Area

The measurement of area was more complicated than the measurement of perimeter which is actually a length. The unit of length could be fixed arbitrarily depending on our convenience. What is important is that the unit should be acceptable to all who are interested in using it. How to fix a unit for measuring area?

Further there are 2 kinds of areas – those of plane surfaces & those of curved surfaces. We will deal with plane surfaces in this chapter & with curved surfaces along with volume.

Square as the Unit of Area

We have seen in the section on measurement that it is very convenient to take the square as a unit for measuring area. Square is one of the figures which require the least information and can be constructed accurately & easily anywhere. It is also a shape that tessellates well to cover any area without leaving gaps. Hence it is an ideal unit of measurement. A unit of area was defined as the area covered by a square whose side was one unit of length. The unit of area was dependent on the unit we choose to measure length.

Square Centimeter and Centimeter Square

If we take the unit of length as a cm, then 1 Square Centimeter can be defined as the area of a square whose sides were 1 cm. There is a subtle difference between the terms “centimetre square” and “square centimetre”.

The “centimeter square” has the shape of a square of side 1 centimetre. Its area is defined as 1 square centimeter. 1 square centimetre itself has no specific shape. It just denotes a specific amount of area.

This difference would be obvious when we look at a statement that the area of a circle is 4 square centimeters. Obviously a circle is not made of 4 centimeter squares! What it means is that if we draw 4 squares each with side 1 cm on a piece of paper, then those 4 squares can be cut into small pieces and spread on the circle so that they will fully cover the circle.

The unit of area could be a Sq cm or Sq Meter or Sq Km depending on the area that we need to measure. The relation between these units can also be worked out since we know the relation between the related lengths.

Area of any Figure

Further, any rectangle can be divided into a series of squares. Mathematically the area of a rectangle is just the product of its length & breadth. Hence the trick for finding the area of any given figure is to see if we can think of a rectangle whose area is equal to that of the given figure.

Area of any polygon

Geometers had shown that all figures bounded by straight lines, could be divided into a series of triangles. They also showed that for any triangle, a rectangle with the same area could be drawn. The area of the polygon is the sum of the areas of the constituent triangles.

Rectangle– Any rectangle with sides a & b linear units can be divided into aXb square area units, by drawing vertical and horizontal lines. Hence its area can be denoted by ab.

Square – by the same logic, the area of a square of side a is a X a or

Triangle– Any triangle can be shown to have half the area of a rectangle whose sides are equal to its base and the height of the base from the opposite vertex. Hence the area of any triangle can be written as ½ x base x height.

Area of a Circle

Circle – like its circumference, the area of a triangle was also studied for several centuries. It was found to be related to its circumference and contained the number π.

The area of a circle can cut into a series of narrow triangles with their vertices at the centre of the circle, By opening out, the area be imagined as that of a rectangle one of whose sides is the circumference and the other side is half the radius. Since circumference, c = 2r, we can write the relation between A & c of a circle as A = ½ rc, which in turn can be written as where r is the radius of the circle.

Area of Complex Figures

Like in the case of perimeter, in real life we come across figures which are a mix of various simple figures. Here again the strategy would be to divide the complex shape into a series of simple figures, compute the areas of the component shapes and then add them together.

Chapter 25.5 gives a visual proof of the area of circle & triangle by paper folding. Chapter 25.3 gives some activities for understanding the inter-related properties of area & perimeter.

Koch Snowflake

Traditional concepts about area & perimeter of plane figures were questioned by the discovery of fractals in the first half of the 20th century.

The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. Its discovery was published Swedish mathematician Helge von Koch in 1904.

It looks like a snowflake which encloses a finite area, but has an infinite perimeter.