Solids
276. The Slice That Never Changes · Uniform cross-sections and volume shortcuts
The cross-section area remains constant as you slide the cut along the prism.
A prism has a uniform cross-section along its length, so its Volume = cross-sectional area × length. Sliding the cut along the prism never changes the cross-section, which is why this shortcut works.
What this lesson covers
Try to break it
Drag the red Cut line along the prism. The purple cross-section keeps exactly the same shape and size at every position — that's what "uniform cross-section" means. Try to find a slice that's a different shape or size; impossible for a prism.
How you build it
Construct a triangular prism and explore its constant cross-section.
- Triangle tool: click three points to draw the cross-section at the front of the prism.
- Triangle tool: draw an identical triangle shifted to the right — the same cross-section appearing at the other end.
- Segment tool: join the top corner of the front triangle to the top corner of the back triangle.
- Segment tool: join the next matching pair of corners (front to back).
- Segment tool: join the last matching pair of corners. The three depth edges complete the prism — every slice across them is the same triangle, so Volume = cross-section area × length.
The proof, step by step
Prove that the cross-section of a prism stays the same all along its length.
- A uniform solid has the same cross-section shape and area at every point along its length.
- Volume = (Area of cross-section) × (Length of the solid).
- Lateral Surface Area = (Perimeter of cross-section) × (Length).
Worked example
A prism has a uniform triangular cross-section with area 24 cm². If its length is 15 cm, what is its volume?
Volume = Area of cross-section × Length = 24 cm² × 15 cm = 360 cm³.
- 144 cm³
- 360 cm³ — correct
- 48 cm³
- 225 cm³