04 // Interactive Formula Compendium // GEOMETRY-TRIG(x − h)² / a² + (y − k)² / b² = 1, Area = π · a · b
Standard Equation of a Circle & Ellipse — Conic Sections Sheet
An ellipse centered at (h, k) with horizontal semi-axis a and vertical semi-axis b satisfies (x − h)²/a² + (y − k)²/b² = 1 and encloses exact area A = πab; when a = b = r, eccentricity e = 0 and it reduces to a circle of radius r.
Formula Reference
Main Formula
(x − h)² / a² + (y − k)² / b² = 1, Area = π · a · b
Rearranged Forms
(x − h)² + (y − k)² = r² (Circle Standard Form, a = b = r)
c = √(a² − b²), e = c / a (Focal Distance & Eccentricity, a ≥ b)
P ≈ π [ 3(a + b) − √((3a + b)(a + 3b)) ] (Ramanujan Perimeter)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| A | Ellipse / Circle Enclosed Area (π·a·b) | m² |
| a | Semi-Major Axis (a) | m |
| b | Semi-Minor Axis (b) | m |
Inputs & Parameters
[m]
[m]
Calculated Result (Ellipse / Circle Enclosed Area (π·a·b))
A =47.12389m²
Formula Notes & Background
- Define an ellipse as the locus of points where the sum of distances to two foci (±c, 0) is constant: d₁ + d₂ = 2a.
- Square the distance sum √((x+c)² + y²) = 2a − √((x−c)² + y²) twice and substitute b² = a² − c² to get x²/a² + y²/b² = 1.
- Scale a unit circle of area π horizontally by a and vertically by b (Jacobian determinant ab) to obtain exact area A = πab.
Step-by-Step Examples
#1: Find the area, focal distance c, and eccentricity e of an ellipse with a = 5 m and b = 3 m.
a = 5 mb = 3 m
A = π × 5 × 3 = 15π; c = √(5² − 3²) = 4; e = 4 / 5 = 0.8
→ Area = 47.1239 m², Foci at ±4 m, Eccentricity e = 0.80
#2: Convert x² + y² − 6x + 4y − 12 = 0 into standard circle form.
Group terms: (x² − 6x) + (y² + 4y) = 12
Complete squares: (x − 3)² − 9 + (y + 2)² − 4 = 12 ⇒ (x − 3)² + (y + 2)² = 25
→ Center (h, k) = (3, −2), Radius r = 5
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Frequently Asked Questions
Why is there no simple elementary formula for the perimeter of an ellipse?+
The arc length integral of an ellipse defines the complete elliptic integral of the second kind E(e), which has no closed form in elementary functions; Ramanujan's approximation is accurate to better than 0.04%.
What does eccentricity e measure in conic sections?+
e = 0 is a perfect circle, 0 < e < 1 is an ellipse, e = 1 is a parabola, and e > 1 is a hyperbola.
What is the optical reflection property of an ellipse?+
Any ray or acoustic wave emitted from one focus F₁ reflects off the elliptical boundary directly through the second focus F₂ with equal travel time.