History

Mensuration developed from practical needs in architecture, agriculture, and trade. Ancient mathematicians studied area and volume formulas, with later refinement by Archimedes and others.

Derivation and reasoning

Why pyramid and cone have one-third factor

A pyramid (or cone) with same base area and height as a prism (or cylinder) has one-third the volume. This can be shown by dissection arguments or calculus in advanced courses.

Checkpoints

Find volume of a cylinder with \(r=3\) cm, \(h=10\) cm.

Answer: \(V=\pi r^2h=90\pi\) cm\(^3\).

Find total surface area of a closed cylinder with \(r=2\) cm, \(h=7\) cm.

Answer: \(2\pi r^2+2\pi rh=8\pi+28\pi=36\pi\) cm\(^2\).

Find volume of a cone with \(r=6\) cm, \(h=9\) cm.

Answer: \(\frac13\pi r^2h=\frac13\pi(36)(9)=108\pi\) cm\(^3\).

Sphere has radius 5 cm. Find its volume.

Answer: \(V=\frac43\pi(5^3)=\frac{500}{3}\pi\) cm\(^3\).

A block is \(4\times3\times2\) m. Density is \(2500\) kg/m\(^3\). Find mass.

Answer: Volume \(=24\) m\(^3\). Mass \(=2500\times24=60000\) kg.

Applications

• Construction: concrete volume and paint area estimation.
• Packaging: minimizing material for fixed volume containers.
• Engineering: tank capacity and material mass calculations.

References

• Standard high-school mensuration units on 3D solids.
• Geometry texts on prism, cylinder, cone, pyramid, and sphere formulas.