Sections of Solids — NCERT Solutions

Correct Answer: (iv) Rectangle

Justification: A plane passing through the axis of a cylinder (axial section) cuts the cylinder along a rectangle. The width of the rectangle equals the diameter of the base ($= 2r$) and the height equals the height of the cylinder ($= r$, since height $=$ base radius). So the section is a rectangle of dimensions $2r \times r$. Since $2r \neq r$, it is a rectangle (not a square).

Correct Answer: (ii) Sphere

Justification: A sphere is the only solid among the given options where every possible plane section is a circle (since all points on a sphere are equidistant from the centre). For a cylinder or cone, the shape of the section depends on the angle of the cutting plane (it can be a circle, ellipse, parabola, or hyperbola). A circular lamina gives a circle only when cut parallel to its face.

Correct Answer: (iv) Triangle

Justification: A plane passing through the apex of a cone and the centre of the base passes through the axis of the cone. This axial section cuts the cone along two slant edges (generators) and the diameter of the base, producing an isosceles triangle (with base equal to the diameter of the cone's base and height equal to the cone's height).

Correct Answer: (iv) Rectangle

Justification: A plane parallel to the axis of a hexagonal prism cuts through the lateral faces (not the bases). Since the lateral faces of a prism are rectangles, the section through one or more lateral faces parallel to the axis produces a rectangle (the height of the rectangle equals the height of the prism and the width depends on how many faces are cut). The answer given in the textbook answer key is (iv) Rectangle.

Given:

• Hexagonal pyramid: base side $= 30$ mm, axis $= 50$ mm
• Resting on HP with axis vertical
• One base edge parallel to VP
• Section plane: parallel to VP (and perpendicular to HP), at $10$ mm in front of the axis

Concept: A section plane parallel to VP appears as a straight line in the Top View. The sectional Front View shows the true shape of the section (since the section plane is parallel to VP).

Construction Steps:

Step 1 – Draw the Top View:

• Draw a regular hexagon of side $30$ mm with one side parallel to X-Y (since one base edge is parallel to VP).
• Label the corners $1, 2, 3, 4, 5, 6$ (with edge $1$-$2$ parallel to XY) and the apex projection $o$ at the centroid.
• The axis projects as point $o$ in the Top View.

Step 2 – Draw the Front View:

• Project the hexagon corners to get the Front View.
• Draw the Front View: a triangle with base equal to the width of the hexagon in the FV direction and height $= 50$ mm.
• Label the apex $o'$ at the top.

Step 3 – Mark the section plane in Top View:

• The section plane is parallel to VP, so it appears as a vertical line in the Top View (perpendicular to X-Y, i.e., parallel to the Y-direction).
• Draw this line at $10$ mm in front of (below) the axis point $o$ in the Top View.
• Label it A-A.
• Mark the Points of Intersection (POIs) of line A-A with the edges of the hexagon in the Top View: these are points $a, b, c, \ldots$ on the edges that the section plane crosses.

Step 4 – Identify cut edges:

• The section plane at $10$ mm in front of the axis will cut through the front portion of the hexagonal pyramid.
• It will intersect the slant edges and/or base edges that lie in front of the axis.
• Mark all POIs carefully.

Step 5 – Draw the sectional Front View:

• Project all POIs from the Top View up to the Front View using vertical projectors.
• On each slant edge in the Front View, mark the projected POI.
• Join these points in sequence to get the section outline.
• Apply hatching (45° lines) to the section area.
• Show the visible edges of the remaining solid (the portion between the section plane and the observer/VP).

Step 6 – Complete the Top View:

• Show the section plane line A-A in the Top View.
• The Top View shows the base hexagon and the projection of the cut solid.

Result: The sectional Front View shows the true shape of the section (a trapezoid or triangle depending on the exact position) with hatching. The Top View shows the hexagonal base with the section plane line.

Given:

• Square pyramid: base side $= 55$ mm, height $= 70$ mm
• Lying on one of its triangular (lateral) faces on HP
• Axis is parallel to VP
• Section plane: parallel to HP, cuts the axis at its midpoint

Concept: The pyramid is in an inclined position (axis parallel to VP but inclined to HP). We use the change-of-position method. The section plane parallel to HP gives a sectional Top View.

