Volume of pyramids, cones and spheres

Volume of Pyramids

A pyramid is formed when the vertices of a polygon are projected up to a common point (called a vertex). A right pyramid is formed when the apex is perpendicular to the midpoint of the base.

We want to be able to calculate the volume of a pyramid. Let's start by thinking about the square based pyramid.

Exploration

Think about a cube, with side length $s$s units. Now lets divide the cube up into 6 simple pyramids by joining all the vertices to the midpoint of the cube.

This creates $6$6 square based pyramids with the base equal to the face of one of the sides of the cube, and height, equal to half the length of the side.

$\text{Volume of Cube }=s^3$Volume of Cube =s3

$\text{Volume of one of the Pyramids }=\frac{s^3}{6}$Volume of one of the Pyramids =s36

Now lets think about the rectangular prism, that is half the cube. This rectangular prism has the same base as the pyramid and the same height as the pyramid.

Now the volume of this rectangular prism is $l\times b\times h=s\times s\times\frac{s}{2}$l×b×h=s×s×s2= $\frac{s^3}{2}$s32

We know that the volume of the pyramid is $\frac{s^3}{6}$s36 and the volume of the prism with base equal to the base of the pyramid and height equal to the height of the pyramid is $\frac{s^3}{2}$s32.

$\frac{s^3}{6}$`s`36	$=$=	$\frac{1}{3}\times\frac{s^3}{2}$13×`s`32	Breaking $\frac{s^3}{2}$`s`32 into two factors
$\text{Volume of pyramid}$Volume of pyramid	$=$=	$\frac{1}{3}\times\text{Volume of rectangular prism}$13×Volume of rectangular prism	Using what we found in the diagrams
	$=$=	$\frac{1}{3}\times\text{Area of base}\times\text{height }$13×Area of base×height	Previously shown

So what we can see here is that the volume of the pyramid is $\frac{1}{3}$13 of the volume of the prism with base and height of the pyramid.

Of course this is just a simple example so we can get the idea of what is happening.

Volume of Pyramid

$\text{Volume of Pyramid }=\frac{1}{3}\times\text{Area of base }\times\text{Height }$Volume of Pyramid =13×Area of base ×Height

Volume of Cones

The volume of a cone has the same relationship to a cylinder as we just saw that a pyramid has with a prism.

That is:

Volume of Right Cone

$\text{Volume of Right Cone }=\frac{1}{3}\times\text{Area of Base }\times\text{Height of cylinder}$Volume of Right Cone =13×Area of Base ×Height of cylinder

$V=\frac{1}{3}\pi r^2h$V=13πr2h

The mathematical derivation of the formula for the volume of a cone is beyond this level of mathematics, so for now it is suffice to know the rule and how to use it.

Where are cones?

Finding out the amount of snowcone that would fit inside one of those conical (cone-like) cups, you can use the volume equation to find the volume inside the cone, right?

Funnels are in the shapes of cones. If you want to find out how much *volume* of water/liquid flows out of it per unit time (say liters per second for example), you will have to know what the volume of a cone is. you can find this volume flow rate, you can even write down an exact equation to describe the flow rate at any given time, that begins for instance with water filled to the brim, showing a slow flow rate (this is at early times), and for later times your equation will show that you have a faster slow.

Traffic cones are cones. Maybe you need to work out the volume of plastic it takes to make one, ten or a thousand.

A whole lot of things are approximately cones, reactor cooling tower caps, thimbles, and so on. These actually are cones, they just have been truncated (i.e. a small cone at the tip has been subtracted from a larger cone), icicles, stalactites/stalagmites in caves, and paper-made megaphones.

The front of pens have a cone shape to them.

Volumes of Spheres

So who cares what the volume of a sphere is. Well...

A toy manufacturer needs to figure out how much plastic they need to make super bouncy rubber balls would need to know the volume of the ball.
An astronomer who wants to calculate how much the sun weighs would need to know its volume.
A pharmaceutical company making round pills needs to know the dosage of medicine in each pill, which is found by its volume.
A party balloon gas hire company needs to know how much gas fits inside each of their balloons, which is found by its volume.

