Sections of Solids

Given/Concept: Sectioning is a drawing technique used in engineering graphics.

Answer: Sectioning is the process of cutting a solid (or an object) with an imaginary cutting plane (called the section plane) in order to reveal its internal details, shape, and construction. The exposed cut surface is shown with hatching lines in the resulting view, which is called a sectional view.

Answer: Two practical examples where sectioning is used:

1. Machine parts / Engineering components – Internal features such as holes, slots, keyways, and recesses of a machine part (e.g., a pulley or a bearing housing) cannot be seen in a normal orthographic view. A sectional view reveals these hidden details clearly. 2.

Answer: The cutting plane is shown as a cutting plane line (a chain line with thick ends and arrows indicating the direction of viewing).

Answer: In sectional view, the portion of the object between the section plane and the observer is assumed to be removed, exposing the internal cut surface.

Answer: The hatching lines are generally drawn inclined to the horizontal at an angle of 45°.

*(They are drawn as thin, equally spaced lines at 45° to the reference line or to the principal edges of the section surface.)*

Note: The actual figures 5.35(a) and (b) are not reproduced in the OCR text. The general procedure to draw sectional views from given orthographic projections is as follows:

Given: Orthographic projections (Front View and Top View) of two solids with section planes A-A and B-B respectively.

Concept/Procedure:

Step 1 – Identify the section plane: Locate the cutting plane line (A-A or B-B) on the given view. Note whether it is horizontal, vertical, or inclined.

Step 2 – Find Points of Intersection (POIs): Mark the points where the section plane cuts the edges/generators of the solid in the view where the section plane appears as a line (edge view).

Step 3 – Project the POIs: Project these points to the adjacent view using standard orthographic projection rules (vertical projectors for Top View, horizontal projectors for Front View).

Step 4 – Join the POIs: Connect the projected points in the correct sequence to obtain the outline of the section.

Step 5 – Apply hatching: Draw hatching lines (thin lines at 45°, equally spaced ~2–3 mm apart) within the section outline to indicate the cut surface.

Step 6 – Show visible edges: Draw all visible edges of the remaining solid (the portion between the section plane and the observer) as continuous thick lines. Hidden lines are generally omitted in sectional views.

Result: The completed sectional view shows the cut surface (hatched) along with the visible outline of the remaining solid.

Given:
- Base diameter of cylinder $= 60$ mm, so radius $= 30$ mm
- Height (axis length) $= 60$ mm
- Cylinder rests on its base on HP with axis vertical
- Section plane: parallel to HP (i.e., parallel to the base), at height $= 40$ mm from HP

Concept: A section plane parallel to HP cuts a vertical cylinder in a circle equal to the base circle. The sectional Top View shows the true shape of the section (a circle), and the Front View shows the cut.

Construction Steps:

Step 1 – Draw the Front View:
- Draw the X-Y line.
- Draw the Front View of the cylinder: a rectangle $60$ mm wide and $60$ mm tall, sitting on X-Y.
- Mark the section plane line (A-A) as a horizontal chain line at height $40$ mm from the base (i.e., $40$ mm above X-Y).
- The section plane cuts the two vertical edges (generators) of the cylinder at points $a'$ and $b'$ at height $40$ mm.
- Hatch the Front View above the section plane line (the cut face visible in FV) — actually, in the Front View, the cut portion above the section plane is removed; the remaining lower portion is shown. The top edge of the remaining solid in FV is the section line A-A.
- Draw hatching on the rectangular cut face visible in FV (the two small rectangles at the top corners are not applicable here since the section is a full horizontal cut).

Step 2 – Draw the sectional Top View:
- Since the section plane is parallel to HP, the sectional Top View shows the true shape of the section.
- The section is a circle of diameter $60$ mm (same as the base).
- Draw a circle of diameter $60$ mm as the sectional Top View.
- Apply hatching (45° lines, equally spaced) throughout the circle to indicate the cut surface.
- Also show the base circle (outline of the cylinder base) as a continuous thin line if required.

Result:
- Front View: Rectangle $60 \times 40$ mm (the lower portion of the cylinder after cutting), with the top edge representing the section plane.
- Sectional Top View: A fully hatched circle of diameter $60$ mm representing the true circular section.

Given:
- Pentagonal pyramid: base side $= 30$ mm, height $= 60$ mm
- Resting on HP with axis vertical
- One edge of base is normal (perpendicular) to VP
- Section plane: horizontal (parallel to HP), at height $= 20$ mm from base

Concept: A horizontal section plane cuts a vertical pyramid parallel to its base. The section is a smaller, similar pentagon. The sectional Top View shows the true shape.

Construction Steps:

Step 1 – Draw the Top View (base):
- Draw a regular pentagon with side $30$ mm such that one side is perpendicular to the XY line (i.e., parallel to VP means the edge is normal to VP, so that edge appears as a point in the side view and as a line perpendicular to XY in the top view).
- Label the base corners $1, 2, 3, 4, 5$ and the apex projection $o$ (centroid of pentagon).

Step 2 – Draw the Front View:
- Project the pentagon corners to get the Front View.
- The Front View of the pyramid is a triangle of base $=$ width of pentagon in FV direction and height $= 60$ mm.
- Draw the section plane line (A-A) as a horizontal chain line at $20$ mm above X-Y.
- The section plane cuts the slant edges at points $1', 2', 3', 4', 5'$ (in FV, only the visible ones are seen).

