Practice drawing cross-sections of 3D shapes cut parallel and diagonally to the base.

Let’s go step by step to figure out what each cross-section looks like for the two 3D shapes: a cube and a square pyramid.

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First Shape: Cube

A cube has 6 square faces, all equal. When you cut it with a plane, the shape of the cross-section depends on how you cut it.

#### 1. Cut parallel to base:
- The “base” is one of the square faces (usually the bottom).
- Cutting parallel to the base means slicing horizontally through the cube.
- Result: You get a square, same size as the base (if cut in the middle) or smaller if near top/bottom — but still a square.
→ Cross-section: Square

#### 2. Cut perpendicular to base:
- Perpendicular means straight up/down relative to the base.
- If you slice vertically through the center (from front to back or side to side), you’ll get a rectangle — actually, since it’s a cube, it will be a square too! Wait — let’s think carefully.
- Actually, if you cut perpendicular to the base *and* pass through opposite edges, you can get a rectangle that’s taller than wide? No — in a cube, all sides are equal.
- Example: Slice from top face to bottom face, going through left and right midpoints → you get a rectangle that’s height = edge length, width = edge length → still a square.
- BUT — if you cut perpendicular to the base but not aligned with faces? For example, diagonally across vertical faces? Then you might get a rectangle or even a hexagon? Hmm… wait — standard school-level interpretation:
- “Perpendicular to base” usually means cutting vertically through the center, parallel to one pair of side faces → gives a rectangle? Actually no — in a cube, if you cut vertically through the middle from front to back, you’re cutting through two opposite vertical faces — the cross-section is a square again.
- Let me clarify: In most textbooks, for a cube:
- Parallel to base → square
- Perpendicular to base (vertical cut through center) → also a square (because cube is symmetric)
- But sometimes they mean “perpendicular to base and passing through opposite edges” — which would give a rectangle only if the cube were stretched — but it’s not.
- Actually, here’s the key: if you cut perpendicular to the base AND along a diagonal of the base, then you get a rectangle that’s longer? No — still same dimensions.
- I think I’m overcomplicating. Let’s use standard answers taught in middle school:

✔ Standard answer for cube:
- Parallel to base → Square
- Perpendicular to base → Rectangle (Wait — why?)

Hold on — maybe “perpendicular to base” means cutting vertically but not necessarily aligned with faces? For example, imagine slicing the cube from top-front-left corner to bottom-back-right corner — that’s diagonal, not perpendicular.

Actually, let’s define clearly:

In geometry problems like this, when they say:

- “Cut parallel to base” → horizontal slice → square
- “Cut perpendicular to base” → vertical slice that goes through the center and is aligned with the sides → still a square? Or do they mean cutting through two opposite vertical edges? That would give a rectangle whose height is the cube’s edge, and width is also the cube’s edge → still square.

I think there’s confusion. Let me check common curriculum:

Upon recalling: In many worksheets, for a cube:

- Parallel to base → square
- Perpendicular to base → rectangle (but only if the cut is not through the center? No)

Wait — perhaps they mean:

If you cut perpendicular to the base but along a direction that’s not parallel to any face — for example, slicing from midpoint of top front edge to midpoint of bottom back edge — that would create a rectangle? Actually, no — in a cube, such a cut would create a parallelogram or rectangle depending on angle.

But for simplicity at this level, let’s assume:

For a cube:

- Cut parallel to base → Square
- Cut perpendicular to base → Rectangle (assuming they mean a vertical cut that doesn’t align with faces — but actually, even if it does, it’s still a square)

This is messy. Let me look at the second shape first.

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Second Shape: Square Pyramid

Base is a square, apex on top.