So apart from having to complete questions at school on volumes of random shapes, volume calculations have a wide variety of applications.

The Formula

The Volume of a sphere with radius $r$r can be calculated using the following formula:

Volume of Sphere

$\text{Volume of sphere }=\frac{4}{3}\pi r^3$Volume of sphere =43πr3

Proof of the formula

Whilst the proof is not typically included as part of the needs of the curriculum, this particular one is a nice introduction to thinking about proofs and abstract proofs. So you don't have to read this if you don't want to but aren't you curious to know where this funny formula came from?

The proof

If four points on the surface of a sphere are joined to the centre of the sphere, then a pyramid of perpendicular height r is formed, as shown in the diagram. Consider the solid sphere to be built with a large number of these solid pyramids that have a very small base which represents a small portion of the surface area of a sphere.

If $A_1$A1, $A_2$A2, $A_3$A3, $A_4$A4, .... , $A_n$An represent the base areas (of all the pyramids) on the surface of a sphere (and these bases completely cover the surface area of the sphere), then,

$\text{Volume of sphere }$Volume of sphere	$=$=	$\text{Sum of all the volumes of all the pyramids }$Sum of all the volumes of all the pyramids
$V$`V`	$=$=	$\frac{1}{3}A_1r+\frac{1}{3}A_2r+\frac{1}{3}A_3r+\frac{1}{3}A_4r$13`A`1`r`+13`A`2`r`+13`A`3`r`+13`A`4`r` $\text{+ ... +}$+ ... + $\frac{1}{3}A_nr$13`An``r`
	$=$=	$\frac{1}{3}$13 $($( $A_1+A_2+A_3+A_4$`A`1+`A`2+`A`3+`A`4 $\text{+ ... +}$+ ... + $A_n$`An` $)$) $r$`r`
	$=$=	$\frac{1}{3}\left(\text{Surface area of the sphere }\right)r$13(Surface area of the sphere )`r`
	$=$=	$\frac{1}{3}\times4\pi r^2\times r$13×4π`r`2×`r`
	$=$=	$\frac{4}{3}\pi r^3$43π`r`3

where $r$r is the radius of the sphere.

Worked Examples

QUESTION 1

Find the volume of the square pyramid shown.

A triangular pyramid is depicted with its faces visible and outlined. A vertical dashed line, representing the height, is drawn from the apex of the pyramid perpendicular to the base and is labeled as $6$ $cm$`cm`. One side of the base is labeled as $10$ $cm$`cm`. All the sides of the base are drawn with a single tick mark indicating that the measurements of the sides are all equal.

QUESTION 2

A small square pyramid of height $4$4 cm was removed from the top of a large square pyramid of height $8$8 cm forming the solid shown. Find the exact volume of the solid.

A square pyramid is depicted. The top portion of the pyramid is removed, indicated by dashed lines, creating a new top base, with its side measuring $4$ cm. The bottom base of the square pyramid has a side measuring $8$ cm. The vertical height of the pyramid is composed of two measurements, the removed top smaller pyramid measuring $4$ cm, and the bottom part measuring $4$ cm.

Give your answer in exact form.

QUESTION 3

Find the volume of the cone shown.

Round your answer to two decimal places.

A cone is depicted with a vertical height labeled as $6$ cm from the apex to the center of the base. The base radius is labeled as $2$ cm. A right-angle indicator is shown where the height meets the base, suggesting the height is perpendicular to the base.

QUESTION 4

QUESTION 5

Find the volume of the sphere shown.

Round your answer to two decimal places.

QUESTION 6

A sphere has a radius $r$r cm long and a volume of $\frac{343\pi}{3}$343π3 cm³. Find the radius of the sphere.

Round your answer to two decimal places.

Enter each line of working as an equation.