8th grade · Math · Unit 9 of 19
Similarity & Scale
Scale factors and similar figures
What your child practices
Using scale factors to find missing sides, solving shadow and scale-drawing problems, and seeing how enlarging a shape changes its area and volume.
New York State Learning Standards learning standard: NY-8.G.4 · similar figures, scale factors and finding missing side lengths
How we explain it
Children can open this short lesson before they practice.
- Similar shapes have the same shape but different sizes.
- The scale factor is new length ÷ old length.
- Multiply every length by the scale factor. Areas change by the factor squared.
Example: A rectangle 4 cm by 6 cm is enlarged by a scale factor of 3. What are the new sides? 4 × 3 = 12. 6 × 3 = 18. Answer: 12 cm by 18 cm
Similarity & Scale in New York
In the New York State Learning Standards, this is unit 9 of 19 in 8th grade Math, after Angle Detective and before Tables, Graphs & Linear Relations.
Gradelings describes a child's practice on this unit as Beginning, Approaching, Meeting or Exceeding (Four practice steps). That reflects practice, not a report-card mark: the teacher decides the mark.
Sample questions
A taste of this unit. In the app, questions change every time and get easier or harder to match your child.
Question 1
Triangle B is an enlargement of triangle A. A side of 4 cm on triangle A matches a side of 16 cm on triangle B. What is the scale factor?
answer box
4
Hint: Scale factor = new length ÷ original length = 16 ÷ 4 = 4.
Question 2
Two rectangles are similar. The small one is 11 cm by 7 cm. The large one is 55 cm long. How wide is the large rectangle?
answer boxcm
35 cm
Hint: The scale factor is 55 ÷ 11 = 5. Multiply the width by 5: 7 × 5 = 35 cm.
Question 3
△ABC ~ △DEF. AB = 7 cm, BC = 5 cm and DE = 21 cm. How long is EF?
answer boxcm
15 cm
Hint: The sides are in proportion: DE/AB = EF/BC, so EF = 5 × (21/7) = 5 × 3 = 15 cm.
Question 4
A 1 m stick casts a 4 m shadow. At the same time, a tree casts a 32 m shadow. How tall is the tree?
answer boxm
8 m
Hint: The triangles are similar, so height ÷ shadow stays the same: tree = 1 × 8 = 8 m.
Question 5
A scale drawing uses 1 cm for every 20 cm in real life. A wall measures 7 cm on the drawing. How long is the real wall in centimeters?
answer boxcm
140 cm
Hint: Multiply by the scale: 7 × 20 = 140 cm.
Question 6
A shape is enlarged by a scale factor of 2. By what factor does its area grow?
answer box
4
Hint: Area uses two lengths, so it grows by 2² = 4.