Development of Surfaces and Intersection of Solids
What the exam tests
Development of surfaces and intersection of solids tests unfolding prisms, cylinders, pyramids and cones, and locating the line where two solids meet. Expect parallel-line development, radial-line development, triangulation, and intersection curves.
Core rules
- Parallel-line development suits prisms and cylinders: stretch the perimeter into a straight line and keep heights perpendicular to it.
- Radial-line development suits pyramids and cones: use true slant lengths from the apex; for a cone, the sector angle is 360° × (base radius ÷ slant height).
- Triangulation is used when surfaces are not developable or when a transition piece joins different shapes; divide the surface into triangles and find true lengths of sides.
- For intersection, project common points where edges or generators of one solid meet the surface of the other; join them smoothly and decide visibility from the views.
- Mark the seam line and do not add extra allowance unless the question specifies a lap or joint.
Worked example
Develop the lateral surface of a cylinder of diameter 50 mm and height 70 mm. Circumference = π × 50 = 157.1 mm. Draw a rectangle 157.1 mm wide and 70 mm high. Divide the circumference into 12 equal arcs; each arc is 157.1 ÷ 12 = 13.1 mm on the development. Draw vertical generators at these divisions. If a hole or intersection curve is marked on the cylinder, transfer its heights from the elevation to the corresponding generators on the development and join the points.
Traps
Using the diameter instead of the circumference is the most common development error. For cones, use the true slant height, not the vertical height. In intersection problems, do not join arbitrary points; each point must lie on both surfaces. Keep the seam at a convenient generator and label it.