# Volume Of Pyramids

**Pyramids** are 3-dimensional solids where the **cross-section reduces in size to a point**.

Examples are **square-based pyramids** (e.g. Egyptian pyramids), **triangle-based pyramids (tetrahedrons)** and **circle-based pyramids (cones)**.

Because the pyramid is one-third of the volume of an equal-dimensioned prism, the general rule for volume of a pyramid is:

Volume of Pyramid = ^{1}⁄_{3} × Area of Cross-section × Height

## Example One - Volume of Stone in an Egyptian Pyramid

The Great Pyramid at Giza in Egypt has a square base with a **side of 230 metres** and a **vertical height of 147 metres**. What is the volume of stone?

**Answer:**

*The cross-section is a square of decreasing size.*

Volume of Pyramid = ^{1}⁄_{3} × Area of Cross-section × Height

= ^{1}⁄_{3} × Area of Square × Height

= ^{1}⁄_{3} × (S × S) × H

= ^{1}⁄_{3} × (230 × 230) × 147

= 2 592 100 m^{3} of stone

## Did You Know That...?

In your retina (the curved back part of your eye) are hundreds of tiny **cone cells for colour vision**. They are mainly concentrated in the middle of the retina.

*(Try looking at something colourful by first looking directly at it, and then looking side on. It should be brighter if you look directly at it.)*

Cone cells also require a higher intensity of light to function than do the other retinal cells called **rod cells for black-and-white vision** which are
mainly around the periphery of the retina.

*(When you are lying in bed tonight with the lights out, look at colourful objects in your room. Do they appear more greyish and lacking in colour?)*

## Example Two - Volume of Crayon

A crayon has 2 parts - a cylinder and a cone-shaped top. The **cylinder has a diameter of 10 mm and a length of 70 mm**.
The **cone has a vertical height of 8 mm**. What is the volume of one crayon? *(Use π = 3.14.)*

**Answer:**

**Volume of Prism (Cylinder)**

*The prism cross-section is a circle of the same size.*

Volume of Prism = Area of Cross-section × Height

= Area of Circle × Height

= π r^{2} × H

= 3.14 × 5 × 5 × 10

= 785 mm^{3}

**Volume of Pyramid (Cone)**

*The pyramid cross-section is a circle of decreasing size.*

Volume of Pyramid = ^{1}⁄_{3} × Area of Cross-section × Height

= ^{1}⁄_{3} × Area of Circle × Height

= ^{1}⁄_{3} × π r^{2} × H

= ^{1}⁄_{3} × 3.14 × 5 × 5 × 8

= 209.3 mm^{3}

## Question - I Scream for Ice-cream

How much ice-cream is in a **cone** with a **diameter of 6 cm** and a **vertical height of 12 cm**?

**Answer**

113.04 ml

*(You took so long to calculate the volume that it melted to a capacity! Haha!)*