A triangle's altitude describes the distance from its highest vertex to the baseline. In right triangles, this is equal to the length of the vertical side. In equilateral and isosceles triangles, the altitude forms an imaginary line that bisects the base, creating two right triangles, which may then be solved using the Pythagorean Theorem. In scalene triangles, the altitude may fall inside the shape at any place along the base or outside the triangle completely. Therefore, mathematicians derive the altitude formula from the two formulas for area instead of from the Pythagorean Theorem.
Equilateral and Isosceles Triangles
Draw the height of the triangle and call it "a."
Multiply the base of the triangle by 0.5. The answer is the base "b," of the right triangle formed by the height and sides of the original shape. For example, if the base is 6 cm, the base of the right triangle equals 3 cm.
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Call the side of the original triangle, which is now the hypotenuse of the new right triangle, "c."
Substitute these values into the Pythagorean Theorem, which states that a^2 + b^2 = c^2. For instance, if b = 3 and c = 6, the equation would look like this: a^2 + 3^2 = 6^2.
Rearrange the equation to isolate a^2. Rearranged, the equation looks like this: a^2 = 6^2 - 3^2.
Take the square root of both sides to isolate the altitude, "a." The final equation reads a = √(b^2 - c^2). For example, a = √(6^2 - 3^2), or √27.
Label the sides of the triangle a, b and c.
Label the angles A, B and C. Each angle should correspond to the name of the side opposite it. For example, angle A should be directly across from side a.
Substitute the dimensions of each side and angle into the area formula: Area = ab(Sin C)/2. For instance, if a = 20 cm, b = 11 cm and C = 46 degrees, the formula would look like this: Area = 20*11(Sin 46)/2, or 220(Sin 46)/2.
Solve the equation to determine the area of the triangle. The triangle's area is approximately 79.13 cm^2.
Substitute the area and the length of the base into a second area equation: Area = 1/2(Base * Height). If side a is the base, the equation would look like this: 79.13 = 1/2(20 * Height).
Rearrange the equation so that the height, or altitude, is isolated on one side: Altitude = (2 * Area)/Base. The final equation is Altitude = 2(79.13)/20.
To solve for the height of a scalene triangle using a single equation, substitute the formula for area into the altitude equation: Altitude = 2[ab(Sin C)/2]/Base, or ab(Sin C)/Base.