Back to: MATHEMATICS JSS3

**Welcome to Class !!**

*We are eager to have you join us !!*

*In today’s Mathematics class, We will continue our lesson on Geometry. We hope you enjoy the class!*

**GEOMETRY CONT’D**

Areas and Volumes of similar shaped

Scale Factor (Length Ratio)

P 9cm Q

A B

2 6cm

D 3 C

S R

$Scalefactor=\frac{9}{3}or\frac{6}{2}=\frac{3}{1}$

Area factor = (Scale factor)^{2}

${\left(\frac{3}{1}\right)}^{2}=\frac{9}{1}or9.1$

If the ratio of the length is X: Y

Then the ratio of the area is X^{2}: Y^{2}

Example 1:

In the diagram below, if the area of triangle ABC is 48cm^{2}, find the area of triangle XYZ to 3s.f

A X

55^{0}

65^{0} 60^{0 }60^{0}

^{ } B 12cm C Y 4cm Z

Solution

In ∆ABC, <A = 180 – (65 + 60) = 55

In ∆XYZ, <Y = 180 – (55-60) = 65^{0}

Since then corresponding angles are equal, ∆ABC and ∆XYZ are similar

$Scalefactor=\frac{4}{12}=\frac{1}{3}\phantom{\rule{0ex}{0ex}}Areafactor={\left(\frac{1}{3}\right)}^{2}=\frac{1}{9}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{Areaof\u2206XYZ}{Areaof\u2206ABC}=\frac{1}{9}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{Areaof\u2206XYZ}{48}=\frac{1}{9}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Areaof\u2206XYZ\times 9=48\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Areaof\u2206XYZ=\frac{48}{9}\phantom{\rule{0ex}{0ex}}=5.3333c{m}^{2}\phantom{\rule{0ex}{0ex}}Areaof\u2206XYZ=5.3c{m}^{2}to2s.f\phantom{\rule{0ex}{0ex}}$

**The volume of Similar Shapes**

Volume factor = (Scale factor)^{3}

If the ratio of the length is X: Y

Then the ratio of the volume is X^{3}: Y^{3}

**Example 2:** The ratio of a cylinder of volume 2970cm^{3} is 30mm. find the volume of a similar cylinder of radius 40mm.

**Solution**

$Scalefactor=\frac{40mm}{30mm}=\frac{4}{3}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Volumeratio={\left(\frac{4}{3}\right)}^{3}=\frac{{4}^{3}}{{3}^{3}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{V}{2970}=\frac{{4}^{3}}{{3}^{3}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Crossmultiply\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}V\times 27=2970\times 64\phantom{\rule{0ex}{0ex}}V=\frac{2970\times 64}{27}=7040\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Thevolumeofthecylinderis7040c{m}^{3}$

*We have come to the end of this class. We do hope you enjoyed the class?*

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*In our next class, we will be talking about the Area of Plane Figures. We are very much eager to meet you there.*

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