Overview
Introduction: "Visualising Solid Shapes" helps students move from flat (2D) figures to three-dimensional (3D) objects — identifying, sketching and reasoning about common solids such as cubes, cuboids, cones, cylinders, spheres, prisms and pyramids. Importance: Developing spatial visualisation builds geometric intuition used in measurement, modelling real objects and solving everyday problems (packing, drawing, making nets). Key themes: recognition of faces, edges and vertices; interpreting and drawing nets; viewing solids from different directions (top, front, side); counting visible and hidden faces; and basic relations among faces, edges and vertices. What the student will learn: how to identify solids from different views and from nets, how to construct and use nets for simple solids, how to count and classify faces/edges/vertices (including the basic Euler relation for simple polyhedra), and how to visualise solids made of cubes and combine reasoning with clear sketches to solve problems.
Learning Objectives
- Define polyhedron, prism, pyramid, cylinder, cone and sphere and give one classroom example of each
- Identify faces, edges and vertices of common solids (cube, cuboid, cone, cylinder, prism, pyramid) and count them accurately
- Distinguish between polyhedra and solids of revolution with two clear examples and reasons
- Classify given solids as prisms, pyramids or other types by examining their faces and bases
- Sketch three-dimensional views of a cube, cuboid, cylinder and cone from given front, top and side projections
- Draw nets for a cube, cuboid, triangular prism and regular tetrahedron and use the nets to visualise the corresponding solid
- Construct physical/models (paper/card) from a given net and verify the number of faces, edges and vertices
- Determine the number of faces, edges and vertices for composite solids formed by joining two or more simple solids
Topics in this chapter
7 topics · tap a topic title to jump straight to it.
Introduction to 3‑D Shapes (Solids)
3‑D shapes (solids) are figures that occupy space and have three dimensions — length, breadth (width) and height (depth). Unlike 2‑D plane figures (which have area only), solids have volume in addition to surface area. Common features used to describe solids are:
- Face: a flat or curved surface of a solid.
- Edge: the line where two faces meet (for polyhedra).
- Vertex (vertices): a corner point where edges meet.
- Base: a face used as a reference (for prisms, pyramids, cones).
- Axis: a line around which a figure is symmetric (for solids of revolution like cylinders, cones, spheres).
Solids are broadly of two types:
- Polyhedra — solids with flat polygonal faces (e.g., cube, cuboid, prism, pyramid). These are made of faces, edges and vertices.
- Curved‑surface solids — solids having curved faces (e.g., cylinder, cone, sphere).
Other important ideas:
- Prism: has two congruent parallel polygonal bases and rectangular lateral faces. Volume = (area of base) × height.
- Pyramid: has a polygonal base and triangular lateral faces meeting at an apex. Volume = (1/3) × (area of base) × height.
- Right vs Oblique: in a right solid, the axis is perpendicular to the base (e.g., right circular cylinder); in an oblique solid it is not.
- Nets: a 2‑D layout of all faces of a solid; useful to find surface area and to visualise assembly.
- Cross‑sections: slices of solids by planes (e.g., slicing a cone horizontally gives circles; vertical slice of a cylinder gives a rectangle).
- Euler's formula for convex polyhedra: V − E + F = 2 (V = vertices, E = edges, F = faces).
Visualising solids involves drawing 3‑D sketches (isometric/perspective), nets, and orthographic projections (front, top, side views). Understanding these helps in computing surface areas, volumes and recognising shapes in real life.
- Cube: a dice, a Rubik's cube.
- Cuboid (rectangular prism): a book, a brick, a shoebox.
- Cylinder: a water tumbler, a can, a pipe section.
- Cone: an ice‑cream cone, a party hat, traffic cone.
- Sphere: a ball (cricket/football/tennis), a marble, the Earth (approx.).
- Triangular prism: a tent with triangular cross‑section, roof truss members.
- Units: surface area in square units (unit^2); volume in cubic units (unit^3).
