- Identify the shape of a face of the solid. - Imagine moving that face inward through the solid without tilting it.

1. The shown solid is a cube, so every face is a square. 2. A plane parallel to a face makes a cross-section with the same shape as that face. 3. Therefore, the cross-section is a square.

A square.

- Compare the direction of the shaded slice with the top face. - What happens to the shape and side lengths when the slice is parallel to that face? - Are the side lengths found from the height of the prism or from its top face?

1. The shaded slice meets all four vertical edges at the same height, so it is parallel to the top face. 2. A slice parallel to the rectangular top face has the same side lengths as that face. 3. The cross-section is a rectangle measuring \(6\,\text{cm}\) by \(4\,\text{cm}\).

A rectangle measuring \(6\,\text{cm}\) by \(4\,\text{cm}\).

- Compare the dashed line's direction with the bottom edge in the side view. - Does the prism's width or depth change at the marked level?

1. The dashed line runs in the same direction as the bottom edge, so the cut is parallel to the rectangular base. 2. A prism keeps the same outline through its height, so the cut surface is a rectangle.

Parallel to the bottom face; the cross-section is a rectangle.

- What is different about the width near the tip and near the base? - What outline does a cut parallel to the base follow?

1. A plane parallel to the square base makes a square cross-section. 2. The pyramid narrows toward the tip, so the square at the dashed level is smaller than the base.

A smaller square.

- Which way does the dashed line run in the side view? - What outline would you see looking straight down at the cylinder?

1. The dashed line runs across the cylinder, so the cut is horizontal and parallel to the circular bases. 2. The flat cross-section is a circle.

A horizontal cut; the cross-section is a circle.

- Compare the three pairs of face dimensions shown on the prism to identify the largest face. - A section parallel to a face has the same dimensions as that face. - Use the rectangle-area formula after identifying the cross-section.

1. The largest face of the prism measures \(8\,\text{cm}\) by \(5\,\text{cm}\). 2. A cross-section parallel to that face is a congruent rectangle with the same dimensions. 3. Its area is \(8\cdot5=40\,\text{cm}^2\).

The cross-section is an \(8\,\text{cm}\) by \(5\,\text{cm}\) rectangle with area \(40\,\text{cm}^2\).

- Focus on the shape and dimensions of a triangular end face. - A plane parallel to that end face creates the same two-dimensional shape. - Use the base and perpendicular height shown for the triangular face.

1. A plane parallel to a triangular end face produces a congruent triangular cross-section. 2. From the diagram, the triangle has base \(6\,\text{cm}\) and height \(4\,\text{cm}\). 3. Its area is \(\frac12\cdot6\cdot4=12\,\text{cm}^2\).

The cross-section is a triangle with area \(12\,\text{cm}^2\).

- Follow the three sides of the shaded region to the prism edges. - Where do those three prism edges meet if extended to their ends? - How many sides does a figure with three vertices have?

1. The shaded plane intersects three prism edges. 2. All three edges meet at the same corner of the prism. 3. Their intersection points with the plane make three vertices, so the cross-section is a triangle.

a) \(3\) edges. b) They all meet at one corner of the prism. c) A triangle.

- Compare the orientation of the shaded region with the base rather than using the outer perspective alone. - Check whether the shaded plane reaches the apex. - Once you identify the plane as base-parallel, connect that orientation to the section shape.

1. The shaded plane cuts all four lateral edges at the same relative height and does not pass through the apex, so it is parallel to the rectangular base. 2. A plane parallel to a pyramid base makes a cross-section similar to the base. 3. Therefore, the section is a smaller rectangle similar to the base.

a) Parallel to the base. b) A rectangle. c) It is a smaller rectangle similar to the base.

- A vertical slice uses the full height of the prism. - Use the stated parallel direction to identify the other cross-section side length. - Once the two side lengths are known, use the rectangle area formula.

1. Because the plane is vertical, one dimension of the cross-section is the prism height, \(3\,\text{cm}\). 2. Because the plane is parallel to the \(5\,\text{cm}\) edges and crosses the prism from one such side to the other, the other dimension is \(5\,\text{cm}\). 3. The cross-section is a rectangle with area \(5\cdot3=15\,\text{cm}^2\).

A \(5\,\text{cm}\) by \(3\,\text{cm}\) rectangle with area \(15\,\text{cm}^2\).

- Which edges join corresponding corners of the two triangular ends? - If the cut stays the same distance from both ends, where does it cross each long edge? - What shape connects those meeting points?

