Deltoid Area Problem: Calculate 'a' Given Area 182 cm² and Side 13 cm

Q: What exactly is a deltoid and how is it different from other quadrilaterals?

A deltoid (or kite) is a quadrilateral with two pairs of adjacent sides that are equal. Unlike rectangles or parallelograms, deltoids have perpendicular diagonals, which is why we can use the simple diagonal formula for area!

Q: Why do we use the diagonal formula instead of base times height?

For deltoids, the diagonals are perpendicular and bisect each other, making them act like the base and height of the shape. The formula $ A = \frac{1}{2} \times d_1 \times d_2 $ is much easier than trying to find a traditional base and height.

Q: How do I know which measurements are the diagonals?

In this problem, AC = 7a and BD = 13 cm are the diagonals because they connect opposite vertices. The diagram shows these as the lines crossing inside the deltoid, not the sides around the perimeter.

Q: Can I use this same formula for other quadrilaterals?

Only for shapes with perpendicular diagonals! This includes deltoids (kites), rhombi, and squares. For rectangles, parallelograms, or trapezoids, you need different area formulas.

Deltoid Area with Diagonal Calculation

Look at the deltoid ABCD.

The side BD is 13 cm long.

The area of the deltoid is 182 cm².

Calculate $a\text{.}$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Let's find A.

00:14 We'll use the formula for the area of a kite.

00:18 Multiply the diagonals, then divide by two.

00:24 Now, let's substitute the given values to solve for A.

00:33 We'll isolate A step by step.

00:52 And that's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the deltoid ABCD.

The side BD is 13 cm long.

The area of the deltoid is 182 cm².

Calculate $a\text{.}$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Use the formula for the area of a deltoid.
Step 2: Set up the equation using given values.
Step 3: Solve for the variable $a$ .

Let's go through each step:
Step 1: For a deltoid, the area $A$ can be calculated using:
$A = \frac{1}{2} \times AC \times BD$

Step 2: Substituting the known values into the expression gives us:
$182 = \frac{1}{2} \times 7a \times 13$

Step 3: Simplify and solve for $a$ :
We first find $\frac{1}{2} \times 13 = 6.5$ , so:
$182 = 6.5 \times 7a$
Solving for $a$ , we have:
$182 = 45.5a$
$a = \frac{182}{45.5} = 4$

Therefore, the solution to the problem is $a = 4$ .

Final Answer

$4$

Key Points to Remember

Essential concepts to master this topic

Formula: Deltoid area equals half the product of diagonal lengths
Technique: Substitute $A = \frac{1}{2} \times 7a \times 13 = 182$
Check: Verify $\frac{1}{2} \times 28 \times 13 = 182$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the one-half factor in area formula
Don't use $A = d_1 \times d_2$ = 7a × 13 = 182, giving a = 2! This ignores the crucial ½ factor in deltoid area formula. Always use $A = \frac{1}{2} \times d_1 \times d_2$ for deltoids.

Practice Quiz

Test your knowledge with interactive questions

Indicate the correct answer

The next quadrilateral is:

FAQ

Everything you need to know about this question

What exactly is a deltoid and how is it different from other quadrilaterals?

A deltoid (or kite) is a quadrilateral with two pairs of adjacent sides that are equal. Unlike rectangles or parallelograms, deltoids have perpendicular diagonals, which is why we can use the simple diagonal formula for area!

Why do we use the diagonal formula instead of base times height?

For deltoids, the diagonals are perpendicular and bisect each other, making them act like the base and height of the shape. The formula $A = \frac{1}{2} \times d_1 \times d_2$ is much easier than trying to find a traditional base and height.

How do I know which measurements are the diagonals?

In this problem, AC = 7a and BD = 13 cm are the diagonals because they connect opposite vertices. The diagram shows these as the lines crossing inside the deltoid, not the sides around the perimeter.

What if I get a decimal answer instead of a whole number?

That's totally normal! Many geometry problems have decimal solutions. Just make sure to check your arithmetic carefully and round appropriately if the problem asks for it.

Can I use this same formula for other quadrilaterals?

Only for shapes with perpendicular diagonals! This includes deltoids (kites), rhombi, and squares. For rectangles, parallelograms, or trapezoids, you need different area formulas.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Deltoid questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime