Lesson Notes By Weeks and Term v3 - Senior Secondary 1
True Lengths and Surface Development
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
True Lengths and Surface Development
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Subject: Technical Drawings
Class: Senior Secondary 1
Term: 2nd Term
Week: 4
Theme: Development Of Geometrical Soldis
This page supports the lesson note with a companion video and a short classroom-ready summary.
For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.
Development: This refers to the process of unfolding or unrolling the entire surface of a three-dimensional solid onto a single plane (a flat sheet) such that all dimensions remain true. The resulting flat pattern, when cut out and folded, will accurately form the original 3D solid.
Applications of Development: Sheet Metal Fabrication: Essential for laying out patterns for ducts, tanks, pipes, funnels, hoppers, and various industrial components fabricated from sheet metal (e.g., in welding shops producing water storage tanks, grain silos).
Packaging Industry: Designing cartons, boxes, and containers.
This topic focuses on common polyhedra and solids of revolution: Prisms: Solids with two identical and parallel polygonal bases, connected by rectangular or parallelogram faces.
Examples: Square prism, Rectangular prism, Triangular prism.
Cylinders: Solids with two identical and parallel circular bases, connected by a curved surface.
Pyramids: Solids with a polygonal base and triangular faces that meet at a common vertex (apex).
Examples: Square pyramid, Triangular pyramid.
Cones: Solids with a circular base and a curved surface tapering to an apex.
Full Solids: The complete, uncut form of the geometrical solid.
Truncated Solids: A solid that has been cut by a plane, resulting in a new, often non-parallel, top surface. The cut portion is removed. When a line is not parallel to the horizontal plane (HP) or vertical plane (VP), its projected length in the plan or front view is foreshortened. Two common methods for determining true length are: Method 1: Rotation Method (Auxiliary Plane Parallel to the Line) This method involves rotating one end of the line (or the entire line) in one view until it becomes parallel to the projection plane, then projecting it to the other view where its true length will be seen.
Example: Finding the true length of an inclined line AB Given: Front view (a'b') and Plan view (ab) of line A
B. Procedure: In Plan View (ab): Keeping point 'a' as center, rotate 'b' about 'a' until the line (ab1) becomes parallel to the X-Y line (HP). This means ab1 is now parallel to the V
P. Project to Front View: From b1 in the plan view, draw a projector perpendicular to the X-Y line upwards.
Locate True Length: From b' in the front view, draw a horizontal line (parallel to X-Y line) to intersect the projector from b1 at b1'.
True Length: The line segment a'b1' represents the true length of line A
B. Angle of Inclination: The angle that a'b1' makes with the horizontal line from a' is the true angle of inclination of line AB to the horizontal plane.
Method 2: Auxiliary View Method This method involves projecting an auxiliary view onto a plane that is parallel to the inclined line.
Given: Front view (a'b') and Plan view (ab) of line A
B. Procedure: Draw a new reference line (X1-Y1) parallel to the plan view of the line (ab). This new plane is parallel to line AB. From a and b, draw projectors perpendicular to X1-Y
1. Measure the distances of a' and b' from the X-Y line in the front view. Transfer these distances onto the respective projectors from X1-Y1 to locate a'' and b''. The line segment a''b'' represents the true length of line A
B. The choice of method depends on the type of solid: Method 1: Parallel Line Development (for Prisms and Cylinders) This method is used for solids with parallel edges or generators.
Principle: The surface is "unrolled" by extending the base perimeter as a straight line and erecting true length heights from points along this line. Worked
Example: Development of a Full Square Prism (Base 40mm, Height 60mm)
1. Draw Orthographic Views: Plan View: A square (40mm x 40mm). Label corners A, B, C,
D. Front View: A rectangle (40mm width, 60mm height). Project corners from plan view. Label a'b' for the bottom edge and d'c' for the top.
