Net Percentage Change in Volume, Geometry or Percentage

Percentage change in area of Polygons – Geometry or Percentage?

PERIMETER:

The continuous line forming the boundary of a closed geometrical figure is called the perimeter. For example, the perimeter of a square with side ‘a’ is 4a and perimeter of a circle (i.e. circumference), with radius ‘r’, is 2πr. The unit of perimeter is same as the unit of the side, thus if there is a change of r% in side then percentage change in perimeter would be r%.

Example 1.

If the length and breadth of a rectangle are increased by 20%, then what is the percentage change in its perimeter?

Solution. Perimeter of rectangle = 2(L + B);

If length and breadth are increased by 20%, the new length and breadth will be, (1.2L) and (1.2B).

The new perimeter = 2 × (1.2L + 1.2B)⇒1.2 × 2(L + B) = 1.2 (Perimeter of the original rectangle)

Thus, the new perimeter is 1.2 times the original perimeter i.e. the perimeter has increased by 20%.

Example 2.

If the radius of a circle is decreased by 30%, then what is the percentage change in its circumference?

Solution. Circumference = 2 πr;

If the radius is decreased by 30%, the new radius will be 0.7r.

The new Circumference = 2π (0.7r)

⇒ = 0.7 × (2 πr) = 0.7 (Circumference of original circle)

Thus, the new circumference is 0.7 times the older circumference i.e. the circumference has decreased by 30%.

“If there is a change of r% in every side of a regular polygon or the radius of a circle, then the overall change in perimeter of the polygon or circumference of the circle would be r%” 

AREA:

The measure of the space enclosed within the boundary of a two dimensional geometrical figure is called its area. For example the area of square is (side)² and area of a circle, with radius ‘r’, is πr². The unit of area is square of the unit of the side in a given figure. 

Example 3.

If the radius of a Sphere is increased by 40%, then what is the change in the surface area?

Solution.  Surface area of sphere = 4 πr²;

If the radius is increased by 40%, the new radius will be 1.4r.

Therefore, surface area of the new sphere = 4π(1.4r)² 

⇒ 1.96 × (4πr²) = 1.96 (Surface area of the original sphere)

Thus, the new surface area is 1.96 times the original surface area i.e. it has increased by 96 %.

NOTE: As seen above, the term ‘4π’ is constant in the surface areas of both the spheres and thus, would cancel out when you compare the areas of two spheres. The change in surface area is due to the term r². i.e. (1.96)² = 1.96, which means an increase of 96%.

“If the side/radius of a figure is changed by r%, then the area will change by (r + r + r2/100)%”

Do remember that you need not take care of the whether there is increase or decrease as the sign will take care of the same. 

VOLUME:

The amount of space that a three-dimensional object occupies is called its volume. For example, the volume of a cube is (side)³ and volume of a sphere, with radius ‘r’, is 4/3 πr³. The unit of the volume is cube of the unit of the side in a given object.

Example 4.

If the radius of a hemisphere is increased by 10%, then what is the percentage change in the volume?

Solution. Volume of sphere = 2/3 πr³; If the radius is increased by 10% then the new radius = 1.1r;

New volume = 2/3 π(1.1r)³ 

⇒1.331 × (2/3 πr³)

The new volume is 1.331 times of the original volume i.e. it has increased by 33.1%.

NOTE: As seen above, the c term ‘2/3π’ is constant for the volumes of the original and the new hemi-spheres and would cancel out when we compare the volumes of two hemi-spheres. The change in volume is due to the term ‘r3’ i.e. (1.1)³ = 1.331, which means an increase of 33.1%.

Example 5.

Find out the percentage change in the area and volume of a sphere, when its radius is increased by 10%.

Solution. You can directly apply the above techniques as:

Change in area = (1.1)² = 1.21 i.e. 21 percent increase.

Change in the volume = (1.1)³ = 1.331 i.e. 33.1 percent increase

Learning:

When one of the side/perimeter/area/volume of a figure/object is changed by a certain percentage, and you have to find the percentage change in any of the remaining three, it is an advanced-level question of Percentages and not Geometry.