Construction Steps:

Stage 1 – Pyramid with base on HP (initial position):

• Draw the Top View: a square of side $55$ mm with diagonals, apex $o$ at centre.
• Draw the Front View: a triangle, base $55$ mm, height $70$ mm.
• Label base corners $1,2,3,4$ and apex $o$.

Stage 2 – Tilt to lie on a triangular face:

• When the pyramid lies on a triangular face, the slant face is on HP.
• The slant height of the triangular face $= \sqrt{h^2 + (a/2)^2} = \sqrt{70^2 + 27.5^2} = \sqrt{4900 + 756.25} = \sqrt{5656.25} \approx 75.2$ mm
• Tilt the Front View so that one triangular face lies on X-Y (on HP), with the axis parallel to VP (axis appears true length in FV).
• Redraw the Front View in the tilted position.
• Obtain the new Top View by projecting from the tilted Front View.

Step 3 – Mark the section plane:

• The midpoint of the axis is at $35$ mm from the base (or apex).
• In the Front View, mark the midpoint $M$ of the axis.
• Draw the section plane line (A-A) as a horizontal line (parallel to X-Y) through $M$ in the Front View.
• Mark the POIs of A-A with all visible edges in the Front View.

Step 4 – Draw the sectional Top View:

• Project the POIs from the Front View down to the Top View.
• Join the projected points to get the section outline in the Top View.
• Apply hatching (45° lines) to the section area.
• Show the visible edges of the remaining solid.

Result: The Front View shows the pyramid lying on its triangular face with the horizontal section line at the midpoint of the axis. The sectional Top View shows the section outline with hatching.

Given:

• Pentagonal pyramid: base side $= 30$ mm, height $= 50$ mm
• Axis vertical, resting on HP
• One base edge perpendicular to VP (i.e., that edge is parallel to the X-Y line in Top View — actually perpendicular to VP means the edge goes into the VP, so it appears as a point in FV and as a line perpendicular to XY in TV)
• Section plane: horizontal (parallel to HP), at height $= 25$ mm from base

Concept: Horizontal section of a vertical pentagonal pyramid gives a smaller similar pentagon.

Construction Steps:

Step 1 – Draw the Top View:

• Draw a regular pentagon of side $30$ mm with one side perpendicular to X-Y (i.e., one edge is perpendicular to VP, so it appears perpendicular to XY in the Top View).
• Label corners $1,2,3,4,5$ and apex projection $o$.

Step 2 – Draw the Front View:

• Project the pentagon to get the Front View.
• Draw the triangular Front View with height $50$ mm.
• Mark the section plane line A-A at height $25$ mm above X-Y (horizontal chain line).
• The section plane cuts the slant edges at $25$ mm height.

Step 3 – Find the section size:

• By similar triangles: distance from apex $= 50 - 25 = 25$ mm.
• Scale factor $= \dfrac{25}{50} = \dfrac{1}{2}$
• Side of section pentagon $= 30 \times \dfrac{1}{2} = 15$ mm
• Mark the cutting points on the slant edges in the Front View at height $25$ 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 $15$ mm, centred on the axis, similar to and parallel to the base.
• Draw this smaller pentagon in the Top View.
• Apply hatching (45° lines) inside the smaller pentagon.
• Show the base pentagon and visible slant edges.

Result:

• Front View: Triangle (pyramid) with horizontal section line at $25$ mm from base.
• Sectional Top View: Hatched regular pentagon of side $15$ mm (true shape of section) with base pentagon and edges shown.

Given:

• Pentagonal prism: base side $= 25$ mm, height $= 65$ mm
• Resting on base on HP, axis vertical
• One side of base inclined at $30°$ to VP
• Section plane: inclined at $60°$ to HP (and perpendicular to VP), passing through midpoint of axis (at $32.5$ mm from base)

Concept: The section plane inclined to HP appears as an inclined line in the Front View. The section is an oblique cut through the prism. True shape is obtained by auxiliary projection.

Construction Steps:

Step 1 – Draw the Top View:

• Draw a regular pentagon of side $25$ mm with one side inclined at $30°$ to X-Y.
• Label corners $1,2,3,4,5$ (base) and $1',2',3',4',5'$ (top, directly above).
• The Top View shows the pentagon (base and top coincide in projection).