- Cube (side = a): Volume = a^3; Total surface area (TSA) = 6a^2; Lateral surface area (LSA) = 4a^2.
- Cuboid (l × b × h): Volume = l × b × h; TSA = 2(lb + bh + hl); LSA (around sides) = 2h(l + b).
- Prism (base area = B, height = h): Volume = B × h; TSA = 2B + perimeter_of_base × h.
- Cylinder (radius r, height h): Volume = πr^2h; Curved surface area (CSA) = 2πrh; TSA = 2πr(h + r).
- Cone (radius r, height h, slant height s = √(r^2 + h^2)): Volume = (1/3)πr^2h; CSA = πrs; TSA = πr(r + s).
Faces, Edges and Vertices
Definitions
In 3D geometry, a solid is described by three basic elements:
Face – a flat (planar) or curved surface that bounds a solid. In polyhedra (like cubes, prisms and pyramids) faces are flat polygons. In solids of revolution (cylinders, cones) the circular bases are called faces and the curved part is called a curved surface.
Edge – the line segment (or curve) where two faces meet. In polyhedra edges are straight line segments. For curved solids the meeting of a curved surface and a base is a curved edge (often thought of as a boundary circle).
Vertex (Vertices) – a point where edges meet. For polyhedra a vertex is the corner point of faces. A cone has one vertex (the apex); many curved solids have no vertices.
Key ideas and distinctions
- Polyhedron vs curved solids: Euler's formula (V − E + F = 2) applies to convex polyhedra (solids made only of flat polygonal faces). It does not apply in the same way to curved solids unless you treat curved boundaries specially.
- When counting faces, include only flat polygonal faces for polyhedra. For solids like a cylinder, you typically say it has two circular faces and one curved surface; for a cone, one circular face and one curved surface.
- Edges are where two faces meet; vertices are where edges meet. If there are no straight-line edges meeting at a point, there is no vertex in the polyhedral sense.
Counting strategy
To count F, E and V for a solid: sketch or build a model, mark each face, trace every line where faces meet to count edges, and identify the corner points where edges meet to count vertices. For prisms and pyramids you can use formulas based on the number of sides of the base (see formulas section).
Common examples (what to expect)
Cube/cuboid: all faces are rectangles (or squares) — F = 6, E = 12, V = 8. Tetrahedron (triangular pyramid): F = 4, E = 6, V = 4. Triangular prism: F = 5, E = 9, V = 6. Square pyramid: F = 5, E = 8, V = 5. Cylinder: 2 circular faces + 1 curved surface; generally treated as having no vertices and no straight edges in elementary discussions. Cone: 1 circular face + 1 curved surface; 1 vertex (apex).
Physical tip
Use cardboard nets or transparent models and color each face differently to make faces, edges and vertices easy to see.
- Cube: faces = 6, edges = 12, vertices = 8.
- Cuboid: faces = 6, edges = 12, vertices = 8 (same counts as cube).
- Regular tetrahedron (triangular pyramid): faces = 4, edges = 6, vertices = 4.
- Triangular prism (base = triangle): faces = 5, edges = 9, vertices = 6.
- Square pyramid (base = square): faces = 5, edges = 8, vertices = 5.
- Cylinder (elementary view): 2 circular faces + 1 curved surface; no vertices and no straight edges (bases meet curved surface along circles).
- Euler's characteristic for convex polyhedra: V - E + F = 2.
- Prism with n-sided base: Faces F = n + 2, Edges E = 3n, Vertices V = 2n. (Example: triangular prism n=3 → F=5, E=9, V=6.)
- Pyramid with n-sided base: Faces F = n + 1, Edges E = 2n, Vertices V = n + 1. (Example: square pyramid n=4 → F=5, E=8, V=5.)
- For common solids: Cube/cuboid: F=6, E=12, V=8; Tetrahedron: F=4, E=6, V=4.