2. Development Setup: Draw a horizontal baseline (stretch-out line). Mark a starting point (A) on the baseline. Measure the perimeter of the base (4 x 40mm = 160mm) along the baseline. Mark points B, C, D, A again. (A-B, B-C, C-D, D-A). At each marked point (A, B, C, D, A), erect vertical lines (generators) perpendicular to the baseline, equal to the true height of the prism (60mm). Connect the top ends of these vertical lines to form the top edge of the development. Attach the two square bases to any two opposite sides of the developed rectangle (e.g., one to A-D, another to A-B). (Often omitted if only lateral surface is required). The resulting rectangle (160mm x 60mm) with the bases attached is the full development. Worked
Example: Development of a Full Cylinder (Diameter 40mm, Height 60mm)
1. Draw Orthographic Views: Plan View: A circle (diameter 40mm). Divide the circle into 8 or 12 equal parts (e.g., 12 parts for better accuracy). Label points 1, 2, 3...
1
2. Front View: A rectangle (width = diameter 40mm, height 60mm). Project points from plan view.
2. Development Setup: Draw a horizontal baseline (stretch-out line). The length of the development will be the circumference of the base: πD = π 40mm ≈ 125.66mm. Mark a starting point (1) on the baseline. Measure the circumference along the baseline. Divide this total length into the same number of equal parts as the plan view circle (e.g., 12 parts). Label these points 1, 2, 3...12, 1 again. At each marked point, erect vertical lines (generators) perpendicular to the baseline, equal to the true height of the cylinder (60mm). Connect the top ends of these vertical lines to form the top edge of the development. Attach the two circular bases (diameter 40mm) to any two opposite sides of the developed rectangle. The resulting rectangle with the bases is the full development. Development of Truncated Solids (Parallel Line) For truncated prisms and cylinders, follow the steps for full solids, but the heights of the generators will vary according to the cutting plane. Worked
Example: Development of a Truncated Cylinder
1. Draw Orthographic Views: Front view of a cylinder cut by an inclined plane, and the plan view (circle).
2. Divide and Project: Divide the plan view circle into 8 or 12 equal parts. Project these points to the base of the front view. Draw generators from the base to the top cutting plane in the front view. Mark where each generator intersects the cutting plane.
3. Development Setup: Draw a baseline representing the circumference (πD) of the cylinder, divided into the same number of parts as the plan. At each point on the baseline, erect a generator. Measure the true heights of each generator from the front view (from the baseline to the cutting plane intersection) and transfer these measurements onto the corresponding generators in the development. Connect the top points on the development with a smooth curve. The base circle is attached as before. The truncated top surface (an ellipse in this case) can also be developed, but it's often not required for SS
1. Method 2: Radial Line Development (for Pyramids and Cones) This method is used for solids where all edges or generators meet at a single apex.
Principle: The development consists of sectors of a circle, with the radius equal plane intersection) and transfer these measurements onto the corresponding generators in the development. Connect the top points on the development with a smooth curve. The base circle is attached as before. The truncated top surface (an ellipse in this case) can also be developed, but it's often not required for SS
1. Method 2: Radial Line Development (for Pyramids and Cones) This method is used for solids where all edges or generators meet at a single apex.
Principle: The development consists of sectors of a circle, with the radius equal to the true length of the slant edge or generator. Worked
Example: Development of a Full Square Pyramid (Base 40mm, Height 60mm)
1. Draw Orthographic Views: Plan View: A square (40mm x 40mm) with diagonals intersecting at the apex (O). Label base corners A, B, C,
D. Front View: A triangle with base 40mm and height 60mm. Project base corners from plan. Label apex O'. Label base corners a'b' and c'd'.
2. Determine True Length of Slant Edge: In the plan view, the diagonal (OA, OB, OC, OD) is foreshortened. To find the true length, use the rotation method: Rotate one of the plan diagonals (e.g., OA) about the apex (O) until it is parallel to the X-Y line (e.g., OA1). Project A1 to the base line in the front view (A1'). Connect O' to A1'. The line O'A1' is the true length of the slant edge. (Alternatively, use the Pythagorean theorem: TL = sqrt(height^2 + (half diagonal of base)^2)).
