Problem 164 - Calculus Volume 3

Question

Find the area of a pentagon with vertices \((0,4),(4,1),(3,0),(-1,-1),\) and (-2,2).

Step-by-Step Solution

Answer

The area of the pentagon is 17 square units.

Step 1: Identify the Coordinates

The vertices of the pentagon are provided. We have: \[ A(0,4), \, B(4,1), \, C(3,0), \, D(-1,-1), \, E(-2,2) \].

Step 2: Order the Vertices

Ensure that the vertices are ordered either clockwise or counterclockwise: We can list them as: \[ A(0,4) ightarrow B(4,1) ightarrow C(3,0) ightarrow D(-1,-1) ightarrow E(-2,2) \].

Step 3: Apply the Shoelace Formula

The Shoelace formula for area is: \[ \text{Area} = \frac{1}{2} \left| \sum (x_i y_{i+1}) - \sum (y_i x_{i+1}) \right| \] First, list the coordinates in order then repeat the first coordinate: \( (0,4), (4,1), (3,0), (-1,-1), (-2,2), (0,4) \).

Step 4: Coordinate Multiplication for the Formula

Multiply the x-coordinate of each vertex by the y-coordinate of the next vertex, sum them: \[(0\times1) + (4\times0) + (3\times-1) + (-1\times2) + (-2\times4) = 0 + 0 - 3 - 2 - 8 = -13.\]

Step 5: Second Part of the Formula

Multiply the y-coordinate of each vertex by the x-coordinate of the next vertex, sum them: \[(4\times4) + (1\times3) + (0\times-1) + (-1\times-2) + (2\times0) = 16 + 3 + 0 + 2 + 0 = 21.\]

Step 6: Calculate the Area

Substitute these sums into the Shoelace formula: \[ \text{Area} = \frac{1}{2} \left| -13 - 21 \right| = \frac{1}{2} \times 34 = 17. \]

Step 7: Final Answer

Therefore, the area of the pentagon is \( 17 \) square units.

Key Concepts

Pentagon Area Calculation

Calculating the area of a pentagon, especially an irregular one, requires knowing the vertices' coordinates. First, determine the precise order of these points, whether listed clockwise or counterclockwise. This order is crucial for applying formulas accurately. One effective method for finding the area when the coordinates are known is the Shoelace Formula, which involves systematic multiplication of coordinates.

• Start by listing the pentagon’s vertices in a complete loop, including repeating the starting vertex at the end.
• Utilize the Shoelace Formula to calculate two sums of products, one for each of the x and y coordinates.
• The absolute difference between these sums gives you a value needed for the final area calculation.

After doing the calculations correctly, you obtain the final area, which should align with any geometric understanding you have based on the shape’s sides overall. This method is valued in mathematics due to its accuracy and systematic approach.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the bridge between algebra and geometry through the use of graphs and coordinates. It allows precise placement of shapes on a coordinate plane, making it easier to compute lengths, areas, and other geometrical properties using formulas. In our exercise, each vertex of the pentagon is represented as a coordinate pair \(x, y\), giving us a visual structure through which we can apply formulas like the Shoelace Formula efficiently.

• Identify the coordinate points of each vertex. Having this list is essential for any calculation.
• These coordinates become your basis for exploring further geometric properties, like distances between points (using the distance formula) or slopes of line segments connecting the points.
• The orientation of these points (whether ordered clockwise or counterclockwise) affects the result measurably in area calculation processes.

Coordinate geometry facilitates not only the computation of areas like in the pentagon example but also dives into more complex geometric investigations undivided by its algebraic prowess.

Polygon Area

To find a polygon's area, different methods apply depending on the polygon type and available information. For regular polygons, straightforward formulas exist since all sides and angles are equal. However, with irregular polygons, such as the one in the exercise, involving pentagons, the Shoelace Formula is particularly useful when coordinates are known. This method converts a geometric shape's challenge into a systematic algebraic process.

• For irregular polygons, like the pentagon, systematically apply a coordinate grid approach.
• The Shoelace Formula’s core is capturing the cross-product of adjacent vertex coordinates which provides an algebraic solution to geometric shapes.
• Determine the area by considering both the positive order of vertex coordinates and their impact on subtractions in cross-product terms, ensuring accuracy.

This technique highlights the power of treating polygons algebraically, enabling deeper insights into geometric concepts while maintaining precision through mathematical means. The derived area gives an understood size representing the level surface the shape occupies.