Visualising solids: profiles, views and isometric drawing

Building in your imagination

Before drawing anything, try picturing it. Read your name backwards by seeing the letters in your mind. Cut the four corners off an imaginary square, each cut joining the midpoints of two neighbouring sides: a smaller, tilted square remains, and the four corner triangles fit together into another square of the same size. Cut the corners of an equilateral triangle at the third-marks and a regular hexagon remains; do the same to a square and an octagon remains.

Profiles change with the viewpoint

When we look at a solid we see its profile, its outline from that viewpoint. An elephant seen from the front, the side and above gives three very different outlines, and simple solids do the same. A cylinder shows a rectangle from the side and a circle from above; a cone a triangle and a circle; a triangular prism a rectangle and a triangle; a bucket shape a trapezium and a circle. More than one solid can fit the same pair of profiles.

Faces, edges and vertices

Solids with flat faces can be counted. A cube has 8 vertices, 12 edges and 6 faces. A prism has two congruent polygons as opposite faces, joined by parallelograms; with n-sided ends it has n + 2 faces, 3n edges and 2n vertices. A pyramid has an n-sided base joined to one apex; it has n + 1 faces, 2n edges and n + 1 vertices. A triangular pyramid is also called a tetrahedron.

Projections

The projection of a point P on a plane is the point O where a line from P meets the plane at right angles. Projecting every point of an object gives its projection. A line’s projection is never longer than the line, because the line is the hypotenuse of a right triangle whose leg is the projection; they are equal only when the line is parallel to the plane. Parallel lines always project to parallel lines, so a parallelogram always projects to a parallelogram (or, edge-on, to a segment). Sunlight shining straight at a wall makes shadows that are practically projections.

Front, top and side views

Since many objects share a projection, engineers use three planes at right angles: the front view on the vertical plane, the top view on the horizontal plane and the side view on the side plane. To build a solid from its views, start from the top view to see where stacks stand, then read the tallest stack in each column from the front view and in each row from the side view.

Isometric drawing

If a cube is balanced on one corner and projected straight down, all 12 edges come out the same length: an isometric projection, a regular hexagon. Tiling the plane with such hexagons gives isometric paper, with lines in three directions for length, depth and height. Draw solids edge by edge along these directions, counting units, and they look right because parallel edges stay parallel and equal steps stay equal.

More: fractals, nets of solids, the chapter notes and the NCERT solutions.