Step 2 – Draw the Front View:

• Project the pentagon to get the Front View.
• The Front View of the prism is a rectangle (or series of vertical lines) of height $65$ mm.
• Draw the section plane line A-A: inclined at $60°$ to X-Y (to HP), passing through the midpoint of the axis at height $32.5$ mm.
• The section plane line cuts the vertical edges of the prism in the Front View at various heights.
• Mark the POIs: $a', b', c', d', e'$ on the respective edges.

Step 3 – Find POIs in Front View:

• The section plane passes through the midpoint of the axis at $32.5$ mm and is inclined at $60°$ to HP.
• Starting from the midpoint on the axis, the section plane line goes up on one side and down on the other at $60°$ to horizontal.
• Mark where this inclined line intersects each vertical edge of the prism in the Front View.

Step 4 – Draw the sectional Top View:

• Project the POIs from the Front View down to the Top View.
• On each edge in the Top View, mark the projected POI.
• Join the POIs in sequence to get the section outline.
• Apply hatching (45° lines) to the section area.
• Show visible edges of the remaining solid.

Step 5 – Draw the true shape of the section:

• Draw an auxiliary reference line $X_1Y_1$ parallel to the section plane line A-A in the Front View.
• Draw projectors from each POI ($a', b', c', d', e'$) perpendicular to $X_1Y_1$.
• Transfer the distances of the corresponding points ($a, b, c, d, e$) from X-Y in the Top View to the respective projectors measured from $X_1Y_1$.
• Join the transferred points $\bar{a}, \bar{b}, \bar{c}, \bar{d}, \bar{e}$ in sequence.
• Apply hatching to the enclosed area.

Result: The true shape of the section is an irregular pentagon (oblique section of a pentagonal prism). The Front View shows the prism with the inclined section line. The sectional Top View shows the apparent section with hatching.

Given:

• Hexagonal prism: base side $= 24$ mm, axis $= 55$ mm
• Resting on an edge of base on HP
• Axis inclined at $60°$ to HP and parallel to VP
• Section plane: inclined at $45°$ to VP (and perpendicular to HP), passing through a point on the axis $25$ mm from the top end

Concept: The prism is in an inclined position (axis inclined to HP). The section plane is inclined to VP, so it appears as an inclined line in the Top View. True shape requires auxiliary projection.

Construction Steps:

Stage 1 – Prism with axis horizontal and parallel to VP:

Step 1 – Initial position (axis vertical):

• Draw Top View: regular hexagon of side $24$ mm.
• Draw Front View: rectangle $\approx 48$ mm wide, $55$ mm tall.

Step 2 – Tilt axis to 60° to HP (axis parallel to VP):

• In the Front View, redraw the prism with its axis inclined at $60°$ to X-Y, with one base edge on X-Y (on HP).
• The axis appears true length in the Front View (since it is parallel to VP).
• Obtain the new Top View by projecting from the tilted Front View.

Step 3 – Mark the section plane:

• The section plane is inclined at $45°$ to VP, so it appears as an inclined line in the Top View.
• Locate the point on the axis $25$ mm from the top end in the Front View. Project this to the Top View to get the corresponding point.
• Draw the section plane line (A-A) through this point at $45°$ to X-Y in the Top View.
• Mark the POIs of A-A with the edges of the prism in the Top View.

Step 4 – Draw the sectional Top View:

• Mark all POIs on the prism edges in the Top View.
• Join the POIs to get the section outline.
• Apply hatching (45° lines) to the section area.
• Show visible edges.

Step 5 – Draw the Front View:

• Project the POIs from the Top View up to the Front View.
• Mark the POIs on the respective edges in the Front View.
• Join them and show the sectional Front View.

Step 6 – True shape of the section:

• Draw auxiliary reference line $X_1Y_1$ parallel to the section plane line A-A in the Top View.
• Draw projectors from each POI perpendicular to $X_1Y_1$.
• Transfer distances from X-Y in the Front View to the projectors measured from $X_1Y_1$.
• Join the transferred points and hatch the area.

