When a plane slices through a cylinder at a 45-degree angle to its base, the cross-section is an ellipse. The major axis of the ellipse is equal to the diameter of the cylinder, and the minor axis is equal to the diameter multiplied by the sine of the angle.

Sat, 02 May 2026 17:34:41 +0000

**Why a 45-Degree Plane Cut Through a Cylinder Creates an Elliptical Cross-Section—What It Means for Design, Science, and Everyday Innovation** Why are experts and enthusiasts discussing how a plane slicing through a cylinder at a 45-degree angle produces an ellipse? This seemingly simple geometric interaction reveals deeper connections between form, space, and function across engineering, design, and digital visualization. More than just a math curiosity, this phenomenon touches on how we understand curves within flat surfaces—especially when angular precision transforms simple shapes into complex outlines. When viewed from directly perpendicular, the intersection of a cylindrical surface and a slicing plane yields a perfect ellipse. The length of the ellipse’s major axis exactly mirrors the cylinder’s diameter. But here’s the precise geometry: the minor axis shrinks by a factor equal to the sine of the 45-degree cut angle. Since sine 45° equals √2/2, the minor axis becomes the cylinder’s diameter multiplied by √2/2, resulting in a balanced elongation that reflects harmonic alignment between plane orientation and shape. This relationship isn’t abstract. It plays a role in technical simulations, architectural modeling, and industrial design, where angular intersections shape real-world components. The mathematical consistency makes it a reliable reference in both academic study and digital design tools, enabling professionals to predict and manipulate spatial relationships with confidence.