Lesson Explainer: Area of a Rhombus Mathematics

In this explainer, we will learn how to find the area of a rhombus using the length of its diagonals.

Recall that a rhombus is any quadrilateral in which all four sides are of equal length. As a rhombus is also a parallelogram, its area can be calculated from the length of its base and perpendicular height using the formula area=π‘β„Ž.

The alternative formula for the area of a rhombus that we derive here instead uses the lengths of its diagonals. The diagonals of any parallelogram bisect one another, but an additional property of the diagonals of a rhombus is that they are perpendicular, as illustrated in the figure below.

Let us consider the diagonal 𝐡𝐷, which divides the rhombus into the congruent triangles 𝐴𝐡𝐷 and 𝐢𝐡𝐷. As the triangles are congruent, each of their areas is half of the area of the rhombus. Equivalently, we can state that areaofrhombusareaoftriangle𝐴𝐡𝐢𝐷=2×𝐴𝐡𝐷.

Suppose also that the lengths of the diagonals are π‘‘οŠ§ and π‘‘οŠ¨ units, as shown in the figure below.

The area of triangle 𝐴𝐡𝐷 can be calculated using the formula areaoftriangle=12π‘β„Ž, where 𝑏 represents the base of the triangle and β„Ž represents the perpendicular height. From the figure, we identify that the base of this triangle is 𝐡𝐷, which has length π‘‘οŠ§ units. The perpendicular height of this triangle is 𝐴𝐸. Recalling again that the diagonals of a rhombus bisect one another, we know that 𝐴𝐸 is half the length of the diagonal 𝐴𝐢 and so has length 𝑑2 units. Hence, areaoftrianglesquareunits𝐴𝐡𝐷=12×𝑑×𝑑2=𝑑𝑑4.

The area of the rhombus is therefore equal to areaofrhombussquareunits𝐴𝐡𝐢𝐷=2×𝑑𝑑4=𝑑𝑑2.

Formula: Area of a Rhombus

The area of a rhombus is equal to half the product of the lengths of its diagonals. For a rhombus with diagonals of length π‘‘οŠ§ and π‘‘οŠ¨ units, areaofrhombussquareunits=𝑑𝑑2.

In our first example, we will apply this formula to calculate the area of a rhombus that has been drawn inside a rectangle.

Example 1: Finding the Area of a Rhombus inside a Rectangle

The figure shows a rhombus within a rectangle. Find the area of the rhombus to two decimal places.

Answer

Upon inspection of the diagram, we observe that each vertex of the rhombus π‘‹π‘‡π‘π‘Œ is at the midpoint of one of the rectangle’s sides. For example, the vertex 𝑋 is at the midpoint of the side 𝐴𝐷. We know this because 𝐴𝑋 and 𝑋𝐷 are of equal length. From this, we can deduce that the rhombus’s diagonal 𝑋𝑍 is parallel to 𝐴𝐡 and 𝐷𝐢, and the diagonal π‘Œπ‘‡ is parallel to 𝐴𝐷 and 𝐡𝐢.

It also follows that 𝑋𝑍=𝐴𝐡=𝐷𝐢 and π‘Œπ‘‡=𝐴𝐷=𝐡𝐢.

The dimensions of the rectangle are given in the question and hence 𝑋𝑍=15.8cm and π‘Œπ‘‡=30.3cm.

We now recall that the area of a rhombus is equal to half the product of the lengths of its diagonals: areaofrhombus=𝑑𝑑2, where π‘‘οŠ§ and π‘‘οŠ¨ are the lengths of the diagonals. In this problem, the lengths of the diagonals are 30.3 cm and 15.8 cm, so we have areaofcmπ‘‹π‘‡π‘π‘Œ=30.3Γ—15.82=478.742=239.37.

Let us briefly note the relationship between the rhombus π‘‹π‘‡π‘π‘Œ and the rectangle 𝐴𝐡𝐢𝐷 in the previous example. The area of any rectangle is equal to its length multiplied by its width, or in other words, the product of its dimensions. In this problem, the diagonals of the rhombus were the same length as the dimensions of the rectangle, and as the area of a rhombus is half the product of its diagonals, this is equivalent to half the product of the rectangle’s dimensions. This illustrates that the area of a rhombus drawn inside a rectangle, such that each vertex of the rhombus is at the midpoint of one of the rectangle’s sides, is half the area of the rectangle enclosing it.

We have now seen one example of how to calculate the area of a rhombus given the lengths of its two diagonals. It is also possible to work in the other direction: if we are given the area of a rhombus and the length of one of its diagonals, we can calculate the length of the other diagonal by forming and solving an equation. Given the area of a rhombus, we can also calculate the length of either or both diagonals if we do not explicitly know the length of either, but we do know the relationship between their lengths, as we will see in our next example.

Example 2: Calculating a Rhombus’s Diagonal Length given Its Area

One diagonal of a rhombus is twice the length of the other diagonal. If the area of the rhombus is 81 square millimetres, what are the lengths of the diagonals?

Answer

We recall that the area of a rhombus is half the product of the lengths of its diagonals π‘‘οŠ§ and π‘‘οŠ¨: area=𝑑𝑑2.

Let the length of the shorter diagonal be π‘‘οŠ§. As the other diagonal is twice the length of the first, we have 𝑑=2π‘‘οŠ¨οŠ§. We can therefore form an equation by substituting 𝑑=2π‘‘οŠ¨οŠ§ in the above formula and setting this expression equal to the known area: 𝑑×(2𝑑)2=81.

Simplifying the expression on the left-hand side by canceling a factor of 2 gives 𝑑=81.

