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 ( 4&pi;r^2) equals 4 times the cross-section of the sphere by a plane passing through the centre (&pi;r^2).

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.