Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

If I have two triangles ABC and ADE and I know that each of the triangles is equilateral, each of the angles is 60 degrees. Is there a way to determine the scale factor of the two triangles, for example:

The ratio:

Area of ADE $:$Area of ABC

Thanks in advance.

enter image description here

share|cite|improve this question
If the ratio of corresponding lengths in two similar figures is $p:q$, then the ratio of their areas is $p^2:q^2$. (The ratio of their volumes would be $p^3:q^3$.) – Blue Jun 5 '13 at 20:23
Yeah, I understand that Area of ADE $:$ Area of ABC $=$ $k^{2}$, where k is the scale factor. – Sujaan Kunalan Jun 5 '13 at 20:26
Usually I am used to finding scale factor by dividing two sides but since no sides are given, I'm not sure how to find it. – Sujaan Kunalan Jun 5 '13 at 20:27
Is there a figure? Without information beyond "each of the triangles is equilateral" (the $60^\circ$ thing is redundant) and that the triangles share vertex $A$, there's no way to determine the scale factor. Where are points $D$ and $E$? – Blue Jun 5 '13 at 20:29
Give me a minute- I'll put up the figure. – Sujaan Kunalan Jun 5 '13 at 20:34
up vote 3 down vote accepted

If side $DC$$ = $ $x$, then $AC$ must be $2x$ because both triangles are equilateral. By Pythagorean theorem, $AD$ is $3^\frac{1}{2}x$. Then $AD:AC=3^\frac{1}{2}:2=\frac{3^\frac{1}{2}{}}{2}:1$. So the scale factor $k$, is $\frac{3^\frac{1}{2}}{2}$. Using the formula:

$\frac{\text{Area of }AED}{\text{Area of }ABC}=k^2$

$\frac{\text{Area of }AED}{\text{Area of }ABC}=(\frac{3^\frac{1}{2}}{2})^2$

$\frac{\text{Area of }AED}{\text{Area of }ABC}=\frac{3}{4}$

Therefore $k^2= \frac{3}{4}$ and so the ratio between triangle $ADE$ and $ABC$ is $1:\frac{3}{4}$

share|cite|improve this answer
(+1) I'd have written the answer as $4:3$, personally, but that's just a matter of taste. – Cameron Buie Jun 5 '13 at 23:20

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.