Perimeter Of Kite

What Is the Perimeter of a Kite?

The perimeter of a kite is the total distance measured around its four outer edges. In geometry, the perimeter of any polygon is found by adding the lengths of all its sides together. A kite is a special type of quadrilateral, a four-sided shape, with a unique property: it has two distinct pairs of equal-length sides that are adjacent to each other. This means two sides of a certain length meet at one corner, and the other two sides of a different length meet at the opposite corner.

Imagine you are walking along the boundary of a kite-shaped field. The total distance you walk to get back to your starting point is the perimeter. Because of a kite's special properties, we don't need to know the lengths of all four individual sides to find the perimeter. If we know the length of one of the shorter sides and the length of one of the longer sides, we have all the information we need.

Let's label the side lengths. We can call the length of the two shorter, equal sides 'a' and the length of the two longer, equal sides 'b'. So, a kite will always have two sides of length $a$ and two sides of length $b$.

What Are the Key Properties of a Kite?

To understand why the perimeter formula works the way it does, it's essential to know the defining characteristics of a kite. Unlike a rectangle or a square, a kite's properties are based on adjacent (neighboring) sides, not opposite sides.

Here are the fundamental properties of a geometric kite:

• Two Pairs of Equal Adjacent Sides: This is the most important property. A kite has two sides of length $a$ that meet at a vertex, and two sides of length $b$ that meet at a different vertex. For example, if the side lengths are $5,5,12,12$, it can form a kite.
• Perpendicular Diagonals: A kite has two diagonals that connect opposite vertices. These two diagonals always intersect at a right angle (${90}^{\circ }$).
• One Diagonal is a Perpendicular Bisector: One of the diagonals cuts the other diagonal into two equal halves. The main diagonal (the one that acts as the axis of symmetry) bisects the other one.
• One Pair of Opposite Angles are Equal: The angles between the sides of unequal length are always equal to each other.

For calculating the perimeter, the first property is the only one we need. The fact that there are two sides of length $a$ and two sides of length $b$ is the key that unlocks the simple perimeter formula.

How Do You Calculate the Perimeter of a Kite?

Calculating the perimeter of a kite is straightforward. We start with the basic definition of perimeter: the sum of all side lengths. For a kite with four sides, we can write:

$$P={\text{side}}_1+{\text{side}}_2+{\text{side}}_3+{\text{side}}_4$$

From the properties of a kite, we know that two of the sides have a length we'll call $a$, and the other two sides have a length we'll call $b$. So, we can substitute these variables into our general formula:

$$P=a+a+b+b$$

Now, we can simplify this expression by combining the like terms. We have two $a$'s and two $b$'s. This gives us:

$$P=2a+2b$$

This formula is perfectly correct and easy to use. However, we can also factor out the $2$ to get an even more compact version. By factoring, we arrive at the most common formula for the perimeter of a kite.

In this formula:

• $P$ stands for the Perimeter.
• $a$ is the length of one of the two equal shorter sides.
• $b$ is the length of one of the two equal longer sides.

This formula tells us to first add the lengths of the two different sides ($a$ and $b$) and then multiply the sum by $2$.

How to Find the Perimeter in 3 Simple Steps

Using the formula $P=2(a+b)$ is very efficient. Let's break down the process into three easy-to-follow steps. We'll use an example to illustrate each step.

1. Identify the lengths of the two different sides. Look at the diagram or read the problem to find the two unique side lengths. Remember, a kite has two pairs of equal sides, so you only need one length from each pair. Let's say a kite has sides measuring $7$ cm and $10$ cm. So, $a=7$ cm and $b=10$ cm.
2. Substitute these values into the perimeter formula. Take the values for $a$ and $b$ and plug them into the formula $P=2(a+b)$.
   $$P=2(7+10)$$
3. Calculate the final result and include the units. First, solve the part in the parentheses. Then, perform the multiplication. Don't forget to write down the units in your final answer.
   $$P=2(17)$$
   $$P=34\text{ cm}$$

Let's Practice: More Perimeter of a Kite Examples

Practice is key to mastering any math concept. Let's work through a couple more examples, including one with decimals and a word problem.

How Do You Find a Missing Side Length Given the Perimeter?

Sometimes, a problem might give you the total perimeter and the length of one side, and then ask you to find the length of the other side. This is a great way to practice your algebra skills! We just need to rearrange our formula to solve for the missing piece.

Our formula is $P=2(a+b)$. Let's say we know $P$ and $a$, but we need to find $b$. Here's how we can isolate $b$:

1. Start with the formula: $P=2(a+b)$
2. Divide both sides by 2: This undoes the multiplication. $\frac{P}{2}=a+b$
3. Subtract the known side length ($a$) from both sides: This will leave $b$ by itself. $\frac{P}{2}-a=b$

So, the formula to find a missing side $b$ is $b=\frac{P}{2}-a$. Let's try an example.

What Are Common Mistakes When Calculating a Kite's Perimeter?

When working with geometric shapes, it's easy to mix up formulas or properties. Here are some common mistakes to watch out for when finding the perimeter of a kite:

• Using the Diagonal Lengths: A very common error is to confuse the formula for perimeter with the formula for area. The area of a kite uses the lengths of its two diagonals ($A=\frac{1}{2}{d}_1{d}_2$). The perimeter only uses the lengths of the outer sides. Always read the question carefully to see if it's asking for perimeter or area.
• Confusing a Kite with a Rhombus: A rhombus is a special type of kite where all four sides are equal ($a=b$). Students sometimes mistakenly use the rhombus formula ($P=4a$) for a general kite. Unless all four sides are stated to be equal, you must use the $P=2(a+b)$ formula.
• Adding Only Two Sides: Some students might add $a+b$ and forget to multiply the sum by $2$. Remember, the perimeter is the distance around all four sides, so you need to account for both pairs of equal sides.
• Multiplying Before Adding: Due to the order of operations (PEMDAS/BODMAS), you must calculate the sum inside the parentheses first before multiplying by $2$. Calculating $2a+b$ or $a+2b$ will give an incorrect answer.
• Forgetting Units: Perimeter is a measurement of length, so the final answer must include units like inches, cm, feet, meters, etc. An answer of '34' is incomplete; the correct answer is '34 inches'.

Perimeter of a Kite: Quick Summary

This lesson covered everything you need to know about calculating the perimeter of a kite. Here is a quick reference table to summarize the key points.

Remember, the perimeter is simply the total length of the boundary of a shape. For a kite, this calculation is made simple because its adjacent sides come in equal-length pairs.