Diagonal To The Base Cutting Off Exactly 1 Vertex

Imagine holding a solid block and slicing off just one corner with a knife held at an angle. The cut isn’t parallel to the bottom; it leans into the block, removing a single vertex while leaving the rest of the shape intact. That simple action touches on a neat geometric idea: a plane that is diagonal to the base and cuts off exactly one vertex of a polyhedron. Though it sounds like a kitchen trick, the concept shows up in design, machining, and even in math puzzles about volume and surface area.

What Is a Diagonal Cut That Removes One Vertex

When we talk about a “diagonal to the base” we mean a plane that is not parallel to the bottom face of a solid. Consider this: a plane that tilts relative to that base can intersect the solid in many ways. But if the plane passes through the three edges that meet at a single corner, it slices away that corner alone. Consider this: in the case of a cube—or any right prism with a rectangular base—the base is one of the faces that sits flat. The piece that falls off is a tetrahedron (a triangular pyramid), and what remains is the original cube with one vertex truncated.

The key is that the plane does not hit any other vertex. Also, it touches exactly three edges, each belonging to the chosen vertex, and then exits the solid through the opposite faces. Because the base stays untouched, the cut is described as “diagonal to the base”: it leans across the base rather than running parallel to it.

Why It Matters / Why People Care

At first glance, cutting off a corner might seem like a trivial exercise, but the idea appears in several practical contexts. In manufacturing, a machinist might need to create a chamfer or a bevel on a workpiece that removes just one sharp edge without affecting the surrounding features. Knowing how to describe the cutting plane helps program CNC machines accurately.

In architecture, truncated corners appear in modern façades where a building’s mass is softened by slicing off a vertex. Engineers calculate the resulting volume to estimate material savings or to understand how the change affects structural load distribution.

From a pure mathematics standpoint, the problem provides a concrete way to apply concepts of three‑dimensional coordinate geometry, linear equations, and volume integration. It bridges the gap between abstract formulas and a tangible shape you can hold in your hand.

How It Works

Visualizing the Cut

Start with a cube sitting on a table. Label the bottom face as the base. Choose the top‑

—front right corner as the vertex to remove. Practically speaking, the chosen vertex is (1,1,1), and the three edges meeting there run along the x-, y-, and z-axes toward (0,1,1), (1,0,1), and (1,1,0), respectively. That's why suppose the cube has edges of length 1 and is positioned with one corner at the origin (0,0,0), extending to (1,1,1). Let’s assign coordinates to the cube to make this concrete. The cutting plane must intersect these three edges at some point before their opposite ends. If we select points one-third of the way along each edge—say, (2/3,1,1), (1,2/3,1), and (1,1,2/3)—the plane passing through these three points will slice off the vertex cleanly.

The equation of this plane can be derived using the general form ( ax + by + cz = d ). This plane intersects the cube precisely where we intended, carving out a tetrahedron with vertices at (1,1,1), (2/3,1,1), (1,2/3,1), and (1,1,2/3). Plugging in the coordinates of the three points yields a system of equations that solves to ( x + y + z = 2 ). The remaining solid retains all original faces except for the new triangular face created by the cut.

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Calculating the Volume Removed

To find the volume of the tetrahedron, we can use the formula for the volume of a pyramid: ( V = \frac{1}{3} \times \text{base area} \times \text{height} ). Here, the base is the triangle formed by the three points on the edges, and the height is the distance from the original vertex to the plane. Alternatively, using coordinates, the volume can be computed via the scalar triple product of vectors from one vertex to the others. For simplicity, if the cube’s edge length is ( a ), the volume of the tetrahedron removed is ( \frac{a^3}{6} ), leaving ( \frac{5a^3}{6} ) of the original cube.

Surface Area Considerations

The cut introduces a new triangular face, increasing the total surface area. The original cube had a surface area of ( 6a^2 ). The triangular face has an area equal to half the product of two edge lengths of the tetrahedron’s base and the sine of the angle between them. For the one-third points chosen earlier, each edge of the triangle is ( \frac{a}{3} ), and the angles between them are all 60° (since they lie on the cube’s edges). The area of the triangle becomes ( \frac{\sqrt{3}}{2} \times \left(\frac{a}{3}\right)^2 \times 3 = \frac{\sqrt{3}a^2}{6} ). Adding this to the original surface area gives a new total of (

6a² + (√3 a²)/6. Even so, the cut also removes portions of the three original faces, each losing a small right triangle. On the flip side, each of these right triangles has legs of length a/3, so their combined area is 3 × (1/2)(a/3)(a/3) = a²/6. The net change in surface area is therefore (√3 a²)/6 - a²/6 = ((√3 - 1)a²)/6, an increase that becomes negligible as the cut is made closer to the original vertex. Consider this: in the limit where the cut passes through points arbitrarily close to (1,1,1), the removed tetrahedron’s volume approaches zero, and the surface area approaches the original 6a², though the shape itself becomes indistinguishable from the original cube. Conversely, if the cut is made at the midpoints of the edges, the removed tetrahedron has volume a³/6, and the surface area increases by a more noticeable amount. The choice of where to cut thus involves a trade-off between altering the volume and altering the surface area, with both being smoothly controlled by the parameter defining the position of the cut along the edges.

This simple geometric operation, known as a corner truncation or vertex truncation, has wide applications in architecture, where beveled edges improve durability and aesthetics; in computer graphics, where it forms the basis of more complex mesh refinement operations like loop subdivision; and in crystallography, where truncated structures appear in certain molecular and crystal lattices. , λ = 1/3 for the one-third points). Still, as λ increases from 0 to 1, the cut moves from the vertex to the opposite ends of the edges, and at λ = 1, the entire cube is theoretically cut away, leaving only the tetrahedron. The surface area of the new triangular face is (√3 λ² a²)/2, and the area removed from the original faces is 3 × (λ a)²/2 = (3λ² a²)/2, leading to the same net change formula in terms of λ. The mathematical description generalizes naturally to higher dimensions, where a similar cut in a hypercube removes a simplex whose volume is given by a straightforward formula involving the edge length and the dimension. In real terms, the full mathematical beauty of this operation lies in how a single parameter can continuously deform the cube, offering a family of related polyhedra that bridge the gap between the original cube and the regular tetrahedron removed at the corner. In three dimensions, the relationship V_removed = a³/6 is exact when the cut is at the midpoints, but for arbitrary cuts, the volume of the removed tetrahedron is (λ³ a³)/6, where λ is the fractional distance from the vertex to the cut points (e.Because of that, g. The intermediate shapes, sometimes called “truncated cubes” in a specific sense, demonstrate how local modifications can produce global changes in a solid’s properties, a principle that resonates throughout geometry and its applications.