1. The dashed segment joins the midpoints of two long edges. 2. A plane parallel to the triangular ends also meets the third long edge halfway along it. 3. The three points form a triangle the same shape and size as either end face.

It meets the third long edge halfway along it. The new face is a triangle the same shape and size as the prism's ends.

- Are the cuts across or along the prism's length? - What do you know about a prism's width and depth from bottom to top? - Does a cut's height change its base measurements?

1. Both cuts are parallel to the rectangular base. 2. Each cut has sides \(6\,\text{cm}\) and \(3\,\text{cm}\), because the prism has the same width and depth all the way up. 3. Both areas are \(6\cdot3=18\,\text{cm}^2\).

Both are rectangles with area \(18\,\text{cm}^2\). Each horizontal cut is parallel to the base, and the prism's width and depth stay the same at every height.

- Which three edges carry the triangular end from one side of the prism to the other? - For a), where would the missing edges of the flat cut meet the solid? - For b), what happens when the two marked segments are connected along the prism's length?

1. In a), a cut across all three long edges is parallel to the triangular ends. Its section is a triangle. 2. In b), joining the matching segments on the two triangular ends makes a section stretching the prism's full length. It has four sides with right angles, so it is a rectangle.

a) Triangle, parallel to the end faces. b) Rectangle, along the prism's length.

- Which cut stays at one height above the base? - Which cut passes through the tip, where the sides meet? - Think about how many boundary edges each plane will meet.

1. In a), the cut is parallel to the square base, so its cross-section is a smaller square. 2. In b), the cut goes through the tip and meets the base along a straight line. It forms a triangular cross-section.

a) A smaller square: the cut is parallel to the square base. b) A triangle: the cut passes through the tip and meets the base along a segment.

- Think about where each side of a section lies on the cube's surface. - Could a single straight side run through the interiors of two different faces? - How many boundary pieces does touching four faces require?

1. The plane meets each of the four cube faces in a straight segment of the cut's boundary. 2. The four face segments make at least four sides, so the cross-section cannot be a triangle. 3. A quadrilateral is possible, for example with a vertical plane passing through the middle of the cube.

No. It must have at least four sides; a quadrilateral is possible.

- Trace where the cutting plane meets each end of the prism. - One cross-section dimension comes from the end-face diagonal, and the other comes from the prism's length. - Identify the shapes of the two parallel bases of each resulting solid.

1. The cut follows a line segment across each end and extends straight along the length of the prism, so the cross-section is a rectangle. 2. One side of the cross-section is the \(5\,\text{cm}\) diagonal of an end. The other side is the \(6\,\text{cm}\) length of the prism, so its dimensions are \(5\,\text{cm} \times 6\,\text{cm}\). 3. Its area is \(5 \cdot 6=30\,\text{cm}^2\). 4. Each end of a resulting piece is a triangle, so each piece is a triangular prism.

a) A rectangle b) \(5\,\text{cm} \times 6\,\text{cm}\) c) \(30\,\text{cm}^2\) d) A triangular prism

- For the parallel cut, compare the slicing plane with the pyramid’s base. - For the second cut, trace where the plane meets the base and then follow it to the apex. - Count the boundary segments of each resulting two-dimensional section.

1. The shown solid is a right rectangular pyramid. 2. In a), a plane parallel to the rectangular base produces a smaller rectangle. 3. In b), the slicing plane meets the solid along the base segment and along two segments that run from the segment’s endpoints to the apex. Those three segments form a triangle. 4. A plane parallel to the base preserves the base’s four-sided shape, while a plane through the apex makes the section narrow to one vertex at the apex.

a) A rectangle. b) A triangle. The parallel cut keeps the base shape, while the cut through the apex has the apex as one vertex of the section.

- Trace the slicing plane on the top, bottom, and vertical faces of the prism. - The vertical sides of the section have the same length as the prism’s height. - Compare a face diagonal with the side lengths of that rectangular face; an exact diagonal calculation is not needed.

1. The plane meets the top and bottom faces along matching diagonals and connects their endpoints vertically, so the cross-section is a rectangle. 2. The vertical sides span the full height of the prism, which is \(3\,\text{cm}\). 3. Each other side is a diagonal of the \(6\,\text{cm}\) by \(4\,\text{cm}\) top or bottom face. 4. A rectangle’s diagonal is longer than either side, so each of those cross-section sides is longer than \(6\,\text{cm}\).

a) A rectangle. b) \(3\,\text{cm}\) c) Longer than \(6\,\text{cm}\), because each is a diagonal of a \(6\,\text{cm}\) by \(4\,\text{cm}\) rectangle.

