Area of a Deltoid (Kite)

There are many geometric shapes that can be found during the solving of engineering problems at all different stages of study, such as in high school, in matriculation exams, and even in psychometry. One of the less trivial shapes is the deltoid and, as part of the questions surrounding it, students are often asked to calculate the area of the deltoid.

A deltoid is a polygon with four sides (that is, a quadrilateral) with two distinct sets of adjacent sides of equal length to each other.

There is a clear distinction between convex deltoid and concave deltoid.

Convex deltoid is a deltoid where the diagonals are inside and cross each other. The longer diagonal acts as a main diagonal, while the shorter diagonal acts as a secondary diagonal.

As you can observe in the following drawing, the main diagonal divides the deltoid into two overlapping triangles, that is, identical, and the secondary diagonal divides the deltoid into two isosceles triangles whose bases are adjacent and, indeed, identical. 

Concave kite is a kite where one of the diagonals (main diagonal) passes inside the kite and the other diagonal (secondary diagonal) passes outside the kite.

The concave deltoid can be described as a shape consisting of two isosceles triangles that share a common base, where one triangle contains the other triangle. The following drawing better describes the concave deltoid:

• When all sides of the kite have the same length, a rhombus is obtained, which is actually a special case of a kite.
• Another special case of a kite is a square, when it is a case where all sides and all angles are of the same size. 

• The angles on the sides, or more precisely, the angles between the different adjacent sides of the kite, are of equal size.
• The diagonals of the kite are perpendicular to each other 
• The main diagonal in the convex kite (or its extension in the concave kite) crosses the secondary diagonal (in both cases), and therefore actually functions as a perpendicular bisector
• The main diagonal equally divides (crosses) the main angles of the kite
• Every convex kite has the possibility of enclosing a circle
• In every kite, there are two sets of adjacent sides of equal size
• As mentioned, a concave kite is characterized by a secondary diagonal located outside of it

Exercise 1

The area of the kite must be calculated according to the attached drawing and the existing data:

• $BK= 3 cm$
• $AC= 7 cm$

Solution:

According to the kite area formula, we are missing the length of the diagonal $BD$.

We know that the diagonal $BK$ is equal to $3$ cm, and according to one of the properties of the kite, the diagonal $AC$ divides the diagonal $BD$ into two equal parts.

That is, from this we can conclude that $BD=6$.

$A=\frac{(AC\times BD)}{2}=\frac{(7\times6)}{2}=21$

At this point, we can place the data in the area formula and obtain:

Answer:

The area of the kite $ABCD$ is $21$ cm².

Exercise 2

Given the kite $KLMN$ whose area is $144$ cm², the meeting point of the diagonals $LN$ and $KM$.

The area of the section KP must be calculated according to the attached drawing and the existing data:

• $LN=18$ cm
• $A=144$ cm²

Solution:

This exercise is a reverse calculation, that is, we know the area and we are asked to calculate the length of the segment $KP$.

In the first step, we replace the data we know in the kite area formula.

We obtain:

$A=(LN\times KM)$

$144=\frac{(18\times KM)}{2}$

We simplify the expression and obtain:

$288=18\times KM$

$KM=16$

In fact, we found the length of the second diagonal of the kite.

According to one of the properties of the kite, the diagonal $LN$ divides the diagonal $KP$ into two equal parts.

From here we obtain that $KP$ is equal to $8$ cm.

Answer:

$KP=8$ cm