What you'll learn
Many everyday objects — boxes, cans, funnels, ducting — are made by folding flat sheets of material into three-dimensional shapes. A development is the flat pattern that, when folded, forms a solid, and an intersection is the line where two solids meet. For CSEC Technical Drawing you need to understand what developments and intersections are, how to construct the developments of common solids, and how to find the line of intersection between solids. This guide covers developments, the developments of prisms, cylinders and cones, intersections, and their practical uses. By the end you should be able to construct a development and explain how intersections are found.
Key terms and definitions
Development — The flat pattern of a solid's surfaces that folds up to form the solid.
Surface — A face of a solid that appears in its development.
Prism — A solid with a constant cross-section along its length.
Cylinder — A solid with a circular cross-section (a "rolled-up" rectangle).
Cone — A solid that tapers from a circular base to a point (apex).
Intersection — The line formed where the surfaces of two solids meet.
True length — The actual length of a line, needed for an accurate development.
Generator — A line on the surface of a cone or cylinder used in construction.
Core concepts
What a development is
A development (sometimes called a "flat pattern" or "net") is the shape you get if you unfold all the surfaces of a solid onto a flat plane. When the development is cut out and folded up, it forms the solid. Developments are essential in industry for making objects from sheet material — for example, a cardboard box, a metal can, or ventilation ducting all start as a flat development that is cut and folded.
For a development to be accurate, every length in it must be the true length of the corresponding edge on the solid, so that the folded shape fits together correctly.
Developing a prism
A prism has flat faces, so its development is straightforward. To develop a prism, you unfold its faces side by side onto the flat plane, then add the top and bottom (the cross-sections). For example, the development of a rectangular box is a cross-shaped arrangement of six rectangles that fold up into the box. Because a prism's faces are flat and their sizes are known, this is the simplest type of development.
Developing a cylinder
A cylinder can be thought of as a rectangle rolled into a tube, with two circular ends. Its curved surface unrolls into a rectangle:
- The height of the rectangle equals the height of the cylinder.
- The width of the rectangle equals the circumference of the circular base (which is π × diameter).
So the development of a cylinder is a rectangle (the curved surface) plus two circles (the top and bottom). This is how a tin can or a tube is made from flat metal.
Developing a cone
A cone develops into a sector of a circle (a "pie slice"). The development is constructed using the slant height of the cone as the radius of the sector, and the length of the arc equals the circumference of the base. The base circle is added separately. Developing a cone is more complex than a prism or cylinder, because it involves the slant height (a true length) and constructing the correct sector angle. Cones are developed to make items such as funnels and traffic cones.
Intersections of solids
An intersection is the line where the surfaces of two solids meet when they pass through or join one another — for example, where a smaller cylindrical pipe joins a larger one, or where two ducts meet. Finding the line of intersection is important because, to make such a joint from sheet material, you need to know exactly where the surfaces meet so the pieces fit together. The line of intersection is usually found by taking a series of points where lines on one solid meet the other, and joining them into a smooth curve.
Why developments and intersections matter
Developments and intersections are highly practical parts of technical drawing. Sheet-metal workers, packaging designers and engineers use developments to cut flat material that folds into the required shape, and use intersections to make accurate joints between shapes such as pipes and ducts. Understanding that a 3D object can be represented as a flat pattern, and that joints require the line of intersection, is the key idea of this topic.
Finding true lengths
Because a development needs true lengths, an important skill is finding the true length of a line that appears foreshortened in a view. A line that is not parallel to a plane of projection appears shorter than its real length in that view. Its true length can be found by construction — for example, by rotating the line until it is parallel to a plane, or by using the difference in heights and the plan length to construct a right-angled triangle whose hypotenuse is the true length. On a cone, the slant height is a true length that must be found before the development can be drawn accurately. Being able to find true lengths is essential, because using a foreshortened length would make the development the wrong size.
Practical applications in industry
Developments and intersections are among the most practical topics in technical drawing, and it helps to know their real uses. Sheet-metal workers use developments to cut flat metal that folds into ducting, hoppers, funnels and containers; packaging designers use them to create nets for boxes and cartons; and engineers use intersections to work out the shapes needed where pipes and ducts join. In each case, an accurate development or intersection means the material can be cut once and folded or joined to fit correctly, saving material and time. Understanding these applications shows why accuracy and true lengths matter so much — a small error in the flat pattern becomes a real fault in the finished object.
Worked examples
Example 1: Developing a cylinder
Explain the shape of the development of a closed cylinder. The curved surface unrolls into a rectangle whose height equals the cylinder's height and whose width equals the circumference of the base (π × diameter). Two circles are added for the top and bottom. Together these form the development.
Example 2: Why true length matters
Why must true lengths be used when constructing a development? When the flat development is folded up, each edge must match the actual edge of the solid. If a length in the development is not the true length, the folded shape will not fit together correctly, so accurate true lengths are essential.
Example 3: Developing a cone
What shape is the development of a cone's curved surface, and what determines its size? The curved surface of a cone develops into a sector of a circle. Its radius is the slant height of the cone, and the length of its arc equals the circumference of the base. The base circle is added separately.
Example 4: Using an intersection
Why do you need to find the line of intersection when one pipe joins another? To make the joint from sheet material, you need to cut each pipe so its edge matches where the two surfaces meet. The line of intersection shows exactly where the surfaces join, so the pieces fit together accurately.
Common mistakes and how to avoid them
A common error is using the wrong measurement for the width of a cylinder's development. The width of the rectangle equals the circumference (π × diameter), not the diameter. Forgetting to use the circumference gives a pattern that is far too narrow.
Students often confuse the developments of the different solids. A prism develops into flat faces, a cylinder into a rectangle plus two circles, and a cone into a sector of a circle. Learn each shape.
Another mistake is not using true lengths. Every length in a development must be a true length, or the folded shape will not fit — this is especially important with the slant height of a cone.
When developing a cone, remember the radius of the sector is the slant height, not the vertical height of the cone. Using the wrong length gives the wrong development.
Finally, for intersections, remember the line of intersection is found by plotting a series of points where the surfaces meet and joining them; it is usually a curve, not a straight line.
Exam technique for "Descriptive Geometry: Developments and Intersections"
Be ready to construct the developments of prisms, cylinders and cones, using true lengths throughout. For a cylinder, remember the width of the rectangle is the circumference; for a cone, the sector radius is the slant height.
Explain what a development is (a flat pattern that folds into the solid) and its practical use in making objects from sheet material. For intersections, explain that they are the line where two solids meet and how this is needed to make accurate joints.
Work accurately with instruments and show your construction lines. Use precise terms — development, true length, slant height, circumference, intersection — throughout, as accurate construction and correct terms earn the marks.
Quick revision summary
- A development is the flat pattern of a solid's surfaces that folds up into the solid, used to make objects from sheet material.
- Every length in a development must be a true length so the folded shape fits.
- Prism: develops into its flat faces; cylinder: a rectangle (height × circumference) plus two circles; cone: a sector of a circle (radius = slant height, arc = base circumference) plus the base.
- The width of a cylinder's development is the circumference (π × diameter), not the diameter.
- An intersection is the line where two solids meet, found by plotting points and joining them into a curve.
- Intersections are needed to make accurate joints between shapes such as pipes and ducts.