Common Solids and Their Properties
Overview
Common solids are three-dimensional shapes encountered in daily life and studied in Class 8 under 'Visualising Solid Shapes'. The main solids are cube, cuboid, cylinder, cone and sphere. Each solid is described by its faces (flat or curved), edges, vertices, base(s), height and dimensions (length, breadth, radius, slant height). Understanding these properties helps compute surface areas, curved/ lateral areas and volumes.
Basic terms
- Face: A flat or curved surface of a solid (e.g. square face of a cube, curved surface of a cylinder).
- Edge: Line segment where two faces meet (no edges on a sphere).
- Vertex (vertices): Point where edges meet (a cone has 1 vertex, a sphere has none).
- Base: The face on which a solid may stand (circular base of cone/cylinder).
- Height (h): Perpendicular distance between base and top or between two bases.
- Slant height (l): Length of sloping side of a cone (l = sqrt(r^2 + h^2)).
- π (pi): Use π ≈ 22/7 or 3.14 in calculations.
Properties of common solids
Cube
All faces are congruent squares. If side = a: faces = 6, edges = 12, vertices = 8.
Cuboid (rectangular box)
Faces are rectangles. If dimensions = l, b, h: faces = 6, edges = 12, vertices = 8.
Cylinder
Has two congruent circular bases and one curved surface. If radius = r and height = h: faces = 3 (2 flat + 1 curved), edges = 0 (no straight edges), vertices = 0.
Cone
Has one circular base and one curved surface tapering to a vertex. If radius = r, height = h and slant height = l: faces = 2 (1 flat + 1 curved), edges = 0, vertices = 1.
Sphere
Perfectly round 3D shape; every point on the surface is equidistant from centre. faces = 1 (a single curved surface), edges = 0, vertices = 0.
Nets and cross-sections
A net is a 2D pattern obtained by 'cutting' and unfolding the surface of a solid. Cross-sections are shapes obtained by slicing a solid with a plane (e.g., a vertical slice of a cylinder gives a rectangle).
Units and conversion
Surface areas use square units (cm², m²). Volumes use cubic units (cm³, m³). Convert lengths before applying formulas.
- Cube: A dice or a Rubik's cube (side = a).
- Cuboid: A brick, a shoe box or cereal box (l × b × h).
- Cylinder: A soup can or a drum (radius r, height h).
- Cone: An ice-cream cone or traffic cone (radius r, height h, slant l).
- Sphere: A cricket ball, a marble or Earth (radius r).
- Net use: Paper box templates (cuboid net), cone made from a sector (cone net).
- Cube (side a): Volume = a^3; Total Surface Area (TSA) = 6a^2; Lateral Surface Area (LSA) = 4a^2.
- Cuboid (l × b × h): Volume = l × b × h; TSA = 2(lb + bh + hl); LSA (around height) = 2h(l + b).
- Cylinder (radius r, height h): Volume = π r^2 h; Curved Surface Area (CSA) = 2π r h; TSA = 2π r (r + h).
- Cone (radius r, height h, slant l): Volume = (1/3) π r^2 h; CSA (lateral) = π r l; TSA = π r (r + l); where l = sqrt(r^2 + h^2).
- Sphere (radius r): Volume = (4/3) π r^3; TSA = 4π r^2.
- Note: Use π ≈ 22/7 or 3.14 as required.
Nets (Developments) of Solids
What is a net (development)? A net (or development) of a solid is a flat 2‑D pattern made by cutting along certain edges of the solid and unfolding its faces so they lie in one plane. When folded along the edges of the net, it reconstructs the original 3‑D solid.
Why nets are useful: Nets help us visualize the faces of a solid, design packaging, and calculate surface area: the total surface area of a solid equals the sum of the areas of the faces shown in its net.
How to draw or use a net:
- Identify all faces of the solid (shape and size).
- Decide which edges to keep as fold lines and which to cut so the faces can lie flat without overlap.
