Key takeaway: Similar figures are two shapes with exactly the same angles and side lengths in the same ratio, so each is a scaled copy of the other, even when one is turned or flipped.
Grade 88.G.A.4Geometry and Measurement
Definition of Similar Figures
Similar figures keep one shape while changing the size. Slide, turn or flip one of them, stretch it by a single factor, and it lands exactly on the other.
Two polygons are similar when every pair of matching angles is equal and every pair of matching sides is in the same ratio. That common ratio is the scale factor. A city map and the real streets it shows are similar figures, and so are a movie poster and the small copy of it on a ticket.
Formula and notation
The symbol \(\sim\) means “is similar to”. Write the vertices in matching order:
\[ABCD \sim EFGH\]
If the scale factor from the first figure to the second is \(k\), then each side of the second equals \(k\) times the matching side of the first, while every matching angle stays equal.
Key facts
- Matching angles are equal, but side lengths are multiplied by the same scale factor.
- A scale factor above 1 enlarges the figure; a factor between 0 and 1 shrinks it.
- When the factor equals 1, the figures are congruent.
- Their areas compare as the factor multiplied by itself.
Similar Figures example
A club logo is a rectangle 14 inches long and 8 inches wide. A banner copy keeps the same shape and is 35 inches long. How wide is the banner?
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Scale factor from length to length: \(\frac{35}{14} = 2.5\).
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Apply the same factor to the width: \(8 \times 2.5 = 20\).
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Check the ratios: \(\frac{8}{14} = \frac{4}{7}\) and \(\frac{20}{35} = \frac{4}{7}\).
The banner is 20 inches wide.
Common mistake
Watch out
Priya sees that the banner is 21 inches longer than the logo (\(35 – 14 = 21\)), so she adds 21 to the width and gets \(8 + 21 = 29\) inches.
Why it is wrong
Similarity works by multiplying, not adding. Adding 21 to both sides would make a long, wide rectangle with a different shape: \(\frac{29}{35} \neq \frac{8}{14}\). Multiply every side by 2.5 to keep the proportions.
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Geometry and Measurement