Geometry

Wastage is not always a Waste

At times, in geometry questions, we have symmetrical figures inscribed in other symmetrical figures e.g. a circle inscribed in a square, a circle inscribed in an equilateral triangle etc. and we have to find the common/wastage areas of such figures. In such cases, the concept of wastage of area may be helpful. Just to explain what this ‘wastage area’ is all about.

Let’s take an example.

Example:

A square paper has an area of 484 sq. cm. The largest possible circle is cut from this paper, what % area of the paper is wasted?

Solution:

In this case the area of square is 484 sq. cm. Hence the side of the square will be 22 cm.

So the diameter of the circle will be 22 cm. Area of the inscribed circle = π r² = 121π cm².

The area wasted at the four corners = (484 – 121 π) cm².

%age area wasted = (484-121π)/484 × 100 = 21.5%

Let me also make it clear to you that as the question was asking about the percentage change, the actual sides or area does not matter i.e. even if you take the side of the square to be 2 or 1, the answer will remain the same. This has universal application, whenever the percentage wastage is asked; any figure that makes your calculations easier can be taken.

Now here the area of the big square is 484 cm². If we take any square with side of any length and a largest circle is cut inside this square, the area wasted would always be 21.5% and the area used/ inside the circle will be 78.5%.

Similarly, if the largest possible square is cut/drawn inside the circle, then the % area wasted will be equal to 36.3%. On the same lines, if the largest possible cube is cut from a sphere, then the percentage volume wasted is 63% or the volume of cube is 37% of the volume of the sphere.

Some other important universal results are as follows: 

Outer figure	Inner figure	Wastage %
Square	Largest circle	21.5
Circle	Largest square	36.36
Equilateral triangle	Largest circle	39
Circle	Largest equilateral triangle	59
Square	Largest equilateral triangle	67
Equilateral triangle	Largest Square	51
Semicircle	Largest Square	49

In case your brain cells are buzzing now with this new tip that you have learnt, and you wish to discover how some of the above percentages were discovered, you have your home task: try to figure out how the above values were derived; this should be fun exercise for those who love mathematical wonders.

*Due to symmetry, all the corners will have equal percentage wastage.