Areas of Rhombuses and Kites
Although a rhombus is a type of parallelogram, whereas a kite is not, they are similar in that their sides have important properties. Recall that all four sides of a rhombus are congruent. Kites, on the other hand, have exactly two pairs of consecutive sides that are congruent. This characteristic of kites does not allow for both pairs of opposite sides to be parallel. Let's look at the image below to examine the properties that make these figures distinguishable.
The key characteristics of rhombus ABCD and kite EFGH are shown in the figure above.
Despite how different they are, when it comes to area, we will see that rhombuses and parallelograms are quite similar. The areas of rhombuses and kites are equal to one half the product of their diagonals. Mathematically, we express this as
where A is the area of the of the quadrilaterals (in square units), d1 is the length of one diagonal, and d2 is the length of the other diagonal.
Recall that every quadrilateral has exactly two diagonals. This is because diagonals are line segments which connect vertices to each other. Since there already exist line segments which connect one vertex to two other vertices in a quadrilateral, the only other line segment to draw is to the vertex diagonal from the chosen vertex.
Let's work on the following exercises, to help us apply the area formula for rhombuses and kites.
Find the area of rhombus PQRS.
The only two parts of the rhombus we need to figure out are the diagonals because that is all that is required when we find the areas of rhombuses. We are given the length of one diagonal of rhombus PQRS, which will be our d1. The length of diagonal PR is 12 centimeters.
Before we can find the area of rhombus PQRS, we need to find the length of d2. We see that the length of SQ, is just the sum of two smaller segments. So, we need to take the sum of segments ST and TQ to find the length of SQ. However, we do not know what the length of TQ is, so we must rely on our knowledge of rhombuses to help us out at this point.
We know that the diagonals of a rhombus bisect each other. This means that the point T is the midpoint of segment SQ. Thus, we know that the length of segment TQ is equal to the length of segment ST; they are both 4 centimeters long. Now, we can find the length of SQ, which is our d2:
So, we know that d2 has a length of 8 centimeters. Now, we have all the requirements we need in order to solve for the area of rhombus PQRS. Let's plug our values of d1 and d2 in:
We see that rhombus PQRS has an area of 48 square centimeters.
Find the value of y given that the area of kite YDOC is 552 square feet.
In this exercise, we will not be solving for area, since it has already been given to us. Rather, we will have to use properties of kites and the area formula to deduce the value of variable x.
We have been given the length of segment CD, which we will use as d1 in our area formula. Therefore, d1 is equal to 24 centimeters.
Now, let's look at the other diagonal of kite YDOC. It has been given a length of 12x+5y centimeters. However, we are not sure what the values of x or y are, so we must deduce key information from the facts that have been given to us. Let's take a look at the area formula of our kite as we have it right now.
At this point in the exercise, we are stuck because we do not know the values of x or y. If we look at the sides of the kite, however, we will notice that we can determine the value of x. We know that kites have two pairs of consecutive sides that are congruent. In this case, we see that CY and DY are congruent, so we can set those values equal to each other to solve for x. We have
Therefore, we have determined that the value of x is 3. We can use this value to plug into our area formula in order to solve for y. We have
After simplifying, we get
Then, we divide both sides of the equation by 12, which yields
So, we have determined that the value of y is 2.
As we progress through the lessons on area, it will be important to recognize the properties of different polygons in order to help us find valuable information about certain problems. While the different classifications of polygons have features which set them apart from each other, it can also help to group polygons up by their relationship with one another. In the same way that parallelograms and triangles have completely different properties but share a distinct relationship in terms of their areas, we see that rhombuses and kites are also two different polygons with the same formula to determine their areas.