Ellipse

Ellipse

Semi-axes: a, b
Eccentricity: e = sqrt(a²-b²)/a
Area: K
Circumference: C

K = p a b
C = 4aE, where E is an elliptic integral with k = e, which can be used to derive the following formulas:
C = p (a+b)[1 + x²/4 + x⁴/64 + ...],
where x = (a-b)/(a+b)
C = p (a+b)(1 + 3x²/[10 + sqrt(4 - 3x²)]), approximately

Segment of a Parabola

Height: h
Chord length: c
Area: K
Length: s

s = c[1+2(2h/c)²/3-2(2h/c)⁴/5+...]
s = sqrt[4h²+c²/4]+[c²/(8h)]
     ln[(2h+sqrt[4h²+c²/4])/(c/2)]
K = 2ch/3
K = 4T/3, where T is the area of
     the triangle formed by the chord
     and the point of tangency of a
     tangent to the parabola parallel
     to the chord

Segment of a Parabola
	Height: h Chord length: c Area: K Length: s
		s = c[1+2(2h/c)²/3-2(2h/c)⁴/5+...] s = sqrt[4h²+c²/4]+[c²/(8h)] ln[(2h+sqrt[4h²+c²/4])/(c/2)] K = 2ch/3 K = 4T/3, where T is the area of the triangle formed by the chord and the point of tangency of a tangent to the parabola parallel to the chord