3. Development Setup: Draw an arc with the true length (O'A1') as the radius, using a point O (representing the apex) as the center. On this arc, starting from a point A, step off the true lengths of the base edges (40mm for a square). Mark points A, B, C, D, A again. Connect these points (A-B, B-C, C-D, D-A) to form the base of the development. Connect the apex O to each of the base points (A, B, C, D, A). Attach the square base (40mm x 40mm) to one of the base edges (e.g., A-D). The resulting figure is the full development. Worked
Example: Development of a Full Cone (Base Diameter 40mm, Height 60mm)
1. Draw Orthographic Views: Plan View: A circle (diameter 40mm).
Front View: A triangle with base 40mm and height 60mm. Label apex O'. Base diameter a'b'.
2. Determine True Length of Slant Generator (TLG): In the front view, the slant edge (O'a' or O'b') is already the true length because the cone's axis is vertical. (Alternatively, use Pythagorean theorem: TLG = sqrt(height^2 + radius^2) = sqrt(60^2 + 20^2) = sqrt(3600 + 400) = sqrt(4000) ≈ 63.25mm).
3. Development Setup: Draw an arc with the true length of the slant generator (TLG) as the radius, using a point O (representing the apex) as the center. Calculate the sector angle (θ) for the development using the formula: θ = (Radius of base / True Length of Generator) 360° θ = (R / TLG) 360° = (20 / 63.25) 360° ≈ 113.8° From apex O, draw a radial line (OA). Measure the calculated angle (θ) from OA and draw another radial line (OB). The sector OAB is the developed lateral surface. Attach the circular base (diameter 40mm) to the arc. Development of Truncated Solids (Radial Line) For truncated pyramids and cones, first determine the true length of all truncated slant edges/generators, then transfer these lengths to the development. Worked
Example: Development of a Truncated Square Pyramid
1. Draw Orthographic Views: Front view of a square pyramid cut by an inclined plane, and the plan view (square).
2. Determine True Lengths of Truncated Edges: First find the true length of the full slant edge (OA') as in the full pyramid example. In the front view, the cutting plane intersects the slant edges at points (e.g., P' on O'A'). * To find the true length of OP, project P' horizontally to the true length line O'A1' (from the full pyramid example). The length from
Sheet Metal Fabrication (Local Welders/Artisans): In Nigerian communities, local welders and fabricators often create water tanks, drainage pipes (culverts), ventilation ducts, funnels, and dustpans. Surface development is the core skill they use, often instinctively or through experience, to cut flat sheets of metal (e.g., galvanized iron, stainless steel) into patterns that can be rolled, bent, and welded to form these objects. Teaching this topic helps formalize their practical knowledge and allows for more complex and precise designs, improving efficiency and reducing material waste. Construction Industry (Roofing and Formwork): For complex roof designs, especially those involving hip and valley roofs, carpenters and roofers implicitly use true length principles to cut roof timbers and sheets (e.g., aluminium roofing sheets) to the correct sizes and angles. Similarly, for casting concrete in non-rectangular shapes (e.g., curved walls, specific culvert designs), formwork carpenters need to develop patterns for their molds. This topic provides the theoretical foundation for these practical tasks, enabling more accurate planning and execution in local building projects. Packaging and Product Design (Small & Medium Enterprises - SMEs): Many Nigerian SMEs produce goods requiring custom packaging, from food items to cosmetics. Understanding surface development allows designers to create cost-effective and aesthetically pleasing carton designs (boxes, pouches, trays) that can be mass-produced from flat cardboard or plastic sheets. This knowledge empowers local entrepreneurs to design innovative and functional packaging solutions for their products, enhancing market appeal.