x2+y2 = 4
x2+y2 = 9
x2+y2 = 16
x2+y2 = 2
Circle centered at origin has what symmetries?
Circle centered at origin (0,0) with r as the radius:
x2+y2 = 4
x2+y2 = 9
x2+y2 = 16
x2+y2 = 2
Circle centered at origin has what symmetries?
Circle centered at (h,k): using distance formula:
√((x-h)2+(y-k)2)=r
Standard form equation of a circle: can read off the center and radius.
Every such equation is a circle.
Radius r is size, (h,k) center is location.
"Tricky": in the equation: positive h or k will be negative, negative h or k will be positive .
Given the center point and the radius of a circle, can make the standard form equation.
Ex. Center (-5,2) radius 3: (x+5)2+(y-2)2 = 9
Ex. Center (-5,-2) radius √3: (x+5)2+(y+2)2 = 3
Ex. Center (5,2) radius 1/3: (x-5)2+(y-2)2 = 1/9
Ex. Center (5,-2) radius 1/√3: (x-5)2+(y+2)2 = 1/3
Might have x and/or y intercept(s).
For x intercepts, set y to 0 and solve for x.
For y intercepts, set x to 0 and solve for y. MP.
Desmos shows all intercepts and min and max y's.
Every one of the infinite number of points on the circle is a solution to its equation.
This circle has the four indicated intercept and min/max points among its solutions: (0,0), (2,2), (4,0), (2,-2)
General form equation of a circle: x2+y2+Dx+Ey+F=0
x and y both squared, and with same coefficient, either both positive or both negative:
Ax2+Cy2+[Dx]+[Ey]+F=0, A=C
The general form relates circles to other conic section (ellipses, parabolas, hyperbolas)
quadratics. Not so useful.
Convert standard form to general form equation:
x2+y2-2hx-2ky+(h2+k2-r2)=0
Hint: MP simplify: (x-h)2+(y-k)2-r2
Convert general form to standard form: complete the square for each variable. Wolfram "Alternate form"
h = D/-2 k = E/-2 r = √(h2+k2-F)
Circumference = 2πr. d=2r, so C = πd
π=C/d the ratio of circumference to diameter [of every circle].
The circumference is a bit more than three times the length of the diameter.
The diameter is a bit less than one-third the circumference.
π, an unintuitive irrational number. We can't wrap our minds around an infinitely expanding number.
The digit sequence is a mathematical constant, not a physical one dependent on universe-specific physics.
The continuous curvature of the circle gives rise to irrational π.
A rational π would imply discrete steps.
Circle has infinite divisibility: everywhere on it has the same amount of curve, and all the way down.
Area of a circle = πr2.
Squared units, so not directly comparable with length unit.
The area is a bit more than three times the area of a square whose side length is r.
Which is more area: an 18" pizza or two 12" pizzas?
Leibniz: π/4 = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - 1/15...
Double the radius → double the circumference and quadruple the area.
Circle is the shape that encloses the most area for a given boundary.
C=1 → A=1/(4π)≈0.079577
(cf. square of side 1/4 has area 1/16=.0625,
equilateral triangle of side 1/3 has area √3/36≈.0481,
isosceles right triangle of sides 1/(2+√2) and
hypotenuse .√2/(2+√2) has area ≈ .0607,
rectangle w/length(1/3) twice its width(1/6) has area ≈ .0555,
regular pentagon of side 1/5 has area ≈ .0688)
Most efficient shape.
Cross-sections of sphere, cylinder, cone.
Stars, planets, plant roots&stems,
Body: cross-section of "tubes": arteries, veins, nerves, intestines, ducts, ear canal... Limbs, digits. pupil, iris
Manufactured: caps, cans, coins, clocks, cables/tubing/hoses, buttons, screws/nails/rivets, balls,
cups/bowls/plates, lenses, wheels, propellers, turbines,
Circle's area is π/4 ≈ 78.54% of square's area.
Circle vs line: two ends of a spectrum?
Line: open-ended, to infinities, lineup straight at every point, same change rate
Circle: closed loop, bounded, curvature/bend at every point, (all) differing change rates
Semicircle functions.
upper half of circle centered at origin with radius r: f(x) = √(r2-x2)
lower half of circle centered at origin with radius r: f(x) = -√(r2-x2)
Arclength of the top unit semicircle = ∫-1 to 1 of √(1+x2/(1-x2))dx = π
upper/lower semicircles centered at (h,k) with radius r: f(x) = ±√(r2-(x-h)2) + k
Disc
Centered at (h,k) with radius r:
(x-h)2+(y-k)2 ≤ r2
Closed disc includes the circle: ≤
Open disc excludes the circle: <
"Unit disc": x2+y2 ≤ 1
Three non-collinear points define a circle.
whose center is the circumcenter of the triangle defined by the 3 points.
Circumcenter of: 1. right triangle is midpoint of hypotenuse;
2. acute triangle is interior; 3. obtuse triangle is exterior.
Circumcenter is the intersection of the perpendicular bisectors
of the triangle's sides. A radius
Archimedes: area of circle equals area of triangle whose base is C and height is r.
Intersecting circles
Equation of line through the two intersection points:
y = (h1-h2)/(k2-k1)x +
(h22+k22-r22-
h12-k12+r12)/
(-2k1+2k2)
