Final answer:
To find the area of an equilateral triangle when given the length of one side, we use the trigonometric area formula. We can find the height of the triangle by dividing the side length by 2 and multiplying it by the square root of 3. Then, we can use the formula for the area of a triangle to calculate the area.
Step-by-step explanation:
The trigonometric area formula can be used to find the area of an equilateral triangle if you are only given the length of one side. An equilateral triangle has all three sides equal in length and all three angles equal to 60 degrees. The formula for the area of a triangle is 1/2 × base × height. In an equilateral triangle, the base is one side and the height is the perpendicular distance from that side to the opposite vertex.
Let's say the length of one side of the equilateral triangle is s. By drawing an altitude from one vertex to the midpoint of the base, we create two congruent right triangles. The base of each right triangle will be half the length of a side, which is s/2. The height of each right triangle can be found using the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse. Since the hypotenuse is the side length s and the legs are each s/2, the height is s √3 / 2.
Now that we know the base and height of each right triangle, we can plug these values into the formula for the area of a triangle:
Area = 1/2 × (base) × (height)
Area = 1/2 × (s/2) × (s √3 / 2)
Area = s2 √3 / 4
Therefore, the area of the equilateral triangle is s2 √3 / 4.