Viewing Different Sections of a Solid

We have already seen that depending on our point of view/orientation, the shape of a 3-D object will accordingly, change. This can be done for any physical entity. A ball, a bicycle, even a building. For example: let's take the Flat Iron Building in New York City.

The Flatiron Building in New York City.

The Flatiron Building seen from the side.

The Flatiron Building seen from the front.

Top view of the Flatiron Building seen from a plane.

While it clearly is a bit inconvenient to view a very large physical entities from different angles, it is much for simpler for small scale objects.

When viewing an object, the different standard views we try to observe include:

Top View

Front View

Side View

Depending on the type of analysis we need to carry out, we can use different methods to understand and view the object at hand.

(1) One Way to View an Object is by Cutting or Slicing

Slicing game

Sliced Bread Loaf

Above we have a loaf of bread which geometrically, is a cuboid with a square face. This cuboid can be cut ("sliced") with a knife into multiple pieces.

When we give a ‘vertical’ cut, we get multiple pieces (as shown).

Each face of the smaller pieces is a square. We call this face the cross-section of the whole bread. The cross section is basically, a square in this case.

Note: If the orientation of the cut is not vertical, we may get a different cross section of the object. Also, the boundary of the cross-section we obtain is a plane curve.

What cross-sections is obtained when a (i) vertical cut (ii) horizontal cut is given to the following solids?

Ice-cream cone (Vertical cut)

Ice-cream cone (Horizontal cut)

Brick (Vertical cut)

Brick (Horizontal cut)

Die (Vertical cut)

Die (Horizontal cut)

Round Apple (Vertical cut)

Round Apple (Horizontal cut)

Horizontal Circular Pipe (Vertical cut)

Horizontal Circular Pipe (Horizontal cut)

Triangle

Circle

Rectangle

Square

(2) Another Way is by Shadow Play

A shadow play

Another easy way to understand 3-D objects in 2-D is using shadows.

All we require is a source of light and the object to be studied. (If an overhead projector is available, place the object under the lamp)

Shadow of Hand

Shadow of hand holding flower

Activity: With the help of a torchlight, view any object of your choice by placing the object in the path of the light beam. What type of shadow is cast on the wall/screen?

The object is three-dimensional; what is the dimension of the shadow? (2D or 3D)

When we carry out this activity we realize that for different positions/orientation of the source of light and the solid object, a variation of results are obtained.

Cup shadow due to Sun light

Different Shadow Orientation

Try this: Place a cup out in the open where there is sunlight. Observe the shadow that gets cast from time to time. How does it change? How does the shadow vary from morning to noon to evening?

Let's Solve

The following objects listed below are placed under the beam of a light bulb. What will be the shape of the shadows for each case:

Ball

Pipe

Book

Hemisphere

Square

Circle

Rectangle

Cylinder

Are the following statements given - True (or) False:

(i) The cube can cast a shadow in the shape of a rectangle.

(ii) The cube can cast a shadow in the shape of a hexagon.

(3) A Third Way is by Looking at it from Certain Angles to Get Different Views

Any physical object existing in our world can be viewed from various directions and angles. As we have seen in the previous section that when we do so , the way we perceive the subject changes.

Here is another example of how an observer gets different views of a given object: