Area of a Deltoid (Kite)

🏆Practice area of a deltoid

How do we calculate the area of a kite?

The area of the kite can be calculated by multiplying the lengths of the diagonals and dividing this product by 2 2 .

Deltoid Area Formula

To facilitate the understanding of the concept of calculus, you can use the following drawing and the accompanying formula:

A=KM×NL2A=\frac{ KM\times NL}{2}

A8 - Area formula of the kite

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Test yourself on area of a deltoid!

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ACBD is a deltoid.

AD = AB

CA = CB

Given in cm:

AB = 6

CD = 10

Calculate the area of the deltoid.

666101010AAACCCBBBDDD

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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.


What is 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.


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Types of Deltoids

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


Convex 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. 

B2 - Convex deltoid


Do you know what the answer is?

Concave Kite

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:

A3 - Concave Kite


  • 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. 

Properties of the Kite

  • 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

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Practice on the Area of a Kite

Exercise 1

Given the kite KLMNKLMN whose area is 144144 cm², the meeting point of the diagonals LNLN and KMKM.

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

  • LN=18LN=18 cm
  • A=144A=144 cm²
A5 - Given the kite KLMN

Solution:

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

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

We obtain:

A=(LN×KM) A=(LN\times KM)

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

We simplify the expression and obtain:

288=18×KM 288=18\times KM

KM=16 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 LNLN divides the diagonal KPKP into two equal parts.

From here we obtain that KPKP is equal to 88 cm.

Answer:

KP=8 KP=8 cm


Examples and exercises with solutions for the Area of the Kite

Exercise #1

ACBD is a deltoid.

AD = AB

CA = CB

Given in cm:

AB = 6

CD = 10

Calculate the area of the deltoid.

666101010AAACCCBBBDDD

Video Solution

Step-by-Step Solution

To solve the exercise, we first need to remember how to calculate the area of a rhombus:

(diagonal * diagonal) divided by 2

Let's plug in the data we have from the question

10*6=60

60/2=30

And that's the solution!

Answer

30

Exercise #2

ABDC is a deltoid.

AB = BD

DC = CA

Given in cm:

AD = 12

CB = 16

Calculate the area of the deltoid.

161616121212CCCAAABBBDDD

Video Solution

Step-by-Step Solution

First, let's recall the formula for the area of a rhombus -

(Diagonal 1 * Diagonal 2) divided by 2

Let's substitute the known data into the formula:

(12*16)/2
192/2=
96

And that's the solution!

Answer

96 cm²

Exercise #3

Shown below is the deltoid ABCD.

The diagonal AC is 8 cm long.

The area of the deltoid is 32 cm².

Calculate the diagonal DB.

S=32S=32S=32888AAABBBCCCDDD

Video Solution

Step-by-Step Solution

First, we recall the formula for the area of a kite: multiply the lengths of the diagonals by each other and divide the product by 2.

We substitute the known data into the formula:

 8DB2=32 \frac{8\cdot DB}{2}=32

We reduce the 8 and the 2:

4DB=32 4DB=32

Divide by 4

DB=8 DB=8

Answer

8 cm

Exercise #4

Look at the deltoid in the figure:

555666

What is its area?

Video Solution

Step-by-Step Solution

To solve the exercise, we need to know the formula for calculating the area of a kite:

It's also important to know that a concave kite, like the one in the question, has one of its diagonals outside the shape - but it's still its diagonal.

Let's plug the data from the question into the formula:

(6*5)/2=
30/2=
15

Answer

15

Exercise #5

Look at the deltoid in the figure:

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What is its area?

Video Solution

Step-by-Step Solution

Let's begin by reminding ourselves of the formula for the area of a kite

Diagonal1×Diagonal22 \frac{Diagonal1\times Diagonal2}{2}

Both these values are given to us in the figure thus we can insert them directly into the formula:

(4*7)/2

28/2

14

Answer

14

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