In an equilateral triangle height of h divides a figure into two identical rectangular triangles. In each of them h is a leg, the party of a — a hypotenuse. It is possible to express a through height of an equilateral figure, and then to find the area.
Instruction
1. Define acute angles of a rectangular triangle. One of them is equal 180 ° / 3 = 60 ° because all corners are equal in the set equilateral triangle. The second is equal 60 ° / 2 = 30 ° because height of h divides a corner into two equal parts. Here are used standard properties of triangles, knowing which, all parties and corners can be found the friend through the friend.
2. Express the party of a through h height. The corner between this leg and a hypotenuse of a — adjacent is also equal 30 ° as it was found out on the first step. Therefore h = a * cos 30 °. The opposite corner is equal 60 ° therefore h = a * sin 60 °. From here a = h/cos 30 ° = h/sin 60 °.
3. Get rid of cosines and sine. cos 30 ° = sin 60 ° = √3/2. Then a = h/cos 30 ° = h/sin 60 ° = h / (√3/2) = h * 2/√3.
4. Determine the area of an equilateral triangle of S = (1/2) * by a * h = (1/2) * (h * 2/√3) * h = h²/√3. The first part of this formula is in mathematical reference books and textbooks. In the second part instead of unknown the expression found on the third step is substituted. As a result the formula at the end of which there are no unknown parts turned out. Now it can be used for finding of the area of an equilateral triangle which is called in a different way correct because at it the parties and corners are equal.
5. Define basic data and solve a problem. Let h = 12 cm. Then S = 12 * 12/√3 = 144/1.73 = 83.24 cm.