Calculating the Area of Quadrilateral ABCD: A Comprehensive SEO-Optimized Guide

The area of a quadrilateral can be calculated using various methods, especially when given the side lengths and a diagonal. In this article, we will discuss how to find the area of quadrilateral ABCD given the side lengths AB 42m, BC 21m, CD 29m, DA 34m, and the diagonal BD 20m. We will explore the steps involved in using Heron's formula to calculate the areas of the triangles formed by the diagonal and then sum them up to get the total area. This guide is optimized for SEO with relevant keywords and a detailed step-by-step explanation for better readability and comprehension.

Step-by-Step Guide to Calculate the Area of Quadrilateral ABCD

In this section, we will break down the process of calculating the area of quadrilateral ABCD into a series of steps. Each step is designed to provide a clear understanding of the method used and the mathematical concepts involved.

Step 1: Divide the Quadrilateral

To find the area of quadrilateral ABCD, we start by dividing it into two triangles: ABD and BCD using the diagonal BD. By breaking down the problem into smaller, more manageable parts, we can apply the appropriate formulas to each triangle.

Step 2: Calculate the Area of Triangle ABD

Side lengths:
AB 42m
AD 34m
B
Diagonal BD 20m
sABD (semi-perimeter of ABD) (AB AD BD) / 2 (42 34 20) / 2 48m

Area using Heron's Formula:

AreaABD √[sABD(sABD - AB)(sABD - AD)(sABD - BD)]
AreaABD √[48 * (48 - 42) * (48 - 34) * (48 - 20)]
AreaABD √[48 * 6 * 14 * 28]

Calculating each term:
48 * 6 288
14 * 28 392

Now multiplying:
288 * 392 112896

Finally, taking the square root:
AreaABD ≈ √112896 ≈ 336 m2

Step 3: Calculate the Area of Triangle BCD

Side lengths:
BC 21m
CD 29m
Diagonal BD 20m
sBCD (semi-perimeter of BCD) (BC CD BD) / 2 (21 29 20) / 2 35m

Area using Heron's Formula:

AreaBCD √[sBCD(sBCD - BC)(sBCD - CD)(sBCD - BD)]
AreaBCD √[35 * (35 - 21) * (35 - 29) * (35 - 20)]
AreaBCD √[35 * 14 * 6 * 15]

Calculating each term:
35 * 14 490
6 * 15 90

Now multiplying:
490 * 90 44100

Finally, taking the square root:
AreaBCD ≈ √44100 ≈ 210 m2

Step 4: Total Area of Quadrilateral ABCD

The total area of quadrilateral ABCD is the sum of the areas of the two triangles ABD and BCD:

AreaABCD AreaABD AreaBCD 336 m2 210 m2 546 m2

Final Answer:
The area of quadrilateral ABCD is approximately 546 m2.

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Conclusion

Understanding how to calculate the area of a quadrilateral, especially when given the side lengths and a diagonal, is a valuable skill in geometry. This comprehensive guide provides a step-by-step approach using Heron's formula and clearly explains the process, making it easier for students and professionals alike to apply these concepts in their work.