- Arrange the faces in the plane so that adjoining faces in the solid are adjacent in the net.
- Label corresponding edges and faces to help fold back into the solid.
Common solids and their nets (typical nets used in class 8):
- Cube: 6 equal squares. There are 11 distinct nets of a cube.
- Cuboid (rectangular box): 6 rectangles (three pairs of equal rectangles).
- Cylinder: one rectangle (lateral surface) and two congruent circles (bases).
- Right triangular prism: 2 congruent triangles (ends) and 3 rectangles (lateral faces).
- Square pyramid: 1 square base and 4 congruent triangular faces attached to its sides.
- Right circular cone: one circular base and one sector of a circle (lateral surface).
Important note: A sphere has no flat net that can be folded into a perfect sphere without distortion (you can approximate it by many small patches, but not an exact single planar net).
Using nets to find surface area: Calculate the area of each face shown in the net and add them up. This is often easier than trying to sum face areas directly on the 3‑D figure.
Tips for students:
- Use grid paper to draw nets to scale; mark equal edges.
- Color matching faces that will meet when folded to avoid mistakes.
- Practice folding paper nets to build physical models — this strengthens spatial visualization.
- A cereal box is a cuboid. Its net is two congruent rectangles (top and bottom) and four side rectangles attached around one of the side rectangles.
- A die is a cube. A common cube net is a T‑shaped arrangement of six squares (four in a row with one square attached above the second and one below the second).
- A soda can is a cylinder. Its net is a rectangle of width equal to the circumference of the base and height equal to the can's height, plus two circles for the top and bottom.
- An ice‑cream cone (plain single scoop) approximates a right circular cone: the net is a sector (lateral surface) plus a circle (base if present).
- A triangular tent can be modeled as a triangular prism: the net has two triangular ends and three rectangles for the sides.
- Area of rectangle = length × breadth (useful for the faces of cuboids and prisms)
- Area of square = a^2 (for cube faces)
- Total surface area of a cuboid = 2(lb + bh + lh) where l = length, b = breadth, h = height
- Total surface area of a cube = 6a^2 where a is the edge length
- Lateral surface area of a cylinder = 2πrh (rectangle of width 2πr and height h)
- Total surface area of a cylinder = 2πr(h + r) = 2πrh + 2πr^2
Cross‑sections and Slices
What is a cross‑section / slice?
A cross‑section (or slice) of a solid is the shape obtained when a plane cuts through the solid. The shape of the cross‑section depends on (a) the type of solid and (b) the orientation and position of the cutting plane.
Basic idea and observations
- If the cutting plane is parallel to a face of a prism or cylinder, the cross‑section is a shape congruent to that face (e.g., rectangle for a cuboid, circle for a circular cylinder).
- If the plane is perpendicular to the axis of a cylinder or cone and cuts across, the cross‑section is a circle (for right circular solids).
- A sphere cut by any plane always gives a circle (or a point if the plane is tangent).
- An oblique plane (not parallel or perpendicular to main directions) can produce different shapes: ellipses, triangles, rectangles, hexagons, etc., depending on the solid and how the plane passes through it.
How to decide the cross‑section shape
- Identify the solid (cube, cuboid, cylinder, cone, sphere, prism,...).
- Decide the plane orientation: horizontal (parallel to base), vertical (perpendicular to base or along an axis), or oblique.
- Visualise intersection curve(s) between plane and faces or curved surface; the boundary of that intersection is the cross‑section.
Useful examples to remember: horizontal cut of a cylinder → circle; vertical cut through axis of cylinder → rectangle; any cut of a sphere → circle; plane through apex of cone and base edge → triangle.
Practical tip: physically slicing a loaf of bread, a cake or an apple helps see how different oriented cuts produce different cross sections.
- Slicing a cylindrical cake horizontally (plane parallel to base) gives a circular cross‑section — every horizontal slice is a circle of the same radius as the cake.
