Volume & Surface Area

For objects that we find in our world, two of the most important characteristics are surface area and volume. The unit of volume can be defined in terms of unit length, as we did in the case of area.

Similar to the case of area, the volume of all simple regular solids can be seen as equal to a cuboid of certain dimensions. This cuboid can be divided into a certain number of cubes with sides made of unit lengths.

Hence the cube was used to define a unit of volume. A unit of volume was defined as the volume occupied by a cube whose side was one unit of length. The unit of volume was dependent on the unit we choose to measure length. Cube is also one of the figures which require the least information and can be constructed accurately & easily anywhere. Hence it is an idea unit of measurement.

But before arriving at this idea, cultures developed many other measures like ounces, pint & gallons for measuring volume. Many of these are still in use in different communities.

Cubic Centimeter and Centimeter Cube

If we take the unit of length as a cm, then 1 Cubic Centimeter was defined as the volume of a cube whose sides were 1 cm. 1 cubic cm itself has no specific shape. The centimeter cube has the shape of a cube. The volume of 1 cubic centimeter is equal to that of a cube with side 1 cm.

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

Surface Area

All solids have several surfaces. The surface area of the sold is the sum of the areas of the individual surfaces. For example a cylinder has a curved surface and 2 circular plane surfaces.

The surface area of these solids will have to be modified if they have follow spaces i.e if they are used as containers.

Volumes & Surface Area of simple Solids

Cuboid – Any cuboid with sides a, b & c linear units can be divided into aXbXc cubic volume units, by drawing vertical and horizontal lines. Hence its volume can be denoted by abc.

Its surface area is the sum of the areas of its 6 faces in which there are 3 sets of equal faces which are opposite to each other. The surface area can be written as 2(ab + bc + ac)

Cube – by the same logic, the area of a square of side a is aXaXa or Its surface area is 6.

Cylinder –Since its cross sectional area is constant, its volume is cross sectional area X height. If the radius is r, then cross sectional area which is a circle is π. Hence the volume is πh.

The curved surface area of a solid cylinder can be imagined to be opened out into a rectangle whole length would be the circumference of the base circle and whose breadth would be the height of the cylinder. Hence its surface area would be 2 π+ 2πrh = 2πr(r + h).

Cone - The volume of a cone was found to be 1/3 rd of a cylinder with the same base and same height. Hence its volume can be written as πh.

The surface are of a cone is a little difficult for primary classes.

Sphere– The volume was found to be π. The surface area of a sphere is 4π

Human Body & the Volume & Surface Area relationships

The human body has many organs which have volume and surface area. The function of these organs decides the relationship between their volumes and surface areas. The shape of these organs is decided by their function in the body. This is an interesting area of study of the relation between biology & mathematics. We will just give one simple example.

The food we eat contains a lot of nutrients which are fully absorbed by the small intestine. Hence the food needs to travel slowly through the small intestine. The surface are of the small intestine also has to have a large surface area since nutrients are absorbed through the walls of the small intestine. Hence the small intestine has to be long and narrow. The small intestine in our bodies is almost 6 meters long and lies curved in a small space in our lower abdomen.