Conic Section Formulas- Complete Reference

Conic Section Formulas: The Complete Reference

If you're working with conic sections, you need formulas you can actually use. This guide cuts through the noise and gives you every equation you need for circles, parabolas, ellipses, and hyperbolas. No derivations, no theory—just the goods.

The Four Conic Sections

Conic sections come from slicing a cone with a plane at different angles. Each shape has its own set of characteristics and equations. Here's what you're dealing with:

• Circle – plane cuts parallel to the base
• Parabola – plane cuts parallel to the slant side
• Ellipse – plane cuts at an angle (but not steep enough to reach the other side)
• Hyperbola – plane cuts through both halves of the cone

Circle Formulas

The circle is the simplest conic section. Every point is equidistant from the center.

Standard Equation (Center at Origin)

x² + y² = r²

Where r is the radius. That's it.

Standard Equation (Center at (h, k))

(x - h)² + (y - k)² = r²

Shift the origin to wherever your center actually is.

General Form

x² + y² + Dx + Ey + F = 0

Convert to standard form by completing the square.

Circle Key Measurements

• Diameter: D = 2r
• Circumference: C = 2πr
• Area: A = πr²
• Arc length (θ in radians): s = rθ
• Sector area: A = (1/2)r²θ

Parabola Formulas

Parabolas are defined by having every point equidistant from a focus and a directrix line.

Vertical Parabola (Opens Up or Down)

(x - h)² = 4p(y - k)

• Opens up if p > 0
• Opens down if p < 0
• Vertex: (h, k)
• Focus: (h, k + p)
• Directrix: y = k - p
• Focal length: |p|

Horizontal Parabola (Opens Left or Right)

(y - k)² = 4p(x - h)

• Opens right if p > 0
• Opens left if p < 0
• Vertex: (h, k)
• Focus: (h + p, k)
• Directrix: x = h - p

Parabola Equation in Terms of Focal Parameter

For a parabola with focus at (a, 0) and directrix x = -a:

y² = 4ax

Here, a equals the focal length.

Ellipse Formulas

An ellipse is the set of points where the sum of distances to two foci is constant.

Standard Equation (Horizontal Major Axis)

((x - h)² / a²) + ((y - k)² / b²) = 1

Major axis is horizontal if a > b.

Standard Equation (Vertical Major Axis)

((x - h)² / b²) + ((y - k)² / a²) = 1

Major axis is vertical if a > b.

Ellipse Key Values

• Semi-major axis: a
• Semi-minor axis: b
• Linear eccentricity (c): c² = a² - b²
• Focus 1: (h ± c, k) for horizontal; (h, k ± c) for vertical
• Eccentricity: e = c / a (always between 0 and 1)
• Latus rectum length: 2b² / a

Ellipse Key Measurements

• Area: A = πab
• Perimeter (approximate): P ≈ π[3(a + b) - √((3a + b)(a + 3b))]
• Vertices: (h ± a, k) or (h, k ± a)
• Co-vertices: (h, k ± b) or (h ± b, k)

Hyperbola Formulas

A hyperbola is the set of points where the absolute difference of distances to two foci is constant.

Standard Equation (Horizontal Transverse Axis)

((x - h)² / a²) - ((y - k)² / b²) = 1

Opens left and right along the x-axis.

Standard Equation (Vertical Transverse Axis)

((y - k)² / a²) - ((x - h)² / b²) = 1

Opens up and down along the y-axis.

Hyperbola Key Values

• Transverse axis length: 2a
• Conjugate axis length: 2b
• Linear eccentricity (c): c² = a² + b²
• Focus 1: (h ± c, k) for horizontal; (h, k ± c) for vertical
• Eccentricity: e = c / a (always greater than 1)
• Asymptotes (horizontal): y - k = ±(b/a)(x - h)
• Asymptotes (vertical): y - k = ±(a/b)(x - h)
• Latus rectum length: 2b² / a

Conic Section Comparison Table

How to Identify a Conic Section From Its Equation

Given a general second-degree equation:

Ax² + Bxy + Cy² + Dx + Ey + F = 0

The discriminant B² - 4AC tells you what you're dealing with:

• B² - 4AC < 0 → Ellipse (or circle if A = C and B = 0)
• B² - 4AC = 0 → Parabola
• B² - 4AC > 0 → Hyperbola

Quick Examples

4x² + 9y² = 36 → Divide by 36: x²/9 + y²/4 = 1 → Ellipse

y² = 8x → Already solved: (y - 0)² = 8(x - 0) → Parabola

x² - 4y² = 16 → Divide by 16: x²/16 - y²/4 = 1 → Hyperbola

Getting Started: Solving Conic Problems

Step 1: Identify the Conic

Check the signs and coefficients. If x² and y² have the same sign → ellipse or circle. If only one squared term exists → parabola. If they have opposite signs → hyperbola.

Step 2: Put It in Standard Form

Complete the square for x and y terms. Group the x terms together, complete the square, then do the same for y. This gives you (h, k) and the key parameters.

Step 3: Extract What You Need

Once in standard form, read off:

• Center (h, k)
• Vertices and co-vertices
• Focus points
• Eccentricity
• Directrix (for parabolas)
• Asymptotes (for hyperbolas)

Step 4: Calculate Desired Measurements

Area, perimeter, arc length—plug your a, b, r values into the appropriate formulas above.

Rotation of Conics

When B ≠ 0 in Ax² + Bxy + Cy² + ... = 0, the conic is rotated. The rotation angle θ eliminates the xy term:

cot(2θ) = (A - C) / B

Use the half-angle formulas to find sin θ and cos θ, then apply the rotation transformation:

x = x'cosθ - y'sinθ
y = x'sinθ + y'cosθ

The rotated equation will have B' = 0, letting you identify the conic type normally.

Polar Equations of Conics

Conics take a clean form in polar coordinates with focus at the origin:

r = (ed) / (1 + e cos θ) or r = (ed) / (1 + e sin θ)

Where:

• e = eccentricity
• d = distance from directrix to focus

Conic type depends on e:

• e < 1 → Ellipse
• e = 1 → Parabola
• e > 1 → Hyperbola

Quick Reference Summary

• Circle: x² + y² = r². Eccentricity = 0.
• Parabola: (x - h)² = 4p(y - k). Eccentricity = 1.
• Ellipse: x²/a² + y²/b² = 1. Eccentricity < 1.
• Hyperbola: x²/a² - y²/b² = 1. Eccentricity > 1.
• Discriminant: B² - 4AC identifies the conic type from general form.

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