- Cutting the same cake vertically along a plane through the axis gives a rectangular cross‑section (height = cake height, width = 2 × radius).
- A slice of an apple (approximate sphere) by any plane gives a circular face. If the plane is at distance d from the center of the apple (radius R), the slice is a circle with radius √(R² − d²).
- A cone cut by a plane parallel to its base produces a smaller circle. A plane passing through the apex and cutting the base edge gives a triangular cross‑section.
- A cuboid cut by a plane parallel to a face gives a rectangle (or square). A diagonal plane passing through three appropriate vertices can produce a triangular cross‑section.
- A triangular prism sliced by a plane perpendicular to its axis gives the triangular base; a plane parallel to the axis and to one side can give a rectangle (or parallelogram if oblique).
- Area of a circle: A = πr². (Used for circular cross‑sections.)
- Perimeter (circumference) of circle: C = 2πr.
- Area of rectangle: A = length × breadth.
- Area of triangle: A = 1/2 × base × height.
- Area of ellipse (for oblique cuts that give an ellipse): A = πab, where a and b are semi‑axes.
- Sphere cross‑section radius: If a sphere has radius R and a plane is at distance d from the sphere centre, the cross‑section is a circle of radius r = √(R² − d²).
Views of 3‑D Shapes (Orthographic/Front, Top, Side Views)
Orthographic views (also called orthographic projections) show a 3‑dimensional object as 2‑dimensional views on principal planes: the front (elevation), the top (plan) and the side (profile). In orthographic projection, projection lines are parallel and perpendicular to the projection plane, so there is no perspective distortion. Each view shows only two of the three dimensions of the object.
Key ideas:
- Front view (elevation): shows height and width.
- Top view (plan): shows length and width.
- Side view (usually right side/profile): shows height and length.
Coordinate/plane mapping (useful for drawing): if we take axes x (length, left↔right), y (width/depth, front↔back) and z (height, up↔down), then
- Top view = projection onto the xy‑plane (shows x and y)
- Front view = projection onto the xz‑plane (shows x and z)
- Side view = projection onto the yz‑plane (shows y and z)
How to draw orthographic views (step‑by‑step):
- Decide orientation of the object (which face is front, which side is right, and what is top).
- Project key points perpendicularly onto each plane (use vertical lines for front↔top alignment and horizontal lines for front↔side alignment).
- Transfer measured dimensions (length, width, height) between views — dimensions stay the same in corresponding directions.
- Include hidden edges using dashed lines (when an edge is not visible in that view but exists in 3‑D).
- Label dimensions and use a consistent scale if the object is large or small.
Reconstructing 3‑D shape from views: If the three principal views are consistent and correctly aligned (same scale and orientation), you can recreate the 3‑D shape by intersecting corresponding features from each view.
Practical notes:
- Always maintain alignment: top is above the front view, side is to the right (common convention) of the front view in drawings.
- Use dashed lines for hidden features and solid lines for visible edges.
- Orthographic views are used in engineering drawings, architecture (elevations and plans) and technical sketches.
- Cuboid with length L = 10 cm, width W = 6 cm, height H = 4 cm: Front view = rectangle H × W = 4 cm × 6 cm; Top view = rectangle L × W = 10 cm × 6 cm; Side view = rectangle L × H = 10 cm × 4 cm.
- Cylinder standing upright (height = h, base diameter = d): Front view = rectangle h × d (with semicircles at top/bottom if showing curved ends) but commonly drawn as rectangle with height h and width d; Top view = circle of diameter d; Side view = same as front view.
- Cone standing on its circular base (height = h, base diameter = d): Front view = isosceles triangle with height h and base d; Top view = circle of diameter d; Side view = same isosceles triangle as the front.
- Sphere with diameter D: Front, top and side views are all circles of diameter D.
- L-shaped block: Front shows one profile (height and width); top shows the plan (length and width) revealing the L shape; side view shows how high the steps of the L are (height and length).
