How can I calculate the area of a rhombus if I know one diagonal and the perimeter?

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Finding the Area of a Rhombus with One Diagonal and Perimeter

The area of a rhombus can be determined easily if you have one diagonal and the perimeter. Let’s break down the steps.

Understanding the Properties of a Rhombus

A rhombus is a type of polygon that is a quadrilateral (four sides) with all sides having equal length. Additionally, the diagonals of a rhombus intersect at right angles and bisect each other.

Step-by-Step Calculation

  1. Identify the Known Variables:

    • Let d1 be the length of one diagonal.
    • Let P be the perimeter of the rhombus.
  2. Calculate the Length of One Side:

    The perimeter of a rhombus can be expressed as:

    P = 4s, where s is the length of one side.

    To find s, rearrange the formula:

    • s = P / 4
  3. Using the Diagonals to Find the Area:

    The area of a rhombus can be calculated using the formula:

    Area = (d1 * d2) / 2, where d2 is the length of the second diagonal.

    To find d2, you can use the relationship between the sides and the diagonals:

    s² = (d1/2)² + (d2/2)²

    This gives:

    P / 4 = √((d1² / 4) + (d2² / 4))

    Squaring both sides:

    (P / 4)² = (d1² / 4) + (d2² / 4)

    Then, isolate d2²:

    d2² = (P² / 16) – (d1² / 4)

    Finally, take the square root to find d2:

    d2 = √((P² / 16) – (d1² / 4))

  4. Calculate the Area:

    Now that you have both d1 and d2, substitute them back into the area formula:

    Area = (d1 * d2) / 2

Example Calculation

Let’s assume the following:

  • One diagonal, d1 = 10 units
  • Perimeter, P = 40 units

Now, calculate:

  1. Side: s = P / 4 = 40 / 4 = 10 units
  2. Find d2:
    • d2 = √((P² / 16) – (d1² / 4))
    • d2 = √((40² / 16) – (10² / 4)) = √((1600 / 16) – (100 / 4)) = √(100 – 25) = √75
    • d2 ≈ 8.66 units
  3. Finally, calculate the area:
    • Area = (d1 * d2) / 2 = (10 * 8.66) / 2 = 43.3 square units

That’s it! You now know how to calculate the area of a rhombus when provided one diagonal and the perimeter. Happy calculating!

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