Students derive the formulas for the area and perimeter of a rhombus, and practice using these formulas.
After completing this tutorial, you will be able to complete the following:
A rhombus is a quadrilateral with four congruent sides. This apparently simple definition actually imposes several other useful properties that will aid in finding formulas for the area and perimeter of rhombuses. Recall that perimeter is the length of the path surrounding a region and that area is the number of square units covered in a region.
The definition of a rhombus provides a simple formula for the perimeter. Recall that the perimeter of any polygon is the sum of the lengths of all its sides. Since all four sides of a rhombus are congruent, they must all have the same length. This common length is called the side length of the rhombus. Say that a rhombus has side length a units. Since the rhombus has four sides, its perimeter is P = a + a + a + a = 4a units. Thus, once the side length of a rhombus is known, its perimeter can be determined by multiplying the side length by 4.
To find the area of a rhombus, a little more information is needed. In particular, a rhombus is always a parallelogram, which is a quadrilateral with two pairs of parallel sides. To see this, we will consider a diagonal of a rhombus. Recall that a diagonal is a line segment connecting two nonadjacent vertices of a polygon. Since a rhombus has four vertices, a diagonal connects two opposite vertices. Since all four sides of a rhombus are congruent and the diagonal is obviously congruent to itself, the diagonal divides the rhombus into two triangles, whose corresponding sides are all congruent. Thus, the diagonal divides the rhombus into two congruent triangles. It follows that opposite angles of a rhombus are congruent. Since the four angles of a quadrilateral sum to 360 degrees, it must be that adjacent angles are supplementary.
Now if we consider two opposite sides of a rhombus and a third side of the rhombus as a transversal to those sides, we see that same-side interior angles are congruent. This in turn implies that the opposite sides of a rhombus are parallel. Repeating this construction with the other diagonal verifies that both pairs of opposite sides are parallel, and therefore proves that a rhombus is a parallelogram.
Since a rhombus is a parallelogram, we can use the formula for the area of a parallelogram to find the area of a rhombus. Recall that the bases of a parallelogram are a set of parallel sides of the parallelogram and that the height of a parallelogram is the shortest distance between two bases. The area of a parallelogram is the product of the length of a base and the height of the parallelogram, since any parallelogram can be divided into a rectangle and two right congruent triangles. Moving one triangle so that the hypotenuses coincide, we form a rectangle with the same base length and height as the parallelogram. Since all four sides of a rhombus are congruent, its bases all have the same length. Therefore, for a parallelogram with side length a units and height h units, its area is given by . We see that, in order to find the area of a rhombus, we must know both the side length and the height of the rhombus.
Somewhat surprisingly, we can also find the area of a rhombus if we know the lengths of the diagonals of the rhombus. Recall that the diagonal of a rhombus divides the rhombus into two congruent triangles. These triangles are actually isosceles because the sides of a rhombus are all congruent. It follows that the diagonals of a rhombus bisect the angles of a rhombus. If we consider the rhombus and both of its diagonals, we see that the diagonals of a rhombus divide the rhombus into four congruent triangles. In particular, the four angles formed by the intersection of the diagonals are all congruent, so they must be right angles. The diagonals of a rhombus divide the rhombus into four congruent right triangles.
In particular, the diagonals of a rhombus perpendicularly bisect each other. Therefore, if a rhombus has diagonals of length e units and f units, then the diagonals of the rhombus divide the rhombus into four congruent right triangles with legs of length units and units.
We can rearrange these right triangles to form a rectangle of length f units and width units . Since the rectangle is composed of the same triangles that compose the rhombus, the areas of the rhombus and the rectangle are equal. The area of the rectangle is the product of its width and its length, so we conclude that the area of the rhombus is also . Thus, to find the area of a rhombus, it suffices to know the lengths of both diagonals of a rhombus.
We have found one formula for the perimeter of a rhombus and two formulas for the area of a rhombus.
|Approximate Time||20 Minutes|
|Pre-requisite Concepts||Students should know the formula for the area of a parallelogram and a rectangle and the perimeter of a polygon, the definition and properties of a rhombus|
|Type of Tutorial||Visual Proof|
|Key Vocabulary||area, base, diagonal|