- If a solid has dimensions length = L, width = W, height = H, then: - Front view dimensions = H × W - Top view dimensions = L × W - Side view dimensions = L × H
- Area of rectangular projection in a view = product of the two visible dimensions (e.g., area(front) = H × W).
- Orthographic projection mapping (coordinates): - Top view: (x,y) = (length, width) - Front view: (x,z) = (length, height) - Side view: (y,z) = (width, height)
Classification and Relationships among Solids
What is a solid? A solid is a three-dimensional object that occupies space and has length, breadth and height. We study their shapes, surfaces, edges and vertices.
Main classification
- Polyhedra (solids with flat faces): Faces are polygons. Examples: cuboid, cube, prism, pyramid. Polyhedra have faces, edges and vertices.
- Solids with curved surfaces: Have one or more curved surfaces. Examples: cylinder, cone, sphere. These do not belong to polyhedra.
Subtypes and relationships
- Prism: A solid with two congruent parallel polygonal faces (called bases) and rectangular lateral faces. Example: triangular prism, rectangular prism (cuboid).
- Pyramid: A solid with a polygonal base and triangular lateral faces meeting at a single vertex (apex). Example: square pyramid.
- Right vs oblique: If the axis is perpendicular to the base, the solid is right (right circular cone, right prism); otherwise oblique.
- Regular solids and symmetry: Solids like cube and regular prisms have equal edges/angles producing symmetry.
Important properties
- Polyhedra: count faces (F), edges (E), vertices (V). For any convex polyhedron Euler's relation holds: V - E + F = 2.
- Solids with curved surfaces: characterize by radius, height, slant height (for cones), and cross-sections (plane intersections).
- Cross-sections: slicing a solid by a plane gives 2D shapes (e.g., slicing a cylinder perpendicular to axis gives a circle; slicing a cone parallel to base gives a circle, oblique slice may give an ellipse).
- Nets: A net is a 2D pattern that can be folded to form the solid. Nets help visualise faces and compute surface area.
How solids relate to each other
- A cube is a special cuboid (all edges equal).
- A circular cylinder can be seen as a prism with circular bases (same cross-section along height).
- A cone is analogous to a pyramid where the base is a circle and lateral faces become a continuous curved surface.
- Many solids transform into each other by changing parameters: shrinking the base of a prism to a point gives a pyramid; shrinking base radius of a cone to zero collapses it to a line segment.
Use of Euler's formula
- For convex polyhedra Euler's relation V - E + F = 2 is useful for checking counts. Example: cube has V=8, E=12, F=6 so 8 - 12 + 6 = 2.
Visualization tips
- Draw nets to count faces and compute surface area.
- Sketch 3D diagrams showing axes, base, height and slant height for cylinders, cones and pyramids.
- Use cross-section drawings to understand intersections and similarity (e.g., vertical cross-section of cone is an isosceles triangle).
- Cube: a dice, Rubik's cube — all faces are congruent squares.
- Cuboid (rectangular prism): a shoebox, book — faces are rectangles.
- Cylinder: a soda can, water pipe — two parallel circular bases and curved lateral surface.
- Cone: an ice-cream cone, traffic cone — circular base and a point (apex).
- Sphere: a basketball, globe — all points on surface are equidistant from centre.
- Prism: a Toblerone chocolate (triangular prism) — two congruent polygonal bases connected by rectangles.
- Cube (side a): Volume = a^3 ; Total Surface Area (TSA) = 6a^2
- Cuboid (l, b, h): Volume = l × b × h ; TSA = 2(lb + bh + hl)
- Right Prism (base area B, height h): Volume = B × h ; TSA = 2B + (perimeter of base) × h
- Cylinder (radius r, height h): Volume = πr^2h ; TSA = 2πr(h + r) [lateral area = 2πrh]
- Cone (radius r, height h, slant height l): Volume = (1/3)πr^2h ; TSA = πr(l + r) [l = √(r^2 + h^2)]
- Sphere (radius r): Volume = (4/3)πr^3 ; TSA = 4πr^2
Key Concepts
- Solid (3D shape)
- A geometric figure that has three dimensions — length, breadth and height — occupying space.
