Two-Dimensional Cross-Sections of Three-Dimensional Solids - Seventh Grade Mathematics
A cross-section is the two-dimensional geometric region exposed when a three-dimensional solid is sliced by an intersecting plane, varying based on the angle and orientation of the cut.
The Nature of Cross-Sections
Imagine slicing cleanly through a solid block of clay or a loaf of bread with a taut wire or thin blade. The exposed flat surface created along the cut is a two-dimensional cross-section.
SLICING A 3D SOLID WITH A PLANE
Cutting Plane
+-------------------+
/ /
+-------------------+ <-- Intersecting 2D Plane
| /======/ |
| / 2D / | <-- Cross-Sectional Region
| /======/ |
+-------------------+
The geometric shape of a cross-section depends on two variables: the specific geometry of the three-dimensional solid, and the angle and position of the intersecting cutting plane.
Cross-Sections of Prisms and Cylinders
Prisms and cylinders have uniform cross-sections along their longitudinal axes, but cutting them at different angles yields distinct geometric polygons.
+-----------------------+-----------------------------+------------------------+
| Solid Form | Cutting Plane Orientation | Resulting 2D Shape |
+-----------------------+-----------------------------+------------------------+
| Rectangular Prism | Parallel to base | Rectangle (congruent) |
| Rectangular Prism | Perpendicular to base | Rectangle |
| Rectangular Prism | Angled through corners | Triangle or Trapezoid |
| Cylinder | Parallel to circular base | Circle (congruent) |
| Cylinder | Perpendicular to base | Rectangle |
| Cylinder | Angled slice through sides | Ellipse (oval) |
+-----------------------+-----------------------------+------------------------+
When a cylinder is sliced parallel to its base, every cross-section is a circle congruent to the base. However, if you slice a cylinder vertically perpendicular to its base, the exposed face is a rectangle whose width is the chord of the circle and whose height matches the cylinder's height.
Slicing a Cylinder:
Horizontal Slice (Parallel) --> CIRCLE
Vertical Slice (Perpendicular) --> RECTANGLE
Angled Slice (Slanted) --> ELLIPSE
Cross-Sections of Pyramids and Cones
Because pyramids and cones taper smoothly to a single vertex (apex), their cross-sections vary in size depending on their distance from the base.
SQUARE PYRAMID CROSS-SECTIONS:
1. Slice Parallel to Base:
Produces a SQUARE that is smaller than the base (similar square).
2. Slice Perpendicular to Base passing directly through Apex:
Produces an ISOSCELES TRIANGLE.
3. Slice Perpendicular to Base NOT passing through Apex:
Produces an ISOSCELES TRAPEZOID.
When slicing a cone: A horizontal cut parallel to the base produces a circle. A vertical cut passing directly through the apex produces an isosceles triangle. A slanted cut slicing through the curved side without touching the base produces an ellipse. A vertical cut that does not pass through the apex produces a hyperbola curve.
The Invariance of Spherical Cross-Sections
The sphere possesses perfect three-dimensional rotational symmetry. Because every point on the surface of a sphere is equidistant from its center, ANY plane intersecting a sphere produces a circle.
SPHERICAL CROSS-SECTIONS
Any planar cut through a sphere yields a CIRCLE.
- Cut through the Center: GREAT CIRCLE (maximum radius R)
- Cut off-center: SMALL CIRCLE (radius r < R)
If the cutting plane passes directly through the exact center of the sphere, the cross-section is known as a great circle, which has the identical radius and circumference as the sphere itself.
Exercises
Work through each problem carefully in your notebook before reading the provided solutions.
Exercise 1. A cube is sliced by a plane that is parallel to its bottom base. What two-dimensional polygon is formed by this cross-section, and how do its dimensions compare to the faces of the cube?
Exercise 2. A right circular cylinder with a radius of five centimeters and a height of twelve centimeters is sliced by a vertical plane passing directly through the centers of both circular bases. Name the resulting two-dimensional shape and state its exact dimensions.
Exercise 3. A right rectangular pyramid with a square base is sliced by a plane perpendicular to its base that passes directly through the top apex. What two-dimensional shape is formed?
Exercise 4. Describe how a cutting plane can slice through a cube to produce a cross-section in the shape of an equilateral triangle.
Solutions
Solution to Exercise 1. A cube has six congruent square faces. Slicing the cube with a plane parallel to its base produces a square cross-section. The dimensions of this square cross-section are identical (congruent) to the square faces of the cube.
Solution to Exercise 2. Slicing a cylinder vertically through the centers of both circular bases produces a rectangle. The height of the rectangle matches the height of the cylinder, which is twelve centimeters. The width of the rectangle corresponds to the diameter of the circular bases. Since the radius is five centimeters, the diameter is two times five, which is ten centimeters. The resulting cross-section is a rectangle measuring ten centimeters wide by twelve centimeters high.
Solution to Exercise 3. A plane cutting vertically through the top apex of a pyramid and down through the base slices through two opposing sloped triangular faces and splits the base along a straight line. The exposed cross-section is an isosceles triangle. The base of this triangle corresponds to the base dimension of the pyramid, and the height of the triangle corresponds to the true perpendicular height of the pyramid.
Solution to Exercise 4. To produce an equilateral triangle cross-section in a cube, orient the cutting plane so that it cuts through one corner (vertex) of the cube, slicing across the three edges that meet at that corner at equal distances from the vertex. The plane intersects three adjacent square faces, creating three straight line segments of equal length connecting the edges. The resulting exposed interior slice is a perfect equilateral triangle.