Result: The true shape of the section is an irregular hexagon (oblique section through the hexagonal prism). The Front View and sectional Top View show the cut solid with hatching.

Given:

• Cone: base diameter $= 50$ mm (radius $= 25$ mm), axis $= 60$ mm
• Axis parallel to VP and inclined at $45°$ to HP
• Section plane: horizontal (parallel to HP), passing through midpoint of axis

Concept: The cone is inclined with axis at $45°$ to HP. The horizontal section plane cuts the cone obliquely (not parallel to the base), producing an elliptical section.

Construction Steps:

Stage 1 – Cone with axis vertical:

• Draw Top View: circle of diameter $50$ mm with centre $o$.
• Draw Front View: isosceles triangle, base $50$ mm, height $60$ mm.
• Divide the base circle into $12$ equal parts and label generators.

Stage 2 – Tilt axis to 45° to HP (parallel to VP):

• In the Front View, redraw the cone with axis inclined at $45°$ to X-Y.
• The base is now inclined; one point of the base circle touches HP (or the base is positioned as given).
• Obtain the new Top View by projecting from the tilted Front View.
• Draw all generators in both views.

Step 3 – Mark the section plane:

• The midpoint of the axis is at $30$ mm from each end.
• In the Front View, mark the midpoint $M$ of the axis.
• Draw the section plane line A-A as a horizontal line (parallel to X-Y) through $M$.
• Mark the POIs of A-A with all generators in the Front View: $a', b', c', \ldots$

Step 4 – Draw the sectional Top View:

• Project the POIs from the Front View down to the Top View.
• On each generator in the Top View, mark the projected POI.
• Join the POIs with a smooth curve (ellipse) to get the section outline.
• Apply hatching to the section area.

Step 5 – True shape of the section:

• Since the section plane is horizontal (parallel to HP), the sectional Top View IS the true shape.
• The true shape is an ellipse.
• Alternatively, draw auxiliary projection parallel to A-A to confirm.

Result:

• Front View: Inclined cone with horizontal section line at midpoint of axis.
• Sectional Top View: Elliptical section with hatching (true shape since section plane is parallel to HP).

Given:

• Cylinder: diameter $= 40$ mm (radius $= 20$ mm), height $= 65$ mm
• Resting on base on HP, axis vertical
• Section plane: inclined at $60°$ to HP (perpendicular to VP), cutting the axis at $20$ mm from the top (i.e., at height $= 65 - 20 = 45$ mm from base)

Concept: An oblique section plane cuts a vertical cylinder to produce an ellipse. The section plane appears as an inclined line in the Front View.

Construction Steps:

Step 1 – Draw the Top View:

• Draw a circle of diameter $40$ mm.
• Divide it into $12$ equal parts and label the points $1, 2, 3, \ldots, 12$.

Step 2 – Draw the Front View:

• Draw a rectangle $40$ mm wide and $65$ mm tall (the cylinder).
• Project the $12$ division points to the Front View as vertical lines (generators).
• Mark the section plane line A-A: inclined at $60°$ to X-Y, passing through the axis at height $45$ mm from base.
• The section plane line cuts each generator at a different height.
• Mark the POIs: $1', 2', 3', \ldots, 12'$ on the respective generators.

Step 3 – Find heights of POIs:

• The section plane passes through the axis at height $45$ mm and is inclined at $60°$ to HP.
• For each generator at horizontal distance $x$ from the axis, the height of the POI $= 45 + x \cdot \tan 60° = 45 + x\sqrt{3}$.
• Calculate for each of the $12$ generators (using their $x$-coordinates from the Top View).
• Mark these heights in the Front View.

Step 4 – Draw the sectional Top View (sectional plan):

• Project the POIs from the Front View down to the Top View.
• On each generator point in the Top View, mark the projected POI.
• Join the POIs with a smooth elliptical curve.
• Apply hatching (45° lines) to the section area.

Step 5 – True shape of the section:

• Draw auxiliary reference line $X_1Y_1$ parallel to the section plane line A-A in the Front View.
• Draw projectors from each POI ($1', 2', \ldots, 12'$) perpendicular to $X_1Y_1$.
• Transfer the distances of the corresponding points ($1, 2, \ldots, 12$) from X-Y in the Top View to the respective projectors measured from $X_1Y_1$.
• Join the transferred points with a smooth curve.
• Apply hatching.

