Representing 3-D in 2-D
ICSE Class 8 · Mathematics · Chapter 19
What this chapter covers
- Solid shapes and flat pictures
- Matching pictures with objects
- 2-D views of 3-D objects
- Drawing solids on paper
- Nets of prisms and pyramids
- Faces, edges and vertices
- Euler's relation
Key points
- A 3-D shape has length, breadth and height; a 2-D shape is flat and has only length and breadth.
- A nested shape has one solid inside another; a joint shape is two solids joined together. Hidden edges are drawn dashed.
- The front view, side view and top view (plan) are the outlines seen from the front, the side and directly above; one view alone may not identify a solid.
- An oblique sketch shows the front face in true shape with slanting depth lines; an isometric sketch, on isometric dot paper, shows every edge at its true length.
- A net is a flat pattern that folds into a solid; a prism's net has 2 bases and a strip of rectangles, and a pyramid's net has 1 base with a triangle on each edge.
- A polyhedron has only flat polygon faces; its parts are faces, edges and vertices.
- A prism with an n-sided base has n + 2 faces, 2n vertices and 3n edges; a pyramid has n + 1 faces, n + 1 vertices and 2n edges.
- Euler's relation: F + V - E = 2 for cubes, cuboids, tetrahedrons, prisms and pyramids.
Questions answered
What is the difference between a 2-D shape and a 3-D shape? Give one example of each.
A 2-D shape is flat and has only length and breadth, such as a square drawn on paper. A 3-D shape has length, breadth and height and takes up space, such as a brick.
A drawing shows a cylinder with a cone on top. Name the type of arrangement and one real object it could be.
It is a joint shape made of two solids. It could be a circus tent or a grain silo.
A box of sweets is packed inside a larger cardboard box. What name is given to this kind of arrangement, and how would the hidden box be shown in a drawing?
It is a nested shape. The edges of the hidden inner box would be drawn as dashed lines.
A solid looks like a rectangle from the front and a circle from the top. Name the solid.
A cylinder standing upright.
Describe the top view of a square pyramid standing on its base.
A square with both of its diagonals drawn. The diagonals are the slanting edges, which meet at the apex in the centre.
How is an isometric sketch different from an oblique sketch?
In an isometric sketch, drawn on isometric dot paper, every edge is drawn to its true length. In an oblique sketch, the front face is drawn in its true shape, but the edges going backwards are slanted and usually shortened.
Which solid does a net of 2 triangles and 3 rectangles make?
A triangular prism.
Find the faces, vertices and edges of a pentagonal pyramid, and check Euler's relation.
n = 5, so F = 6, V = 6 and E = 10. Then F + V - E = 6 + 6 - 10 = 2, which agrees with Euler's relation.