Calculate the Diagonal Length: Deltoid with Area 39 cm² and Side 13 cm

Q: What exactly is a deltoid shape?

A deltoid is a kite-shaped quadrilateral with two pairs of adjacent sides that are equal. It has two diagonals that are perpendicular to each other, which is why we use the diagonal formula for area.

Q: Why do we multiply by ½ in the area formula?

The diagonals of a deltoid split it into four right triangles. The formula ½ × d₁ × d₂ calculates the total area of all four triangles combined.

Q: Can I use this formula for any quadrilateral?

No! This formula only works for deltoids (kites) and rhombuses where the diagonals are perpendicular. For other quadrilaterals, you need different area formulas.

Deltoid Area Formula with Diagonal Calculation

Given the deltoid ABCD

Side length AC equals 13 cm

The area of the deltoid is equal to 39 cm².

Find the length of the side BD

❤️ 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

00:00 Find BD

00:03 We'll use the formula for calculating the area of a kite

00:07 (diagonal times diagonal) divided by 2

00:12 We'll substitute appropriate values according to the given data and solve for BD

00:20 We'll isolate BD

00:38 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the deltoid ABCD

Side length AC equals 13 cm

The area of the deltoid is equal to 39 cm².

Find the length of the side BD

Step-by-step solution

To find the length of diagonal $BD$ , we will apply the formula for the area of a deltoid:

$\text{Area} = \frac{1}{2} \times \text{Diagonal 1} \times \text{Diagonal 2}$

In this problem, Diagonal 1 is $AC = 13$ cm, and Diagonal 2 is $BD$ , which we are trying to find. The area is given as $39$ cm². Substituting these values into the formula, we get:

$39 = \frac{1}{2} \times 13 \times BD$

To solve for $BD$ , multiply both sides by 2 to eliminate the fraction:

$78 = 13 \times BD$

Now, solve for $BD$ by dividing both sides by 13:

$BD = \frac{78}{13}$

Simplify to find:

$BD = 6$

Therefore, the length of diagonal $BD$ is $6$ cm.

Final Answer

$6$ cm

Key Points to Remember

Essential concepts to master this topic

Formula: Area of deltoid equals half the product of both diagonals
Technique: Substitute known values: $39 = \frac{1}{2} \times 13 \times BD$
Check: Verify by calculating area with found diagonal: $\frac{1}{2} \times 13 \times 6 = 39$ ✓

Common Mistakes

Avoid these frequent errors

Using the wrong formula for deltoid area
Don't use base times height or side squared = wrong area formula! A deltoid isn't a rectangle or square. Always use the diagonal formula: Area = ½ × diagonal₁ × diagonal₂.

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 shape?

A deltoid is a kite-shaped quadrilateral with two pairs of adjacent sides that are equal. It has two diagonals that are perpendicular to each other, which is why we use the diagonal formula for area.

Why do we multiply by ½ in the area formula?

The diagonals of a deltoid split it into four right triangles. The formula ½ × d₁ × d₂ calculates the total area of all four triangles combined.

How do I know which diagonal is which?

It doesn't matter! The area formula works the same way regardless of which diagonal you call 'diagonal 1' or 'diagonal 2'. Just make sure you have both diagonal lengths to calculate the area.

Can I use this formula for any quadrilateral?

No! This formula only works for deltoids (kites) and rhombuses where the diagonals are perpendicular. For other quadrilaterals, you need different area formulas.

What if I get a decimal answer?

That's perfectly fine! Many geometry problems have decimal answers. Just make sure to round appropriately based on the precision given in the problem, and always check your work.

🌟 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