Similar Triangles
Area & Volume Scale Factor
OCR Module 9
D J Millward 2008
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Stage 7 Recap
AREA
VOLUME
Similar shapes
FROM STAGE 7
• Are Enlargements of each other
• Corresponding angles are equal
• Sides are related by the same scale factor
Similar Triangles
Triangles are similar if matching
angles remain the same size.
100º
30º
50º
100º
30º
50º
To calculate a length
15
5x
5
3
4
x3
6
Scale factor 3
15
1
3
18
Scale factor 1/3
12
Harder example
A
3
4
D
E
6
C
B
Triangle ABC is similar to triangle ADE.
DE is parallel to BC.
Calculate the length of BC
Harder example
A
D
3
E
9
4
6
B
C
12
9x
3
3
…and then…
AB & DE are parallel
Explain why ABC is similar to CDE
5
A
B
<CED = <BAC Alternate Angles
<EDC = <ABC Alternate Angles
<ECD = <ACB Vert Opp Angles
3
C
6
E
D
?
Triangle ABC is similar to Triangle CDE
…and then…
Calculate the length of DE
AC corresponds to CE
Scale Factor = 2
5
A
B
AB corresponds to DE
DE = 2 x AB
3
C
DE = 10cm
6
E
D
?
Summary – Similar shapes
• To calculate missing sides, we first of all need the scale factor
• We then either multiply or divide by the scale factor
• To show that 2 shapes are similar we can either show that all
of the sides are connected by the scale factor or show that
matching angles are the same
AREA SCALE FACTOR
10cm²
2cm
5cm
ENLARGE
rectangle
Scale Factor 2
40cm²
4 cm
10 cm
AREA SCALE FACTOR = 4
AREA SCALE FACTOR
10cm²
2cm
5cm
ENLARGE
rectangle
Scale Factor 3
90cm²
6 cm
15 cm
AREA SCALE FACTOR = 9
AREA SCALE FACTORS
• Reflect the Length Scale Factor
Length Scale Factor
2
3
Area Scale Factor
4 2²
9 3²
AREA SCALE FACTOR = LENGTH SCALE FACTOR SQUARED
ASF = LSF²
Example Questions - AREA
Calculate the unknown areas
Q1
Q2
10 cm²
40 cm²
4 cm
12 cm²
Q3
4 cm²
108 cm²
2 cm
9 cm²
Example Questions - SIDES
Calculate the unknown lengths
Q1
Q2
10 cm²
250 cm²
2 cm
12 cm²
6 cm
Q3
8 cm²
48 cm²
2 cm
18 cm²
VOLUME SCALE FACTOR
1 cm
3 cm
6 cm³
2 cm
ENLARGE
cuboid
Scale Factor 3
162 cm³
6 cm
VOLUME SCALE FACTOR = 27 = 3³
VOLUME SCALE FACTOR
4 cm
8 cm³
64cm³
ENLARGE
cylinder
Scale Factor 2
8 cm
Volume Scale Factor = 2³
VOLUME SCALE FACTOR = LENGTH SCALE FACTOR CUBED
VSF = LSF³
Example Questions - VOLUME
Q2
Q1
5 cm³ 2 cm
8 cm³
3 cm
135 cm³
27 cm³
4.5 cm
6 cm
Q3
40 cm³
5 cm³
4 cm
2 cm
Example Questions - SIDES
Q2
Q1
5 cm³ 2 cm
6 cm³
1.5 cm
40 cm³
384 cm³
a cm
4
6bcm
Q3
120 cm³
15 cm³
3.6 cm
1.8c cm