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

Practice Area of a Deltoid

Examples with solutions for Area of a Deltoid

Exercise #1

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

Exercise #2

Look at the deltoid in the figure:

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What is its area?

Video Solution

Step-by-Step Solution

To solve the exercise, we first 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 now substitute the data from the question into the formula:

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

Answer

15

Exercise #3

ACBD is a deltoid.

AD = AB

CA = CB

Given in cm:

AB = 6

CD = 10

Calculate the area of the deltoid.

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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 #4

ABDC is a deltoid.

AB = BD

DC = CA

AD = 12 cm

CB = 16 cm

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

Now we will substitute the known data into the formula, giving us the answer:

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

Answer

96 cm²

Exercise #5

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 #6

The deltoid below has an area of 60 cm².

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What is the value of X?

Video Solution

Step-by-Step Solution

To solve the problem, we need to remember the formula for the area of a rhombus:

The product of the diagonals multiplied together and then divided by 2.

Let's substitute in our data into the formula:

(8*X) = 60
2

Note that we can simplify the fraction, thus eliminating the denominator:

4X = 60

Let's finally divide the equation by 4 to get our answer:

X = 15

Answer

15

Exercise #7

The kite ABCD shown below has an area of 42 cm².

AB = BC

DC = AD

BD = 14

The diagonals of the kite intersect at point 0.

Calculate the length of side AO.

S=42S=42S=42141414DDDAAABBBCCCOOO

Video Solution

Step-by-Step Solution

We substitute the data we have into the formula for the area of the kite:

S=AC×BD2 S=\frac{AC\times BD}{2}

42=AC×142 42=\frac{AC\times14}{2}

We multiply by 2 to remove the denominator:

 14AC=84 14AC=84

Then divide by 14:

AC=6 AC=6

In a rhombus, the main diagonal crosses the second diagonal, therefore:

AO=AC2=62=3 AO=\frac{AC}{2}=\frac{6}{2}=3

Answer

3 cm

Exercise #8

Given the deltoid ABCD

The main diagonal is equal to 2a+2

Secondary diagonal is equal to a

The area of the deltoid equals 6a

Calculate a a

S=6aS=6aS=6a2a+22a+22a+2aaaAAABBBCCCDDD

Video Solution

Step-by-Step Solution

To solve the question, we first need to remember the formula for the area of a kite:

Diagonal * Diagonal / 2

This means that if we substitute the given data we can see that:

a(2a+2)/2 = area of the kite

Let's remember that we are also given the area, so we'll put that in the equation too

a(2a+2)/2 = 6a

Now we have an equation that we can easily solve.

First, let's get rid of the fraction, so we'll multiply both sides of the equation by 2

a(2a+2)=6a*2
a(2a+2)=12a

Let's expand the parentheses on the left side of the equation

2a²+2a=12a

2a²=10a

Let's divide both sides of the equation by a

2a=10

Let's divide again by 2

a=5

And that's the solution!

Answer

5 cm

Exercise #9

A deltoid-shaped stage is to be built in a rectangular field.

The length of the field is 30 m and the width is 20 m.

Determine the area of the stage shaded in orange?

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Video Solution

Step-by-Step Solution

We can calculate the area of rectangle ABCD as follows:

20×30=600 20\times30=600

Now let's divide the deltoid along its length and width and add the following points:

202020303030PPPMMMNNNKKKAAABBBCCCDDDFinally, we can calculate the area of deltoid PMNK as follows:

PMNK=PN×MK2=20×302=6002=300 PMNK=\frac{PN\times MK}{2}=\frac{20\times30}{2}=\frac{600}{2}=300

Answer

300 m

Exercise #10

Below is a deltoid with a length 2 times its width and an area equal to 16 cm².


Calculate x.

1616162x2x2xxxx

Video Solution

Step-by-Step Solution

Given the problem, we are tasked to find the value of x x for a deltoid where the length is twice the width and the area is given. Let's proceed as follows:

  • Step 1: In this deltoid problem, the diagonals correspond to length 2x 2x and width x x . The formula for the area of a deltoid in terms of its diagonals is A=12×d1×d2 A = \frac{1}{2} \times d_1 \times d_2 .
  • Step 2: Substitute the values. Thus, the area 16=12×(2x)×x 16 = \frac{1}{2} \times (2x) \times x .
  • Step 3: Simplify the equation: 16=12×2x2=x2 16 = \frac{1}{2} \times 2x^2 = x^2 .
  • Step 4: Solve for x x : We find x2=16 x^2 = 16 , so x=16 x = \sqrt{16} .
  • Step 5: Conclude x=4 x = 4 .

Therefore, the solution to the problem is x=4 x = 4 .

Answer

x=4 x=4

Exercise #11

Look at the kite ABCD below.

Diagonal DB = 10

CB = 4

Is it possible to calculate the area of the kite? If so, what is it?

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Video Solution

Answer

It is not possible.

Exercise #12

Given the deltoid ABCD

Find the area

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Video Solution

Answer

12 12 cm².

Exercise #13

Given the deltoid ABCD

Find the area

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Video Solution

Answer

17.5 17.5 cm².

Exercise #14

Given the deltoid ABCD

Find the area

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Video Solution

Answer

27 27 cm².

Exercise #15

Given the deltoid ABCD

Find the area

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Video Solution

Answer

35 35 cm².