axis.
Ellipse is not rotated.

Sphere V = ⁴/₃πr³
  A = 4πr²
Volume of a partial sphere (cap)
  V = (πh²/3)(2r–h)
  V = (πh/6)(3a²+h²)
where r is the radius of the sphere, h is the height of the cap,
  a is the radius of the cap.

Cylinder  V = πr²h
  A = 2πr² + 2πrh = 2πr(r+h)
but skipping the ends, A = 2πrh

Triangle
equalaterial triangle
  A = (s²√3)/4)

Isosceles triangle
A = (b/4)√(4a²–b²) = (a²/2)sinθ
a = length of equal sides, b the other
θ = angle between equal sides

any triangle
A = bh/2
b is base, h is height
A = √[s(s–a)(s–b)(s–c)]
where s = (a+b+c)/2
a,b,c are the sides of the triangle
Σ interior angles 180º

Pythagorean theorem applies to only right
triangles, where c is the side opposite the
right angle, and a and b are the otehr two
sides
c = √(a² + b²)

Cube
  Volume = s³
  SA = 6s²

Pyramid
The volume of a pyramid is V = (1/3)Bh where B is the
area of the base and h the height from the base to the
apex. This works for any location of the apex, provided
that h is measured as the perpendicular distance from
the plane which contains the base.
Surface area square base pyramid
  A = Aʙ + ps/2
  Aʙ is area of base
  p is the perimeter of base
  s is slant height

regular tetrahedron
made up of 4 equilateral triangles
A = s²√3
V = s³/(6√2)

Cone
surface area A = πr² + πrs = πr(r+s)
where s is slant height
volume V =  ⅓πr²h, h is height of cone (not slant)
  h² + r² = s²
A = πr² + πr√(h² + r²)

Torus
volume of a torus is 2π²Rr²
where R is the distance from the center of the tube to the
center of the torus, r is the radius of the tube.

A conical frustum is a frustum created by slicing the top
off a cone (with the cut made parallel to the base). For
a right circular cone, let h be the height (not slant)
and R₁ and R₂ the base and top radii. Then
V = (1/3)πh(R₁² + R₁R₂ + R₂²)

regular polygon with n sides
area = s²(n/4)(cot(π/n))
where s is length of one side

Pentagon (5), with sides all equal,
Area is A = (s²/4)√(25+10√5) = 1.720s²
where s is length of one side
Σ interior angles 540º

Hexagon (6), with sides all equal,
Area = (3√3/2)(s²) = 2.598s²
where s is length of one side
Σ interior angles 720º

Octagon (8), with sides all equal,
Area = 2(√2–1)s² ≈ 4.828s²
where s is length of one side
Sum of the interior angles of a convex polygon with n sides
  180(n–2)

Square, Rectangle, Quadrilateral
square A = s²
rectangle A = L•W
cube V = s³
A = 6s²
cuboid (rectangular prism)
V = L•W•H
A = 2LW + 2LH + 2WH
Simple Quadrilateral Σ interior angles 360º

Rhombus
A rhombus (equilateral quadrilateral) is a parallelogram
with 4 equal sides.

parallelogram
A = sh = pq/2
where h is the height, s is a side and p,q are the diagonals.

Trapezoid
A = (h/2)(b₁+b₂)
h is the height
b₁,b₂ are the lengths of the two parallel sides

Parallelogram
A = hb
h is the height
b is the base

Parabola
y = a(x–h)² + k
vertex is at (h, k)
focus relative to vertex is 1/4a
If the chord has length b, and is perpendicular to the parabola's axis
of symmetry, and if the perpendicular distance from the parabola's
vertex to the chord is h, the parallelogram is a rectangle, with sides
of b and h. The area, A, of the parabolic segment enclosed by the
parabola and the chord is therefore:
A = (2/3)bh


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