Free math worksheets

- 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.

- Trace the four sides of the shaded region rather than relying only on the prism outline. - Compare the directions of those sides with the corresponding edges of the top face. - Decide whether the section is congruent to the top face.

1. The shaded plane meets all four vertical edges at the same relative height and its boundary runs in the same directions as the top-face edges. 2. Therefore, the section is parallel and congruent to the \(6\,\text{cm}\) by \(4\,\text{cm}\) top face. 3. The cross-section is a rectangle with side lengths \(6\,\text{cm}\) and \(4\,\text{cm}\).

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

- 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 \(12\,\text{cm}^2\).

- Trace the boundary of the shaded region and note where it meets prism edges. - Check whether those intersected edges share a prism vertex. - The intersection points become vertices of the cross-section.

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

a) \(3\) edges. b) They all meet at one prism vertex. 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 \(15\,\text{cm}^2\).

- 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.

- 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 the shaded boundary and identify the prism face containing each boundary segment. - For a cutting plane that is not the same plane as a prism face, think about how two planes can intersect. - Check the separate case where the cutting plane lies exactly in a prism face before claiming a universal maximum.

1. In diagram a), the shaded plane meets all six faces of the prism. Its six boundary segments form a hexagon. 2. In diagram b), the shaded plane meets five faces. Its five boundary segments form a pentagon. 3. If the cutting plane does not coincide with a prism face, its intersection with the plane containing any one rectangular face is at most one straight line. Within that face, the cross-section can therefore contribute at most one line segment. 4. A rectangular prism has only six faces, so any non-face-coincident plane section can have at most six sides. A heptagon is therefore impossible. 5. If the cutting plane does coincide with a prism face, the cross-section is that rectangular face itself, which has only four sides. This special case also cannot produce a heptagon.

a) Diagram a): \(6\) faces; diagram b): \(5\) faces. b) Diagram a): hexagon; diagram b): pentagon. c) A non-face-coincident cutting plane meets the plane of any one prism face in at most one line, so that face contributes at most one section side. With only six prism faces, such a section has at most six sides. If the cutting plane coincides with a prism face, the section is a rectangle. Therefore, a heptagonal cross-section is impossible.