A triangle has two corners with angles of $\frac{π}{3}$ $( pi ) / 3$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $12$ $12$ , what is the largest possible area of the triangle?

Question

A triangle has two corners with angles of $\frac{π}{3}$ $( pi ) / 3$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $12$ $12$ , what is the largest possible area of the triangle?

1 Answer

Impact of this question

1843 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-12-07T23:33:16

$72 \sqrt{3}$ $72sqrt3$ square units

Explanation:

First, we know that the angels of a triangle has to add up to $π$ $pi$ . Therefore, we know that the leftover angle is $π - (\frac{π}{3} + \frac{π}{6}) = \frac{π}{2}$ $pi-(pi/3+pi/6)=pi/2$ which is a right angle. Therefore, we know that the triangle is a right triangle. We also know that this is the special "30-60-90" triangle. Therefore, we can figure out the sides while assuming 12 being one of each: the shortest, the middle, the longest.

Remember that in a "30-60-90" triangle, the shortest side is $a$ $a$ , the second longest (middle) side is $a \sqrt{3}$ $asqrt3$ and the longest side is $2 a$ $2a$ .
Also, you multiply the two legs and then divide it by two to find the area of a right triangle.

When 12 is the hypotenuse, we know that the shortest side is 6 and the middle side is $6 \sqrt{3}$ $6sqrt3$ with the area of $18 \sqrt{3}$ $18sqrt3$

When 12 is the shortest side, we don't really care about the hypotenuse. The middle side is $12 \sqrt{3}$ $12sqrt3$ with the area of $72 \sqrt{3}$ $72sqrt3$ .

When 12 is the middle side, we again don't care about the hypotenuse. The shortest side would be $\frac{12}{\sqrt{3}}$ $12/sqrt3$ which really is $4 \sqrt{3}$ $4sqrt3$ with the area of $24 \sqrt{3}$ $24sqrt3$ . Out of these three, we see that $72 \sqrt{3}$ $72sqrt3$ is the largest possible area.

Science

Math

Humanities

... and beyond

A triangle has two corners with angles of $\frac{π}{3}$ $( pi ) / 3$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $12$ $12$ , what is the largest possible area of the triangle?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

A triangle has two corners with angles of π3 ( pi ) / 3 and π6 ( pi )/ 6 . If one side of the triangle has a length of 1212 , what is the largest possible area of the triangle?

1 Answer

Explanation:

Related questions

Impact of this question

A triangle has two corners with angles of $\frac{π}{3}$ $( pi ) / 3$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $12$ $12$ , what is the largest possible area of the triangle?