Result: The true shape of the section is an ellipse. The major axis of the ellipse $= \dfrac{\text{diameter}}{\sin 60°} = \dfrac{40}{\sin 60°} = \dfrac{40}{0.866} \approx 46.2$ mm. The minor axis $=$ diameter $= 40$ mm.

Given:

• Sphere: diameter $= 50$ mm, radius $R = 25$ mm
• Resting on HP (centre is at height $25$ mm above HP)
• Section plane: perpendicular to HP, inclined at $45°$ to VP, at distance $10$ mm from the centre of the sphere

Concept: Any plane cutting a sphere gives a circular section. The radius of the section circle $r = \sqrt{R^2 - d^2}$ where $d$ is the perpendicular distance from the centre to the cutting plane.

Radius of section:
$$r = \sqrt{25^2 - 10^2} = \sqrt{625 - 100} = \sqrt{525} = 5\sqrt{21} \approx 22.9 \text{ mm}$$

Construction Steps:

Step 1 – Draw the Front View:

• Draw the Front View of the sphere: a circle of diameter $50$ mm with centre $o'$ at height $25$ mm above X-Y.

Step 2 – Draw the Top View:

• Draw the Top View of the sphere: a circle of diameter $50$ mm with centre $o$.
• The section plane is perpendicular to HP and inclined at $45°$ to VP, so it appears as an inclined straight line in the Top View.
• Draw the section plane line A-A at $45°$ to X-Y in the Top View, at a perpendicular distance of $10$ mm from the centre $o$.
• The section plane line intersects the base circle (Top View of sphere) at two points. Mark these as $a$ and $b$.

Step 3 – Draw the sectional Front View:

• The section plane is perpendicular to HP, so it appears as a vertical line in the Front View (if it were parallel to VP) or as an inclined line. Since it is inclined at $45°$ to VP and perpendicular to HP, it appears as a vertical line in the Front View (its edge view in FV is vertical).
• Project points $a$ and $b$ from the Top View up to the Front View.
• The section in the Front View is bounded by the chord $a'b'$ (vertical line) and the arc of the sphere.
• The sectional Front View shows the portion of the sphere on the observer's side of the section plane, with the cut face shown as a chord.
• Apply hatching to the cut face (the chord area in FV).

Step 4 – True shape of the section:

• Draw auxiliary reference line $X_1Y_1$ parallel to the section plane line A-A in the Top View.
• The section is a circle of radius $r \approx 22.9$ mm.
• Draw projectors from $a$ and $b$ perpendicular to $X_1Y_1$.
• Transfer the heights of $a'$ and $b'$ from X-Y in the Front View to the projectors.
• The true shape is a circle of radius $\approx 22.9$ mm (diameter $\approx 45.8$ mm).
• Draw this circle and apply hatching.

Result: The true shape of the section is a circle of diameter $\approx 45.8$ mm. The sectional Front View shows the sphere with the cut face (chord) hatched.

Given:

• Triangular pyramid (tetrahedron): base side $= 45$ mm, slant height $= 70$ mm
• Base on HP, axis vertical
• One side of base perpendicular to VP
• Section plane: inclined at $60°$ to VP (perpendicular to HP), intersecting the axis at $35$ mm from base

Note: Slant height $= 70$ mm refers to the slant edge (lateral edge) of the triangular pyramid.

Concept: The section plane is perpendicular to HP and inclined to VP, so it appears as an inclined line in the Top View. The sectional Top View shows the apparent section. True shape requires auxiliary projection.

Construction Steps:

Step 1 – Draw the Top View:

• Draw an equilateral triangle of side $45$ mm with one side perpendicular to X-Y (one edge perpendicular to VP).
• Label base corners $1, 2, 3$ and apex projection $o$ (centroid).

Step 2 – Draw the Front View:

• Project the triangle to get the Front View.
• Calculate the height of the pyramid:
• For equilateral triangle base, centroid to vertex $= \dfrac{45}{\sqrt{3}} = 25.98$ mm
• Height $h = \sqrt{70^2 - 25.98^2} = \sqrt{4900 - 675} = \sqrt{4225} = 65$ mm (approx., using slant edge $= 70$ mm)
• Draw the triangular Front View with height $\approx 65$ mm.

