Determine the elliptical shape and perimeter created when cutting a cylindrical surface at an angle | Step-by-Step Solution

Hello! 🌟

What We're Solving:

We need to figure out what happens when we slice through a cylinder at an angle - specifically, we want to understand the shape we get and calculate its perimeter. This is a beautiful example of how 3D geometry creates 2D curves!

The Approach:

Think of this like slicing a soup can with a tilted knife. When we cut straight across (perpendicular), we get a circle. But when we tilt our cut, something magical happens - we get an ellipse! Our strategy is to:

• 1. Visualize the 3D situation
• 2. Use coordinate geometry to describe the cut mathematically
• 3. Apply ellipse formulas to find the perimeter

Step-by-Step Solution:

Step 1: Set up our coordinate system

• Place the cylinder so its axis runs along the z-direction
• Let the cylinder have radius r
• The cutting plane makes an angle θ with the horizontal

Step 2: Understand why we get an ellipse When you slice a cylinder at an angle, you're essentially "stretching" a circle in one direction. The cross-section perpendicular to the cylinder axis is always a circle of radius r, but the angled cut makes this circle appear elongated.

Step 3: Find the ellipse dimensions

• The minor axis (shorter dimension) equals the cylinder diameter: 2r
• The major axis (longer dimension) gets stretched by the angle: 2r/cos(θ)

As the cutting angle gets steeper (θ approaches 90°), cos(θ) approaches 0, making the major axis much longer!

Step 4: Apply the ellipse perimeter formula For an ellipse with semi-major axis a and semi-minor axis b:

• a = r/cos(θ) (half the major axis)
• b = r (half the minor axis)

The exact perimeter formula is complex, but we can use Ramanujan's approximation: P ≈ π[3(a + b) - √((3a + b)(a + 3b))]

The Answer:

The perimeter of the elliptical cut is: P ≈ π[3r(1/cos(θ) + 1) - √((3r/cos(θ) + r)(r/cos(θ) + 3r))]

For specific values of r and θ, you'd substitute and calculate numerically.

Memory Tip:

Remember "CATS" - Cylinder Angled cuts make Tilted ellipses that get Stretched! The steeper the angle, the more stretched the ellipse becomes. At 0° you get a circle, and as you approach 90°, the ellipse becomes infinitely long! 🐱

Keep exploring these 3D-to-2D relationships - they show up everywhere from architecture to astronomy!