- Face
- A flat surface of a solid (for polyhedra) or a curved/flat surface portion of a solid.
- Edge
- A line segment where two faces of a solid meet.
- Vertex (Vertices)
- A point where three or more edges of a solid meet.
- Polyhedron
- A solid made up entirely of flat polygonal faces, edges and vertices.
- Prism
- A polyhedron with two congruent, parallel faces called bases and other faces that are parallelograms (usually rectangles).
- Cuboid
- A prism whose six faces are rectangles; opposite faces are equal.
- Cube
- A special cuboid with all six faces congruent squares; all edges equal.
- Pyramid
- A polyhedron with a polygonal base and triangular lateral faces that meet at a single vertex (apex).
- Base
- The face of a solid on which it is considered to stand; for prisms/pyramids it is the chosen polygonal face(s).
- Lateral face / Lateral surface
- The faces of a solid excluding the base(s); collectively called the lateral surface (may be flat or curved).
- Cylinder
- A solid with two congruent parallel circular bases joined by a curved surface; cross-sections perpendicular to axis are circles.
- Cone
- A solid with a circular base and a curved surface that tapers smoothly to an apex.
- Sphere
- A perfectly round 3D object where every point on the surface is at the same distance (radius) from the centre.
- Net
- A 2D pattern obtained by unfolding the faces of a solid so they lie flat without overlap.
- Cross-section
- The intersection shape obtained when a plane cuts through a solid.
- Axis
- An imaginary straight line about which a solid (like a cylinder, cone or sphere) is symmetric; for cylinder/cone it joins centers of circular faces or apex to center of base.
- Slant height
- The length of the sloping side from the apex to a point on the edge of the base of a cone or pyramid.
- Curved Surface Area (CSA) / Lateral Surface Area
- Area of the curved portion (or lateral faces) of a solid, excluding the base(s).
- Total Surface Area (TSA)
- The sum of the areas of all faces (flat and curved) of a solid.
End-of-Chapter Trial Paper & Test Questions
Topic-wise questions to test your understanding of every concept in this chapter.
-
How many faces, edges and vertices does a cube have? / एक घन में कितने फलक, किनारे और शीर्ष होते हैं? (a) F=6, E=12, V=8 / F=6, E=12, V=8 (b) F=4, E=6, V=4 / F=4, E=6, V=4 (c) F=5, E=9, V=6 / F=5, E=9, V=6 (d) F=8, E=12, V=6 / F=8, E=12, V=6
Show answer
(a) — A cube has 6 square faces, 12 edges and 8 vertices. You can verify with Euler's formula: V − E + F = 8 − 12 + 6 = 2. / घन में 6 वर्गाकार फलक, 12 किनारे और 8 शीर्ष होते हैं। ऑयलर सूत्र: 8 − 12 + 6 = 2 से जाँच करें।
-
A triangular prism has how many faces? / एक त्रिभुजाकार प्रिज्म में कितने फलक होते हैं? (a) 3 / 3 (b) 4 / 4 (c) 5 / 5 (d) 6 / 6
Show answer
(c) — A triangular prism has 2 triangular bases + 3 rectangular lateral faces = 5 faces. Using the prism formula F = n + 2 with n = 3: F = 5. / त्रिभुजाकार प्रिज्म में 2 त्रिभुजाकार आधार + 3 आयताकार पार्श्व फलक = 5 फलक होते हैं।
-
