Same Math › Geometry · grades 9–10 › Similar triangles and scale factor
How do similar triangles work?
Similar triangles have the same shape but not the same size: matching angles are equal and matching sides grow by the same scale factor. If a side of 4 matches a side of 8, the scale factor is 2, so a side of 5 matches 10. Kids set up a proportion like 4/8 = 5/x and solve it.
- Scale factor: big side ÷ matching small side.
- Missing side: multiply (or divide) by the scale factor.
- Match sides by their position, not by how they look.
Why schools teach it
Measuring what you can't reach
Geometry students define similarity with scale factors and use it to solve problems (HSG.SRT.A.2 and B.5). It's how you find a tree's height from its shadow.
The common mistake is pairing the wrong sides. Classes have kids redraw the triangles facing the same way and label matching sides first.
At the kitchen table
What to say, and what to skip
Say this
Which side in the big triangle matches this one?
Same position, across from the same angle.What's the scale factor?
Big ÷ small.Can you write it as a proportion?
small/big = small/big.Does your answer make sense? Bigger triangle, bigger side?
Skip this
- Adding instead of multiplying.4 → 8 is × 2, not + 4. The 5 becomes 10, not 9.
- Mixing the order in a proportion.Keep small/big on both sides.
- Assuming triangles are similar because they look it.You need matching angles (AA) or sides in proportion.
See it in action
A 6-foot person casts a 4-foot shadow; a tree casts a 30-foot shadow
Set up height over shadow for both
6/4 = h/30Cross-multiply
4h = 180Solve
h = 45 feet
The sun hits both at the same angle, which is what makes the triangles similar.
Free practice sheet
Make a similar triangles sheet
Loading practice sheet…
Fridge cheat sheets
Math Fridge Cheat Sheets
Every K–8 math method in one printable book (159 pages): a cheat sheet page like this guide, plus a practice sheet and its answer key for each, in Classic and handwritten Notebook styles. $9 one time, or all 7 subjects for $29.99. See all prices.
Teacher? This method has a classroom pack on TeachersPayTeachers: an anchor chart, a parent letter and practice sheets with answer keys, $4. See the pack
Parents: tell your child's teacher about this guide
Parents also ask
How do you prove triangles similar?
AA (two matching angles) is the usual way. SSS and SAS similarity use sides in proportion.
Is the scale factor always bigger than 1?
No. Going from big to small, it's less than 1, like 1/2.
What happens to area?
It grows by the scale factor squared. Double the sides and the area is 4 times as big.
Is similar the same as congruent?
Congruent is similar with a scale factor of 1: same shape and same size.