How to Find the Perimeter of a Rhombus When Given the Area

When a rhombus is a rectangle it is also a square.
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A rhombus is a four-sided shape where all of the sides are of equal length. Depending on the skew of the interior angles, rhombi are sometimes called rectangles or diamonds. Like other quadrilaterals, you can use stable formulas to calculate the properties of rhombi such as tilt, size and area if there is enough given information. For example, there are three ways to calculate the area of a rhombus: With the product of the base and height; with the sin of the angles, or with the product of the diagonals. If the area is known, you can rearrange these same formulas to produce the the length of the sides or the perimeter of the shape.

Base-Height Method

    Ensure all of your measurements are in the same units. For example, if the area is square inches, the lengths should be in inches.

    Divide the area of the rhombus by the height to find the length of one side. For example, if the area is 50 and the height is 5, the quotient of the equation is 10.

    Multiply the quotient by 4. The product of 10 and 4 is 40.

    Label the solution with the same unit used for the height. In this case, the solution is 40 inches.

Sin of Angle Method

    Write down the following formula and fill in the known information: perimeter = 4[area/sin(interior angle)]

    Calculate the sin of one of the angles of the rhombus by entering the value into a calculator and pressing the "Sin" key. The adjacent angles within a rhombus are supplementary which means they add up to 180 degrees and have the same sin so it does not matter which angle you use. For example, if the angles are 30 and 150 the sin will be .5 either way.

    Divide the area by the sin of the angle. For example, if the area is 50 square inches and the angle is 30 degrees, the quotient is 100.

    Multiply the quotient by 4 to get the solution, 400. Label the solution with the proper unit measurement, 400 inches.

Diagonal Formula

    Find the length of the diagonals: X and Y. If only one diagonal is known, calculate the value of the other diagonal using the following formula: (2 * area)/X = Y. Multiply the area by 2 and then divide it by the known diagonal.

    Write down and fill in the following formula with the known information: (1/2X)^2 + (1/2Y)^2 = side^2. If the diagonals are 10 and 20 the formula would read: [(1/2 * 10)^2 + (1/2 * 20)^2 = side^2. Solve the equation starting with the parenthetical phrases and exponents. Ten times .5 is 5. Five squared is 25. Twenty times .5 is 10, squares is 100. Twenty-five plus 100 is 125. The square root of 125 is the value of one side of the rhombus, 11.18.

    Multiply the value of one side by 4 to find the perimeter. For example, 11.18 times 4 is 44.72. Label the solution appropriately based on the units of the diagonals.


About the Author

Sylvia Cini has written informative articles for parents and educators since 2009. Her articles appear on various websites. Cini has worked as a mentor, grief counselor, tutor, recreational leader and school volunteer coordinator. She holds a Bachelor of Arts in psychology from Clark University of Worcester, Massachusetts.

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