#### 1. Cut parallel to base:
- Horizontal slice somewhere between base and apex.
- Result: A smaller square (similar to base, scaled down).
→ Cross-section: Square

#### 2. Cut perpendicular to base:
- Vertical cut through the apex and center of base.
- This cuts through two triangular faces.
- Result: An isosceles triangle (with base = side of square base, height = height of pyramid).
→ Cross-section: Triangle

#### 3. Cut diagonally:
- Diagonal could mean different things. Usually, for a pyramid, “diagonally” means cutting from one edge of base to another non-adjacent edge, possibly through the apex or not.
- Common interpretation: Cut from one corner of base to opposite corner of base, passing through apex → that’s the same as perpendicular cut? No.
- Better: Cut diagonally across the base (along diagonal of square base) and up to apex → still gives a triangle, but now the base of the triangle is the diagonal of the square base → so it’s an isosceles triangle with base = diagonal of square.
- Alternatively, if you cut diagonally without going through apex — e.g., from one side edge to another — you might get a trapezoid or quadrilateral.
- But standard answer for “cut diagonally” in pyramid often means: cutting from one vertex of base to opposite vertex of base, through the interior — which for a square pyramid, if you cut along the diagonal of the base and up to apex, you get a triangle. If you cut diagonally but not through apex, you might get a quadrilateral.

Actually, let’s stick to typical textbook answers:

For square pyramid:

- Parallel to base → Square
- Perpendicular to base (through apex and center) → Triangle
- Diagonally (e.g., from one base edge to adjacent base edge, slanting) → often results in a trapezoid or quadrilateral

Wait — I recall now: In many sources, for a square pyramid:

- Cut parallel to base → square
- Cut perpendicular to base (through apex) → triangle
- Cut diagonally (meaning from one lateral edge to another, not through apex) → trapezoid

Yes, that makes sense.

Example: Imagine slicing the pyramid from halfway up one triangular face to halfway up the adjacent triangular face — the cross-section would be a trapezoid.

So:

Pyramid:
- Parallel to base → Square
- Perpendicular to base → Triangle
- Diagonally → Trapezoid

Now back to cube.

After checking standard educational resources (mentally):

For a cube:

- Cut parallel to base → Square
- Cut perpendicular to base → Rectangle — wait, why rectangle? Because if you cut vertically but not through the center? No.

Actually, I found the issue: In some contexts, “perpendicular to base” for a cube means cutting vertically but along a plane that intersects four faces — for example, cutting from top front edge to bottom back edge — but that’s diagonal.

Let me redefine based on common worksheet answers:

Upon reflection, here’s what is typically expected:

Cube:

- Cut parallel to base → Square
- Cut perpendicular to base → Rectangle (this is incorrect for a cube unless specified otherwise — but perhaps they mean cutting perpendicular to base and parallel to a pair of opposite edges — which still gives square)

I think there’s a mistake in my reasoning. Let me think differently.

Perhaps for the cube:

- "Cut perpendicular to base" means cutting vertically through the center, but if the cut is made such that it goes through two opposite vertical edges, then the cross-section is a rectangle with dimensions: height = edge of cube, width = edge of cube → still square.

Unless... ah! Maybe they consider "perpendicular to base" as cutting in a direction perpendicular to the base but not necessarily aligned with the faces — for example, cutting at an angle that creates a rectangle.

But in reality, for a cube, any vertical cut that is planar and perpendicular to the base will result in a rectangle only if the cut is not parallel to any face — but even then, in a cube, due to symmetry, it might still be a square or rhombus.

I recall now: In many middle-school materials, for a cube:

- Parallel to base → square
- Perpendicular to base → rectangle (they might be thinking of a rectangular prism, but for cube it's square)
- Diagonally → hexagon or rectangle or parallelogram

Actually, let's look for a reliable mental model:

Standard answers for cube cross-sections:

- Horizontal cut (parallel to base) → square
- Vertical cut through center, parallel to faces → square
- Vertical cut through center, diagonal to faces (i.e., cutting from midpoint of one top edge to midpoint of opposite bottom edge) → rectangle? No, in cube, that would be a rectangle with sides: height = edge, width = distance between those points — which is √2 times edge if cutting diagonally across face, but wait.

If you cut the cube with a plane that is perpendicular to the base and passes through two opposite vertical edges (say, front-left and back-right vertical edges), then the cross-section is a rectangle with:
- Height = edge length of cube
- Width = distance between those two edges = diagonal of the base = s√2, where s is edge length.

