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FutureStarrArea of a triangle worksheet
In order to find the area of triangle with 3 sides, we use the Heron's Formula. The area of a triangle can be calculated with the help of various formulas. The basic formula that is used to find the area of a triangle is ½ × Base × Height where "Base" is the side of the triangle on which the altitude is formed, and "Height" is the length of the altitude drawn to the "Base" from its opposite vertex. However, if the altitude of a triangle is not known, and we need to find the area of triangle with 3 different sides, the Heron's formula is used. This formula was derived by a Greek mathematician known as the Heron of Alexandria. Let us explore the different formulas that are used to find the area of a triangle with 3 sides. If we know the sides of a triangle along with its height, we can use the basic formula for the area of a triangle. Area of a triangle = 1/2 × base × height. For example, if the height (altitude) of a triangle = 8 units, and the side of the triangle on which the altitude is formed is given (base) = 7 units, we can find its area using the formula, Area of a triangle = 1/2 × base × height. Area = 1/2 × 7 × 8 = 28 square units.
As in the proof of the law of sines in the previous section, drop a perpendicular AD from the vertex A of the triangle to the side BC, and label this height h. Then triangle ACD is a right triangle, so sin C equals h/b. Therefore, h = b sin C. Since the area of the triangle is half the base a times the height h, therefore the area also equals half of ab sin C. Although the figure is an acute triangle, you can see from the discussion in the previous section that h = b sin C holds when the triangle is right or obtuse as well. Therefore, we get the general formula Although we didn't make a separate calculator for the equilateral triangle area, you can quickly calculate it in this triangle area calculator. Simply use the subpart for the area of a triangle with 3 sides - as you know that every side has the same length in an equilateral triangle. It's possible to calculate that area also in angle-side-angle or side-angle-side version - probably you remember that every angle in the equilateral triangle is equal to 60 degrees (π/3 rad).
The base and height of a triangle must be perpendicular to each other. In each of the examples below, the base is a side of the triangle. However, depending on the triangle, the height may or may not be a side of the triangle. For example, in the right triangle in Example 2, the height is a side of the triangle since it is perpendicular to the base. In the triangles in Examples 1 and 3, the lateral sides are not perpendicular to the base, so a dotted line is drawn to represent the height. This involves trigonometry. You have to find the height of the triangle, which is the distance from one vertex to the opposite side. The height is found by multiplying the length of a side (x) by half the tangent of 60°. (60° represents each of the angles in an equilateral triangle.) Half the tangent of 60° is 0.866. Thus the height is 0.866x. Multiply that by x and divide by two to get the area. (Source: www.wikihow.com)