Cholula is the largest Pyramid in the world (by volume)(Mexico)

StarterThe Louvre is an Art museum in Paris, part of which is in the shape of a Pyramid. The Pyramid is 20.6m high and has a Square base of side 35m. Calculate the amount of litres of air the Pyramid would contain. 1 Litre = 1000cm3Volume of a Pyramid = 1/3 x base x height35m = 3500cm20.6m = 2060cmChange the units first!Volume = 1/3 x base x heightVolume = 1/3 x (3500 x 3500) x 2060Volume = 8,411,666,667cm3Litres = 8,411,666.667 1000

FrustumsA Frustum is effectively a Cone or Pyramid with a smaller Pyramid or Cone cut off

FrustumsWe have looked at the Volume of a number of 3D solids

Today we will be looking at calculating the Volume of a Frustum

Learning ObjectivesAll will be able to calculate the Volume of Pyramids and Cones (Grade B/A)

Most will be able to for equations to calculate the missing length required for the Volume of a Frustum (Grade A/A*)

Some will be able to calculate the Volume of a Frustum from start to finish (Grade A*)

FrustumsTo calculate the Volume of a Frustum, you must consider the Full Cone, as well as the smaller one which has been removed.

You will need to work out the missing height, which is part of the full cone.

The RATIO of the height to the base will be the same for both cones as they are similar.6cm5cm4cmxcmSmall Cone (Height divided by base radius)Big Cone (Height divided by base radius)x 4x + 5 6x 4x + 5 6=6x 244x + 20 24=6x4x + 20=2x20=x10=Multiply all of left fraction by 6 and all of right fraction by 4 (remember this doesnt change their actual values)Multiply both sides by 24Subtract 4xDivide by 2

FrustumsTo calculate the Volume of a Frustum, you must consider the Full Cone, as well as the smaller one which has been removed.

Volume of a Pyramid/Cone = 1/3 x base x height

Big Cone

Small Cone

Big Cone Small Cone6cm5cm4cmxcm10cmVb = 1/3 x x 62 x 15Vs = 1/3 x x 42 x 10Vb = 180Vs = 160/3Vf = 180 - 160/3Vf = 397.94 cm3

FrustumsTo calculate the Volume of a Frustum, you must consider the Full Cone, as well as the smaller one which has been removed.

You will need to work out the missing height, which is part of the full cone.

The RATIO of the height to the base will be the same for both cones as they are similar.8cm9cm5cmxcmSmall Cone (Height divided by base radius)Big Cone (Height divided by base radius)x 5x + 9 8x 5x + 9 8=8x 405x + 45 40=8x5x + 45=3x45=x15=Multiply all of left fraction by 8 and all of right fraction by 5 (remember this doesnt change their actual values)Multiply by 40Subtract 5xDivide by 3

FrustumsTo calculate the Volume of a Frustum, you must consider the Full Cone, as well as the smaller one which has been removed.

Volume of a Pyramid/Cone = 1/3 x base x height

Big Cone

Small Cone

Big Cone Small Cone8cm7cm5cmxcm15cmVb = 1/3 x x 82 x 22Vs = 1/3 x x 52 x 15Vb = 1408/3Vs = 125Vf = 1408/3 - 125Vf = 1081.76 cm3

PlenaryTo the right is a picture of The Temple of Kukulcn, in Mexico. By modelling it as a Frustum of a Square-based Pyramid, estimate its Volume. (Ignore the top bit!)24m55.3m19.52mSmall Pyramid (Height divided by base width)Big Pyramid (Height divided by base width)x 19.52x + 24 55.319.52 x 55.3 = 1079.456x 19.52x + 24 55.3=55.3x 1079.45619.52x + 468.48 1079.456=55.3x19.52x + 468.48=35.78x468.48=x13.09=Make Fractions EquivalentRemove the FractionSubtract 19.52xDivide by 35.78x

PlenaryTo the right is a picture of The Temple of Kukulcn, in Mexico. By modelling it as a Frustum of a Square-based Pyramid, estimate its Volume. (Ignore the top bit!)

24m55.3m19.52m13mVb = 1/3 x 55.32 x 37.09Vs = 1/3 x 19.522 x 13.09Vb = 37,811.599 m3Vs = 1,662.988 m3Vf = 37811.599 1662.988Vf = 36,148.61 m3

SummaryToday we have seen how to calculate the Volume of a Frustum

This involved the fact that a Frustum is made by removing a smaller cone/pyramid from a bigger one, and that both are similar shapes

This also recapped your knowledge of equations containing fractions

*