Calculating Area: Composite figure
Calculating the area of composite figures involves breaking down complex shapes into simpler geometric components, such as triangles, rectangles, and circles. A composite figure is a shape made up of two or more basic geometric forms, allowing for more accurate modeling of everyday objects. For example, a simple sketch of a house combines a triangle and a square to create a composite figure. The process is intuitive and has practical applications, as seen in historical contexts like ancient Egypt, where land boundaries were determined using triangular and rectangular sections after flooding altered property lines.
The ease of calculating areas for basic shapes, such as using the formulas for triangles (A = 1/2bh), rectangles (A = bh), and circles (A = πr²), makes them ideal for creating composite figures. Ancient mathematicians, including Archimedes, used innovative methods to estimate areas, such as viewing circles as combinations of triangles to refine their calculations. Understanding composite figures is particularly useful in real-world scenarios, such as measuring space for flooring, painting, or gardening, where irregular shapes are common. This method of area calculation is essential for effectively managing various practical tasks in daily life.
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Calculating Area: Composite figure
Few objects have a shape as simple as a triangle, rectangle, or circle. However, by composing several of these simple shapes, it is possible to create complex shapes that can model almost any two-dimensional object. Such a complex shape, made up of two or more simpler geometric shapes, is known as a composite figure. A basic sketch of a house, like the one in Figure 1, is a composite figure that combines a square and a triangle. Modeling complex objects with composite figures is not only rather intuitive, but also quite useful in everyday situations. For these reasons, composite figures have been used since antiquity.
The three shapes shown in Figure 2 have two advantages that make them particularly suitable for use in composite figures. They have formulas that are relatively easy to derive, remember, and use in complex calculations, and they are simple and flexible enough to approximate virtually any shape. In fact, it is for these reasons that rectangles are used to approximate the area between the graphs of functions in integral calculus.
Overview
One of the earliest recorded uses of composite figures occurred in ancient Egypt in the farmlands of the Nile River Delta. Although flooding made the soil extremely fertile, it also occasionally washed away property boundaries. Eventually, the pharaoh assigned teams of men known as "rope stretchers," to divide the irregularly shaped regions bordering the river into triangles and rectangles to calculate the area and distribute farmland consistently.
The area of a circle was more complicated for ancient mathematicians to determine because they did not have many of the geometric and algebraic techniques available to us today. Interestingly, whereas circles (and sectors of a circle) are now used to calculate the area of other composite figures, ancient mathematicians viewed the circle itself as a composite figure made of triangles in order to estimate its area. In the third century BCE, Archimedes of Syracuse had the creative idea of dividing a circle into triangular wedges. Although a portion of the circle’s area always remained outside of the triangles, Archimedes realized that this leftover area got smaller and smaller as he increased the number of triangles. Using a technique known now as the method of exhaustion, Archimedes increased the number of triangles arbitrarily until he correctly determined the area of a circle. Rather than the A = πr2 formula we have today, he stated that the area of a circle with radius r is equal to the area of a triangle with height r and a base with length equal to the circle’s circumference.
Composite figures are more common in the real world than basic geometric figures. A room could be in the shape of a perfect square or rectangle, but it could also be in an L shape or some other kind of layout. Finding the area of composite figures assists in life when carpeting a house or purchasing an area rug, painting a room, or creating a garden in your backyard.
Bibliography
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