Step 3 – Find the section in Front View:
- At height $20$ mm, the section plane cuts all five slant edges.
- By similar triangles, the distance of the section from apex $= 60 - 20 = 40$ mm from apex.
- Scale factor $= \dfrac{40}{60} = \dfrac{2}{3}$
- Side of section pentagon $= 30 \times \dfrac{2}{3} = 20$ mm
- In the Front View, mark the cutting points on the visible slant edges at height $20$ mm.

Step 4 – Draw the sectional Top View:
- Project the cutting points from the Front View down to the Top View.
- The section is a regular pentagon of side $20$ mm, similar to the base and centred on the axis.
- Draw this smaller pentagon in the Top View.
- Apply hatching (45° lines) inside the smaller pentagon.
- Show the base pentagon outline as a thin line.
- Show the visible slant edges from the section pentagon to the base corners.

Result:
- Front View: Triangle showing the pyramid with a horizontal section line at $20$ mm from base. The lower portion (frustum) is retained; the top face is the section.
- Sectional Top View: A hatched regular pentagon of side $20$ mm (the true shape of the section) centred on the axis, with the base pentagon and visible edges shown.

Given:
- Cube: side $= 55$ mm
- One edge of the cube rests on HP
- Axis (body diagonal or the axis of the cube) is inclined at $60°$ to HP
- Section plane: vertical, parallel to VP, perpendicular to HP, cuts the axis at its midpoint

Concept: When a cube has its axis inclined to HP, we use the change-of-position method (two-stage projection). The section plane parallel to VP gives a sectional Front View showing the true shape of the section.

Construction Steps:

Stage 1 – Cube with axis vertical (initial position):
- Draw the Top View as a square of side $55$ mm.
- Draw the Front View as a square of side $55$ mm with base on X-Y.
- Label corners: base $1,2,3,4$ and top $5,6,7,8$.

Stage 2 – Tilt the axis to 60° to HP:
- Redraw the Front View so that the axis (vertical line through centre) makes $60°$ with X-Y.
- One edge of the cube rests on HP (on X-Y line).
- Obtain the new Top View by projecting from the tilted Front View.

Step 3 – Draw the section plane:
- The section plane is parallel to VP (appears as a vertical line in the Top View) and cuts the axis at its midpoint.
- In the Top View, draw a vertical line (parallel to Y-axis / perpendicular to X-Y) through the midpoint of the axis projection. This is the section plane line.
- Mark the Points of Intersection (POIs) of this line with the edges of the cube in the Top View: label them $a, b, c, \ldots$

Step 4 – Draw the sectional Front View:
- Project the POIs from the Top View up to the Front View.
- The section plane is parallel to VP, so the sectional Front View shows the true shape of the section.
- Join the projected points in sequence.
- Apply hatching (45° lines) to the section area.
- Show the visible edges of the remaining half of the cube.

Step 5 – Complete the drawing:
- Show the Top View with the section plane line and the cut portion indicated.
- The sectional Front View shows the cross-section (which will be a rectangle or irregular polygon depending on the orientation) with hatching.

Result: The projections show the cube inclined at $60°$ to HP with one edge on HP. The sectional Front View shows the true shape of the vertical section through the midpoint of the axis, with the cut surface hatched at $45°$.

Answer: True shape can be obtained on a plane parallel to the section plane.

Reason: When an auxiliary reference plane is drawn parallel to the section plane, and the section points are projected onto it (with distances transferred from the adjacent view), the resulting shape is the true shape of the section without any distortion.

Answer: The true shape of a section of a sphere cut by any plane (at any angle) is always a circle.

Reason: Any plane cutting a sphere produces a circular cross-section. The size of the circle depends on the distance of the cutting plane from the centre of the sphere. If the plane passes through the centre, the section is a great circle (maximum diameter); otherwise it is a smaller circle.

Answer: When a vertical pentagonal pyramid is cut by a horizontal section plane, the true shape will be a pentagon (a smaller regular pentagon similar to the base).

Reason: A horizontal section plane is parallel to the base of the vertical pyramid. Since the base is a regular pentagon, every horizontal cross-section is also a regular pentagon (smaller in size, depending on the height of the cut), similar to the base.

Correct Answer: (i) Top View

Justification: The projection of any object (or its cut portion) onto the Horizontal Plane (HP) gives the Top View. When the solid is sectioned and the cut portion is projected onto HP, the resulting view is called the Sectional Top View.

Correct Answer: (iii) frustum

Justification: When a horizontal plane cuts a vertical cone parallel to its base (but not through the apex), the portion between the cutting plane and the base is a frustum of a cone — a solid with two parallel circular faces (top and bottom) and a tapering lateral surface.

Correct Answer: (i) When the cutting plane is parallel to HP and perpendicular to VP

Justification: The Top View is the projection onto HP. When the section plane is parallel to HP, the section surface is also parallel to HP. Therefore, its projection onto HP (the sectional Top View) gives the section without any foreshortening — i.e., the true shape. A plane parallel to HP is automatically perpendicular to VP.