Step 3 – Mark the section plane in Top View:

• The section plane is perpendicular to HP, so it appears as a straight line in the Top View.
• It is inclined at $60°$ to VP, so the section plane line makes $60°$ with X-Y in the Top View.
• The section plane intersects the axis at $35$ mm from the base. Project this point from the Front View to the Top View to locate it on the axis projection $o$.
• Draw the section plane line A-A through this point at $60°$ to X-Y in the Top View.
• Mark the POIs of A-A with the edges of the pyramid in the Top View.

Step 4 – Draw the sectional Top View:

• Mark all POIs on the pyramid edges in the Top View.
• Join the POIs to get the section outline.
• Apply hatching (45° lines) to the section area.
• Show visible edges of the remaining solid.

Step 5 – Draw the Front View:

• Project the POIs from the Top View up to the Front View.
• Mark the POIs on the respective edges in the Front View.
• Show the sectional Front View.

Step 6 – True shape of the section:

• Draw auxiliary reference line $X_1Y_1$ parallel to the section plane line A-A in the Top View.
• Draw projectors from each POI perpendicular to $X_1Y_1$.
• Transfer the heights of the corresponding points from X-Y in the Front View to the projectors measured from $X_1Y_1$.
• Join the transferred points and hatch the area.

Result: The true shape of the section is an irregular triangle or quadrilateral (depending on how many edges the section plane cuts). The Front View and sectional Top View show the cut pyramid with hatching.

Given:

• Cone: base diameter $= 42$ mm (radius $= 21$ mm), axis $= 54$ mm
• Resting on base on HP, axis vertical
• Section plane: vertical (perpendicular to HP), inclined at $60°$ to X-Y (i.e., inclined at $60°$ to VP), at $10$ mm from the axis in the Top View

Concept: The section plane is perpendicular to HP, so it appears as an inclined straight line in the Top View. The sectional Front View shows the apparent section. True shape requires auxiliary projection.

Construction Steps:

Step 1 – Draw the Top View:

• Draw a circle of diameter $42$ mm (the base of the cone) with centre $o$.
• Divide the circle into $12$ equal parts and label the points $1, 2, 3, \ldots, 12$.
• Mark the apex projection at $o$ (centre).

Step 2 – Draw the Front View:

• Draw the Front View of the cone: an isosceles triangle, base $42$ mm, height $54$ mm.
• Draw all $12$ generators in the Front View.

Step 3 – Mark the section plane in Top View:

• The section plane is perpendicular to HP (vertical), inclined at $60°$ to X-Y.
• Draw the section plane line A-A at $60°$ to X-Y in the Top View, at a perpendicular distance of $10$ mm from the centre $o$.
• The line A-A intersects the base circle at two points and also cuts some generators.
• Mark the POIs of A-A with the base circle and generators: $a, b, c, \ldots$

Step 4 – Draw the sectional Front View:

• Project the POIs from the Top View up to the Front View.
• On each generator in the Front View, mark the projected POI at the appropriate height.
• Join the POIs with a smooth curve to get the section outline in the Front View.
• Apply hatching (45° lines) to the section area.
• Show visible edges of the remaining solid.

Step 5 – True shape of the section:

• Draw auxiliary reference line $X_1Y_1$ parallel to the section plane line A-A in the Top View.
• Draw projectors from each POI ($a, b, c, \ldots$) perpendicular to $X_1Y_1$.
• Transfer the heights of the corresponding points from X-Y in the Front View to the projectors measured from $X_1Y_1$.
• Join the transferred points with a smooth curve.
• Apply hatching to the enclosed area.

Nature of the section:

• Since the section plane is $10$ mm from the axis (which is less than the radius $21$ mm) and does not pass through the apex, the section is an ellipse (a conic section where the cutting plane is not parallel to any generator and does not pass through the apex).

Result: The true shape of the section is an ellipse. The Top View shows the base circle with the inclined section plane line. The sectional Front View shows the cone with the elliptical section outline and hatching.