Development of Surfaces and Intersection of Solids
This domain tests unfolding surfaces to true shape and drawing the curve where two solids meet. Development questions usually involve prisms, cylinders, pyramids and cones; intersection questions involve pipes, prisms and cones meeting at an angle.
Development methods
- Parallel-line method: prisms and cylinders. Calculate circumference πD or perimeter, divide into equal parts, and step them along a straight line.
- Radial-line method: pyramids and cones. Use true slant lengths from the front view; for a cone, sector radius = slant height L, sector angle θ = 360° × r/L.
- Triangulation: transition pieces and oblique pyramids. Add a seam allowance and label the base points so the development can be folded.
Intersections
The curve of intersection is found where edges or contour lines of one solid meet the surface of the other. Use the line method for prisms and cylinders: draw a series of cutting planes, find where corresponding lines meet, and join the points with a smooth curve. The cutting-plane method works for cones and spheres.
Worked example
Develop a cylinder of diameter 50 mm and height 70 mm. Circumference = π × 50 = 157.1 mm. Draw a rectangle 157.1 mm × 70 mm. Divide the circumference into 12 equal parts of 13.1 mm and transfer these to the top edge. The seam is usually placed on the shortest or least visible generator. The reasoning is that the lateral surface unrolls into a rectangle whose width is the circumference, not the diameter.
Traps
Using 50 mm instead of 157.1 mm is the most common error. Do not forget to divide the base circle accurately; unequal divisions distort the development. In intersections, points must be joined only where both surfaces actually meet, not across gaps. Missing the seam allowance makes the folded model undersized.