Oh! That's it!

So if "cut perpendicular to base" means cutting with a plane that is perpendicular to the base and contains two opposite vertical edges (not faces), then the cross-section is a rectangle with dimensions s by s√2.

Similarly, "cut diagonally" might mean cutting with a plane that is not parallel or perpendicular to any face — for example, cutting off a corner — which can give a triangle or hexagon.

But for simplicity, let's use the following commonly accepted answers for this level:

Cube:

- Cut parallel to base → Square
- Cut perpendicular to base → Rectangle (assuming cut through opposite vertical edges)
- Cut diagonally → Hexagon or Parallelogram — but often in basic problems, "diagonally" for cube means cutting from one face to opposite face diagonally, giving a rectangle or parallelogram

I found a better approach: Let's use the most straightforward interpretations taught in schools.

After reviewing typical homework answers:

For Cube:

- Parallel to base: Square
- Perpendicular to base: Rectangle (yes, because if you cut vertically but not aligned with faces, you get a rectangle; but in cube, if cut through center perpendicular to base and parallel to a diagonal of the base, you get a rectangle with sides s and s√2)
- Diagonally: Often means cutting from one vertex to opposite vertex through the interior — which for a cube gives a hexagon if cut properly, but sometimes they expect triangle or parallelogram

This is confusing. Let me search my memory for exact standard answers.

I recall that in many online worksheets, for a cube:

- Cut parallel to base: square
- Cut perpendicular to base: rectangle
- Cut diagonally: hexagon

For pyramid:

- Cut parallel to base: square
- Cut perpendicular to base: triangle
- Cut diagonally: trapezoid

Yes, that seems consistent.

So let's go with that.

Final decision:

Cube:

- Cut parallel to base: Square
- Cut perpendicular to base: Rectangle (plane perpendicular to base and containing two opposite vertical edges)
- Cut diagonally: Hexagon (plane cutting through six faces)

Square Pyramid:

- Cut parallel to base: Square
- Cut perpendicular to base: Triangle (plane through apex and center of base)
- Cut diagonally: Trapezoid (plane cutting two adjacent lateral faces and the base)

Now, to draw them — but since we're describing, we'll state the shapes.

The task is to "draw the cross-section", but since this is text, we describe what to draw.

So for the table:

First row: Cube

- Original shape in 3D: given
- Cut parallel to base: draw a square
- Cut perpendicular to base: draw a rectangle (taller than wide or wider than tall? In cube, if cut through opposite vertical edges, the rectangle has height = s, width = s√2 ≈ 1.414s, so wider than tall)
- Cut diagonally: draw a regular hexagon (if cut symmetrically)

Second row: Square Pyramid

- Cut parallel to base: draw a square (smaller than base)
- Cut perpendicular to base: draw an isosceles triangle
- Cut diagonally: draw a trapezoid (with two parallel sides: one is part of the base, the other is a line segment inside the pyramid)

To confirm, let's verify with logic:

For cube diagonal cut: if you cut with a plane that intersects all six faces, you get a hexagon. Yes, that's standard.

For pyramid diagonal cut: if you cut from one point on a lateral edge to another point on an adjacent lateral edge, not through apex, you get a trapezoid. Yes.

So, final answer description:

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Final Answer:

For the cube:
- Cut parallel to base: Draw a square.
- Cut perpendicular to base: Draw a rectangle (wider than it is tall, with width = diagonal of cube’s face, height = edge of cube).
- Cut diagonally: Draw a regular hexagon (six-sided polygon with equal sides and angles).

For the square pyramid:
- Cut parallel to base: Draw a square (smaller than the base).
- Cut perpendicular to base: Draw an isosceles triangle (base = side of pyramid’s base, height = height of pyramid).
- Cut diagonally: Draw a trapezoid (four-sided figure with one pair of parallel sides; the parallel sides are segments from the base and from inside the pyramid).

Note: Since the instruction is to "draw", in practice, the student should sketch these shapes in the respective boxes. The descriptions above tell exactly what to draw.