Triangles are extremely common shapes, found both on their own and within larger shapes. Therefore, being comfortable with triangle calculations is critical to an understanding of geometry. Many of the properties of a triangle are interrelated. For example, the area of a triangle is related to its base and height, which are in turn related to its sides and angles. Therefore, finding the horizontal base of a triangle can be accomplished through a number of approaches.
Find the Base from the Area
Measure the height of the triangle. If it is a right triangle (a triangle with one 90 degree angle), the height is perpendicular to the base. If not, the height is a line that is perpendicular base that also touches the vertex (corner) opposite the base.
Write down the formula for a triangle's area, substituting your known area and height. For example, A = (1/2) * b * h. For an area of 42 square inches and a height of 14 inches, use 42 = (1/2) * 14 * b.
Solve for b in the area formula. For example, 42 = (1/2) * 14 * b, then 42 = 7 * b and finally 6 = b. In this case, the length of the base is 6 inches.
Measure the angles to see if your triangle is a right triangle. If the triangle has a 90 degree angle, it is a right triangle.
Measure the lengths of the sides other than the base.
Enter the measurements into the Pythagorean theorem. The Pythagorean theorem is a special formula that applies to the sides of right triangles only. It states that a^2 + b^2 = c^2, where a and b are the height and base of the triangle and c is the hypotenuse -- the side opposite the 90 degree angle.
For example, for a height of 24 inches and a hypotenuse of 30 inches:
a^2 + b^2 = c^2
(24)^2 + b^2 = (30)^2
Solve for b in the Pythagorean Theorem. For example, (24)^2 + b^2 = (30)^2, then 576 + b^2 = 900, b^2 = 324 and finally b = 18 inches.
The Law of Sines
Calculate the base using the Law of Sines if you know two angles and a side or two sides and an opposite angle of your triangle. The Law of Sines is as follows -- (sin A)/a = (sin B)/b = (sin C)/c, where A, B, and C are the angles opposite the sides a, b and c, respectively.
Enter the measurements you know into the Law of Sines. There are certain situations where the Law of Sines does not work, or yields an ambiguous case (a case with more than one solution). If you are given two sides and an opposite angle, make sure you know whether the triangle is acute (contains no angle greater than 90 degrees) or obtuse (contains an angle greater than 90 degrees). For example, if you know the two sides other than the base and one of the opposite angles, and you know that the triangle is acute, then [sin (30)]/10 = (sin C)/19.
Solve for the missing values. At worst, if you know the two sides other than the base and one of the opposite angles, you will have to solve for the second and third angles first, remembering that the angles in a triangle total 180 degrees. For example, [sin (30)]/10 = (sin C)/19, or 0.05 = (sin C)/19, 0.95 = sin C and finally, C approximately equals 71.8 degrees. Because the total number of degrees in the three angles of a triangle must equal 180, you can determine the angle opposite the base -- A + B + C = 180, 30 + 71.8 + B = 180, 101.8 + B = 180 and therefore B = 78.2 degrees.
Solve for the base once you have the angle opposite the base. For example, (sin A)/a = (sin B)/b, then [sin (30)]/10 = [sin (78.2)]/b, 0.05 = [sin (78.2)]/b, 0.05b = sin (78.2), b = [sin (78.2)]/0.05 and finally, b equals approximately 19.6.