Cross Section
A cross section is the flat, two-dimensional shape you get when you slice through a three-dimensional solid with a plane. For example, slicing a cylinder straight across gives you a circle.
A cross section is the intersection of a three-dimensional solid and a plane. The resulting figure is a two-dimensional shape whose size and form depend on the orientation of the cutting plane relative to the solid. Different angles and positions of the cut through the same solid can produce different cross-sectional shapes.
Worked Example
Problem: A right circular cone has a height of 12 cm and a base radius of 6 cm. A horizontal plane cuts the cone parallel to the base at a height of 8 cm from the base. What is the shape and area of the cross section?
Step 1: Identify the shape. A plane cutting a cone parallel to the base always produces a circle.
Step 2: Find the distance from the apex to the cut. The cone is 12 cm tall and the cut is 8 cm above the base, so the cut is 4 cm below the apex.
Step 3: Use similar triangles. The radius at the cut scales proportionally with the distance from the apex. The full cone has radius 6 at distance 12 from the apex.
Step 4: Calculate the area of the circular cross section.
Answer: The cross section is a circle with radius 2 cm and area cm².
Why It Matters
Cross sections appear throughout engineering, medicine, and design. CT scans and MRI images are cross sections of the human body, giving doctors a view of internal structures without surgery. On the SAT and in geometry courses, you need to identify cross-sectional shapes when planes intersect common solids like prisms, pyramids, cones, and spheres.
Common Mistakes
Mistake: Assuming a cross section of a solid is always the same shape as its base.
Correction: The shape depends on the angle of the cut. For instance, cutting a cylinder at an angle produces an ellipse, not a circle. Only a cut parallel to the base gives the same shape as the base.
Mistake: Confusing cross sections with projections or shadows.
Correction: A cross section is the shape at the actual cut surface inside the solid. A projection is the outline cast onto an external surface, which can differ from the cross section.