The net of a cylinder consists of which shapes? / एक बेलन का जाल किन आकृतियों से बना होता है? (a) Two squares and a rectangle / दो वर्ग और एक आयत (b) Two circles and a rectangle / दो वृत्त और एक आयत (c) One circle and a triangle / एक वृत्त और एक त्रिभुज (d) Six squares / छः वर्ग
Show answer
(b) — The net of a cylinder unfolds to one rectangle (the curved lateral surface of width 2πr and height h) and two circles (the top and bottom bases). / बेलन का जाल एक आयत (वक्र पार्श्व सतह, चौड़ाई 2πr, ऊँचाई h) और दो वृत्तों (ऊपरी और निचला आधार) से बनता है।
-
For any convex polyhedron, Euler's formula states V − E + F = _____. / किसी भी उत्तल बहुफलक के लिए ऑयलर का सूत्र V − E + F = _____ है।
Show answer
2 / 2 — Euler's formula: Vertices − Edges + Faces = 2, valid for all convex polyhedra. E.g., for a tetrahedron: 4 − 6 + 4 = 2. / ऑयलर सूत्र: शीर्ष − किनारे + फलक = 2, सभी उत्तल बहुफलकों के लिए मान्य। उदा. चतुष्फलक: 4 − 6 + 4 = 2।
-
When a horizontal plane cuts a right circular cone parallel to its base, the cross-section obtained is a _____. / जब एक क्षैतिज तल एक लंब वृत्ताकार शंकु को उसके आधार के समानांतर काटता है, तो प्राप्त परिच्छेद _____ होता है।
Show answer
Circle / वृत्त — A plane parallel to the base of a cone always gives a circular cross-section (a smaller circle than the base). / शंकु के आधार के समानांतर तल हमेशा वृत्ताकार परिच्छेद देता है (आधार से छोटा वृत्त)।
-
True or False: The front view of a sphere is a circle regardless of the direction from which it is viewed. / सत्य या असत्य: एक गोले का सामने का दृश्य किसी भी दिशा से देखने पर वृत्त ही होता है।
Show answer
True / सत्य — A sphere looks circular from every direction (front, top, side), because every cross-section through its centre is a circle. / गोला हर दिशा से वृत्ताकार दिखता है क्योंकि इसके केंद्र से होकर जाने वाला प्रत्येक परिच्छेद वृत्त होता है।
-
Name a real-life object that resembles a triangular prism and identify how many faces, edges and vertices it has. / एक वास्तविक जीवन की वस्तु का नाम बताएँ जो त्रिभुजाकार प्रिज्म से मिलती-जुलती हो और उसके फलक, किनारे व शीर्षों की संख्या लिखें।
Show answer
A Toblerone chocolate box or a tent / टोबलेरोन चॉकलेट बॉक्स या तंबू — Faces = 5 (2 triangular + 3 rectangular), Edges = 9, Vertices = 6. These satisfy Euler's formula: 6 − 9 + 5 = 2. / फलक = 5 (2 त्रिभुजाकार + 3 आयताकार), किनारे = 9, शीर्ष = 6। ऑयलर सूत्र: 6 − 9 + 5 = 2 सत्यापित होता है।
-
A square pyramid has a square base. Find its number of faces, edges and vertices and verify Euler's formula. / एक वर्गाकार पिरामिड का आधार वर्ग है। उसके फलक, किनारे और शीर्षों की संख्या ज्ञात करें और ऑयलर का सूत्र सत्यापित करें।
Show answer
F=5, E=8, V=5; Euler check: 5 − 8 + 5 = 2 ✓ / F=5, E=8, V=5; ऑयलर जाँच: 5 − 8 + 5 = 2 ✓ — Using pyramid formula: F = n+1 = 5, E = 2n = 8, V = n+1 = 5, where n = 4 (square base). / पिरामिड सूत्र: n=4 (वर्गाकार आधार), F = n+1 = 5, E = 2n = 8, V = n+1 = 5।
Related Laws & Principles
Explore allFoundational laws & principles behind this chapter. Each one opens a full page — what it says, why it matters, five practice questions and the mistakes to avoid.