Geometric Solids

For primary school, we restrict to the study of solids to spheres, cylinders, cones, cubes & cuboids.

And two important concepts - volume & surface area.

Volume can be thought of as “filling” a box

Surface can be thought of as “wrapping” a box

Archimedes Relation of Volumes

Archimedes brought out the beautiful relation between the volumes of a cone & sphere which can fit snugly into a cylinder.

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Imagine a cylinder whose radius is "r" and height is "2r".

A sphere of radius "r" can be fitted snugly into the cylinder.

A cone with a base radius "r" and height "2r" can also be fitted into this configuration.

In the above configuration, Archimedes proved the following relations.

Volume of the Sphere = 2/3* Volume of the Cylinder &

Volume of the Cone = 1/3 * Volume of the Cylinder.

Hence the Volume of the Cylinder = Volume of Inscribed Sphere + Volume of Inscribed Cone.

He further showed that the surface area of the sphere equals the surface area of the curved face of the cylinder. Theoretically this means that a sphere can be completely "covered" by a sheet of paper, whose height is equal to the diameter of the sphere and which just wraps around it.

Archimedes also showed that the surface area of a sphere ( 4πr^2) equals 4 times the cross-section of the sphere by a plane passing through the centre (πr^2).

Ratio of Surface Area to Volume

The ratio of the surface area of a solid to its volume is an important ratio in physical & biological sciences.

We are not sure why this has not acquired a separate name like density, which is the ratio of weight of a solid to its volume.

Physics

Law of Floatation states that a body will float in a liquid if the weight of the liquid displaced by it is more than its weight.

The hull of a ship (or its surface area) is shaped in such a way that it displaces a volume of water whose weight is more than the weight of the ship & its cargo.

We can see the ratio of the volume/weight of the ship to its surface area playing a role.

It is also the reason, tea poured in a saucer cools much faster than tea in a cup.

Biology

The small intestine needs to absorb all the nutrients in the food which passes through it. Hence it needs to have a large surface area. A long tube with a small diameter gives a high ratio of surface area to volume. At the same time a long tube can curl and fit in a small space.

Amoeba take in nutrition through its surface area to support the nutritional needs of its volume/weight. The growth of the surface area ( a square function) is slower than the volume (a cubic function).

Hence after some growth, the surface area is not sufficient enough to absorb the needed nutrition to support its volume. Hence the amoeba splits into two smaller ones.

There are many other examples in biology on the criticality of this ratio.

Conic Sections

Apollonius showed that by slicing a cone with a plane at different angles, we can get all the standard geometrical plane figures - circle, ellipse, parabola & hyperbola. The sections are called Conic Sections.