Final answer:
The area of a triangle formula is derived from the area of a rectangle by considering a triangle as half of a rectangle when cut along its height. This explanation shows the logical connection and consistency of mathematical formulas. The Pythagorean theorem further exemplifies the reliability of such formulas when applied correctly.
Step-by-step explanation:
The formula for the area of a triangle is 1/2 × base × height, which makes logical sense when you consider the properties of a triangle. This formula is derived from the area formula of a rectangle, which is base × height. Since a triangle can be thought of as half of a rectangle when it is divided by its height from the base, the area is also half that of the rectangle formed by the same base and height.
Additionally, in terms of practical application, using this formula provides consistency and reliability, akin to the principles observed in physics and trigonometry. The reliability of mathematical formulas is predicated on their logical derivation and universal applicability. For instance, whether using the Pythagorean theorem to find the distance between two points or calculating the hypotenuse of a right triangle, the correct application of these formulas will always yield the same results when done accurately without any arithmetic errors.
The area of a triangle can be calculated by converting all measurements to the same unit. As an example, if its base is 1.007 m and its height is 0.665 m, the area would be 1/2 × 1.007 m × 0.665 m = 0.334 square meters, which needs to be expressed with the proper number of significant figures depending on the context.