The area of a regular triangle.

We can fold the regular triangle into two congruent right triangles.

Then, the side which got folded is $\displaystyle \frac \pi 2$ .

We can use the Pythagorean theory.

If I let the height of the right triangle to $a$ , then $\displaystyle a ^ { 2 } + \left ( \frac { \pi } { 2 } \right ) ^ { 2 } = \pi ^ { 2 }$ .

Then $\displaystyle a ^ { 2 } = \frac { \pi ^ { 2 } \times 3 } { 4 }$ .

$\displaystyle a = \pm \sqrt { \frac { 3 \pi ^ { 2 } } { 4 } }$ , but the length cannot be negative, so $\displaystyle a = \frac { \sqrt { 3 } \pi } { 2 }$ .

Then the area of a triangle is $\displaystyle \frac { d \times h } { 2 }$ , $d$ is a down side of the triangle, and $h$ is a height of the triangle.

We get $\displaystyle \frac { \pi ( \frac { \sqrt { 3 } \pi } { 2 } ) } { 2 }$ $\displaystyle = \frac { \frac { \pi ^ { 2 } \sqrt { 3 } } { 2 } } { 2 } =$ $\displaystyle \frac { \pi ^ { 2 } \sqrt { 3 } } { 2 } \div 2 =$ $\displaystyle \frac { \pi ^ { 2 } \sqrt { 3 } } { 2 } \times \frac { 1 } { 2 } =$ $\boxed { \displaystyle \frac { \pi ^ { 2 } \sqrt { 3 } } { 4 } }$ .