- Follow the shaded boundary where it crosses the top and bottom faces. - Identify which two sides of the section are vertical prism edges. - Compare a face diagonal with the longer side of that rectangular face.

1. The shaded boundary crosses each of the top and bottom faces from one corner to the opposite corner, so it follows a diagonal on each face. 2. Those two matching diagonals are joined by two vertical prism edges, forming a rectangle. 3. The vertical sides span the full prism height, so each is \(3\,\text{cm}\). 4. Each other side is a diagonal of a \(6\,\text{cm}\) by \(4\,\text{cm}\) rectangle, so it is longer than \(6\,\text{cm}\).

a) A diagonal. b) A rectangle. c) \(3\,\text{cm}\) d) Longer than \(6\,\text{cm}\).

- For Plane P, count the apex as one cross-section vertex. - For Plane Q, trace where the plane enters and leaves the base and lateral faces after it misses the apex. - Compare the number of boundary intersection points for the two planes.

1. Plane P crosses the base in a segment and also passes through the apex. Its two remaining boundary segments run from the endpoints of the base segment to the apex, so the cross-section is a triangle. 2. Plane Q misses the apex. It crosses the base in one segment and a lateral face in a shorter parallel segment, while the other two sides lie in two other lateral faces. 3. Those four boundary segments form a trapezoid. 4. Shifting the plane away from the apex replaces the single apex vertex with two separate intersection points on lateral edges, increasing the section from three vertices to four.

a) A triangle. b) A trapezoid. c) Missing the apex replaces one apex vertex with two separate lateral-edge intersection points, so the cross-section has four vertices instead of three.

- Work backward from the fact that a triangle has three vertices. - Which three prism edges naturally meet in one local region? - A small corner cut can avoid every other edge of the prism.

1. Choose one vertex of the rectangular prism. 2. Position the plane close to that vertex so it intersects each of the three edges that meet there before reaching any other edge. 3. The three intersection points are the vertices of the cross-section, so the section is a triangle.

Slice off one corner: the plane should intersect the three edges that meet at a single vertex, producing three intersection points and therefore a triangular cross-section.

- A cut through the cylinder at a fixed height follows what kind of outline? - In the vertical cut, how wide is the cylinder all the way across its center? - Which area formulas match the two flat regions?

1. Cut A is parallel to the circular base, so it is a circle with area \(\pi\cdot3^2=9\pi\,\text{cm}^2\approx28.26\,\text{cm}^2\). 2. Cut B runs through the cylinder's axis, so its section is a rectangle \(6\,\text{cm}\) wide and \(8\,\text{cm}\) tall. 3. Its area is \(6\cdot8=48\,\text{cm}^2\), larger than cut A.

A: Circle, \(9\pi\,\text{cm}^2\approx28.26\,\text{cm}^2\). B: Rectangle, \(48\,\text{cm}^2\). B is larger.

- Imagine a plane that cuts off only one corner of the solid. - Each side of the cross-section lies on one face of the solid. - Use the total number of faces to reason about the largest possible number of cross-section sides.

1. The shown solid is a rectangular prism. 2. For a), place a plane near one vertex so that it intersects the three edges that meet at that vertex. The plane cuts off that corner, and its intersection with the prism is a triangle. 3. For b), each side of a plane cross-section lies in one face of the prism. Because a rectangular prism has six faces and the intersection with any one rectangular face is at most one line segment, a cross-section can have at most six sides. 4. Therefore, a seven-sided cross-section is impossible.

a) Slice off one corner with a plane that intersects the three edges meeting at that vertex; the cross-section is a triangle. b) No. A rectangular prism has only six faces, so a plane cross-section can have at most six sides.

- Follow each shaded boundary segment and find which prism face it lies on. - How many faces does the prism have altogether? - Can a straight, flat slice cross one rectangular face in two different straight pieces?

1. In diagram a), the shaded section meets six prism faces. It has six sides, so it is a hexagon. 2. In diagram b), the section meets five prism faces. It has five sides, so it is a pentagon. 3. When a flat cut passes through a rectangular face rather than lying on it, the cut leaves at most one straight boundary segment on that face. 4. There are six faces, so there can be at most six such boundary segments. A seven-sided section is impossible. 5. If the cut lies along one whole prism face, the section is a rectangle, with four sides, not seven.

a) Diagram a): \(6\) faces; diagram b): \(5\) faces. b) Diagram a): hexagon; diagram b): pentagon. c) Each crossed rectangular face contributes at most one straight section side, and there are only six faces. A cut along a whole face gives a rectangle. Neither makes seven sides.