We solve for π‘‘οŠ§ by square rooting each side of the equation, taking only the positive value as π‘‘οŠ§ represents a length: 𝑑=√81=9.mm

We have found the length of the shorter diagonal π‘‘οŠ§ to be 9 mm. The second diagonal is twice as long, so we have 𝑑=2𝑑=2Γ—9=18.mm

The lengths of the diagonals of the given rhombus are 9 mm and 18 mm.

In our next example, we will solve a problem involving a rhombus and a square that have the same area. Given the square’s perimeter and the length of one diagonal of the rhombus, we will calculate the length of the rhombus’s other diagonal.

Example 3: Finding the Length of the Diagonal of a Rhombus given the Length of the Other Diagonal and a Shape with the Same Area

A rhombus and a square have the same area. If the square’s perimeter is 44 and one of the diagonals of the rhombus is 10, how long is the other diagonal, to two decimal places?

Answer

In order to relate the two shapes, we will need to find a value or expression for each of their areas, which we are told are equal. Let us begin by considering the square.

We are given that the perimeter of the square is 44 units. We recall that the perimeter of a shape is the distance around its edge. In the case of a square, which has four equal sides each of length 𝑠 units, the perimeter is equal to 4𝑠. Setting this expression equal to 44 gives an equation that can be solved to determine the side length of the square: 4𝑠=44𝑠=11.units

The area of the square can be calculated from its side length using the formula areaofsquare=π‘ οŠ¨. Substituting 𝑠=11 gives areaofsquaresquareunits=11=121.

We now know that the area of the rhombus is also 121 square units. Recall that the area of a rhombus is equal to half the product of the lengths of its diagonals π‘‘οŠ§ and π‘‘οŠ¨: areaofrhombus=𝑑𝑑2.

We are given that one diagonal is of length 10 units, and so we can form an equation: 10×𝑑2=121.

To solve this equation for π‘‘οŠ¨, we first simplify the left-hand side by canceling a factor of 2: 5𝑑=121.

We then divide each side of the equation by 5 to give 𝑑=1215=24.2.

To two decimal places, the length of the other diagonal of the rhombus is 24.20 units.

In the previous example, we used our knowledge of how to calculate the area of a square given its side length. However, a square is actually a particular type of rhombus in which the two diagonals are of equal length. Suppose each diagonal of a square is of length 𝑑 units. Using the formula for the area of a rhombus, the area of the square is equal to half the product of the lengths of its diagonals, leading to areaofsquare=𝑑×𝑑2=𝑑2.

Formula: Area of a Square

The area of a square is equal to half the square of the length of its diagonal. For a square with diagonal of length 𝑑 units, areaofsquaresquareunits=𝑑2.

In our next example, we consider how to find the length of the diagonal of a chessboard, by first considering the relationship between the area and the diagonal of the smaller squares from which the chessboard is composed.

Example 4: Finding the Diagonal Length of a Square given its Area

Given that the area of each square on the chessboard is 81 cm2, find the diagonal length of the chessboard.

Answer

The chessboard is composed of 64 congruent squares, arranged in 8 rows of 8. We observe that the diagonal length of the chessboard, which we will denote by 𝐷, is equal to 8 times the diagonal length of each individual square, which we will denote by 𝑑: 𝐷=8𝑑.

We are given that the area of each square on the chessboard is 81 cm2. We recall also that the area of a square can be calculated from the length of its diagonal 𝑑 using the formula areaofsquare=𝑑2.

Hence, we have 𝑑2=81.

To solve for 𝑑, we begin by multiplying each side of this equation by 2, leading to 𝑑=162.

We then square root and simplify the radical, giving 𝑑=√162=√81Γ—2=√81Γ—βˆš2=9√2.cm

Finally, we are able to calculate the length of the diagonal of the chessboard (𝐷) by recalling that 𝐷=8𝑑. Hence, 𝐷=8Γ—9√2=72√2.cm

In the previous example, an alternative approach would be to

  • calculate the side length of each of the smaller squares by recalling that areaofsquare=π‘ οŠ¨,
  • calculate the length of the diagonal of each of the smaller squares by applying the Pythagorean theorem,
  • multiply this by 8 to give the length of the diagonal of the chessboard.

While this is a perfectly valid method, it involves a similar number of steps to the method we presented, so it is no more or less efficient.

In our final example, we will find the difference between the areas of a square and a rhombus, each calculated using the lengths of their diagonals.

Example 5: Finding the Areas of a Square and a Rhombus given Their Diagonals

Determine the difference in area between a square having a diagonal of 10 cm and a rhombus having diagonals of 2 cm and 12 cm.

Answer

We begin by calculating the area of each shape. The area of a square can be calculated from the length of its diagonal 𝑑 using the formula areaofsquare=𝑑2.

Hence, for a square with a diagonal of 10 cm, areaofsquarecm=10Γ—102=50.

The area of a rhombus is equal to half the product of the lengths of its diagonals π‘‘οŠ§ and π‘‘οŠ¨: areaofrhombus=𝑑𝑑2.

Hence, for a rhombus with diagonals of 2 cm and 12 cm, areaofrhombuscm=2Γ—122=12.

Finally, we calculate the difference in area by subtracting the area of the smaller quadrilateral (the rhombus) from the area of the larger quadrilateral (the square): differenceinareaareaofsquareareaofrhombuscm=βˆ’=50βˆ’12=38.

Let us finish by recapping some key points.

Key Points

  • The area of a rhombus is equal to half the product of the lengths of its diagonals.
  • For a rhombus with diagonals of length π‘‘οŠ§ and π‘‘οŠ¨ units, areaofrhombussquareunits=𝑑𝑑2.
  • The area of a square is equal to half the square of the length of its diagonal.
  • For a square with diagonals of length 𝑑 units, areaofsquaresquareunits=𝑑2.

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