A triangle is the simplest figure closed on a plane, consisting of only three interconnected segments. In geometry problems, it is often necessary to determine the area of this figure. What do you need to know for this? In the article we answer the question of how to find the area of a triangle on three sides.
General formula
Every student knows that the area of a triangle is calculated as the product of the length of either side - a and half the height - h, lowered to the selected side. Below is the corresponding formula: S = a * h / 2.
This expression can be used if at least two sides and the value of the angle between them are known. In this case, the height h is easy to calculate using trigonometric functions, for example, a sine. But not everyone knows how to find an area on three sides of a triangle.
Heron's formula
This formula is the answer to the question of how to find the area of a triangle on three sides. Before writing it, we denote the lengths of the segments of an arbitrary figure as a, b, and c. The Heron formula is written as follows: S = √ (p * (pa) * (pb) * (pc)).
Where p is the semiperimeter of the figure, that is: p = (a + b + c) / 2.
Despite the apparent cumbersomeness, the above expression for the area S is easy to remember. To do this, you must first calculate the half-perimeter of the triangle, then subtract from it the length of the side of the figure, multiply all the differences obtained and the half-perimeter itself. At the end, take the square root of the work.
This formula is named after Heron of Alexandria, who lived at the beginning of our era. Modern history believes that it was this philosopher who first applied the indicated expression to perform the corresponding calculations. This formula is published in his work "Metric", which dates from the 60th year of our era. Note that some of the works of Archimedes, who lived two centuries earlier than Heron, contain signs that the Greek philosopher already knew the formula. In addition, the ancient Chinese also knew how to find the area of a triangle, knowing three sides.
It is important to note that the problem can be solved without knowing the existence of the Heron formula. To do this, draw a couple of heights in the triangle and use the general formula from the previous paragraph, making up the corresponding system of equations.
The Heron expression can be used to calculate the areas of arbitrary polygons, previously breaking them into triangles and calculating the lengths of the resulting diagonals.
Problem solving example
Knowing how to find the area of a triangle on three sides, we consolidate the knowledge obtained by solving the following problem. Let the sides of the figure be 5 cm, 4 cm and 3 cm. You need to find the area.
Three sides of the triangle are known, which means you can use the Heron formula. We calculate the half-perimeter and the necessary differences, we have:
- p = (a + b + c) / 2 = 6 cm;
- pa = 1 cm;
- pb = 2 cm;
- pc = 3 cm.
Then we get the area: S = √ (p * (pa) * (pb) * (pc)) = √ (6 * 1 * 2 * 3) = 6 cm 2 .
The triangle given in the condition of the problem is right-angled, which is not difficult to verify using the Pythagorean theorem. Since the area of such a triangle is equal to half of the leg product, we get: S = 4 * 3/2 = 6 cm 2 .
The obtained value coincides with that for the Heron formula, which confirms the validity of the latter.