Calculate Side Length DB in Deltoid with Area 32 cm² and Height 8

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

A deltoid is a special quadrilateral with two pairs of adjacent sides that are equal in length. Unlike rectangles or parallelograms, deltoids have perpendicular diagonals that intersect, making the area formula unique.

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

The diagonal formula $ Area = \frac{1}{2} \times d_1 \times d_2 $ works because deltoid diagonals are perpendicular. This creates four right triangles, making the calculation much easier than finding a traditional base and height.

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

Diagonals connect opposite vertices (corners) of the deltoid. In this problem, AC and DB are diagonals because they go from one corner to the opposite corner, crossing inside the shape.

Q: Can deltoids have diagonals of different lengths?

Yes! Most deltoids have unequal diagonals. In this problem, it turns out both diagonals equal 8 cm, but that's just a special case. The formula works for any deltoid regardless of diagonal lengths.

Deltoid Area Formula with Diagonal Calculation

Below is the deltoid ABCD.

C = 8

The area of the deltoid is equal to 32 cm².

Calculate the side DB.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate diagonal DB

00:03 We'll use the formula for calculating rhombus area

00:07 (diagonal times diagonal) divided by 2

00:12 We'll substitute appropriate values according to the given data and find DB

00:21 Divide 8 by 2

00:28 Isolate BD

00:34 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

Below is the deltoid ABCD.

C = 8

The area of the deltoid is equal to 32 cm².

Calculate the side DB.

Step-by-step solution

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

Step 1: Set up the equation using the area of the deltoid formula.
Step 2: Substitute the known values and solve for the unknown diagonal $DB$ .

Here's the step-by-step solution:

Step 1: The area of a deltoid can be calculated using the formula:
$\text{Area} = \frac{1}{2} \times d_1 \times d_2$
Given that $\text{Area} = 32$ cm² and $d_1 = AC = 8$ cm, we place these values into the equation:

Step 2: Substitute into the formula:
$32 = \frac{1}{2} \times 8 \times DB$

Step 3: Simplify the equation:
$32 = 4 \times DB$

Step 4: Solve for $DB$ :
$DB = \frac{32}{4} = 8$

Therefore, the length of diagonal $DB$ is 8 cm.

Final Answer

8 cm

Key Points to Remember

Essential concepts to master this topic

Formula: Deltoid area equals half the product of both diagonals
Technique: $32 = \frac{1}{2} \times 8 \times DB$ simplifies to $DB = 8$
Check: Verify $\frac{1}{2} \times 8 \times 8 = 32$ cm² matches given area ✓

Common Mistakes

Avoid these frequent errors

Confusing deltoid area formula with triangle area formula
Don't use Area = ½ × base × height for deltoids = wrong formula entirely! This gives incorrect results because deltoids need both diagonal lengths. Always use Area = ½ × diagonal₁ × diagonal₂ for deltoids.

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FAQ

Everything you need to know about this question

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

A deltoid is a special quadrilateral with two pairs of adjacent sides that are equal in length. Unlike rectangles or parallelograms, deltoids have perpendicular diagonals that intersect, making the area formula unique.

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

The diagonal formula $Area = \frac{1}{2} \times d_1 \times d_2$ works because deltoid diagonals are perpendicular. This creates four right triangles, making the calculation much easier than finding a traditional base and height.

How do I identify which measurements are the diagonals?

Diagonals connect opposite vertices (corners) of the deltoid. In this problem, AC and DB are diagonals because they go from one corner to the opposite corner, crossing inside the shape.

What if I'm given the area and need to find the other diagonal?

Use the same formula but solve for the unknown diagonal! If you know Area = 32 and one diagonal = 8, then $32 = \frac{1}{2} \times 8 \times d_2$ , so $d_2 = 8$ .

Can deltoids have diagonals of different lengths?

Yes! Most deltoids have unequal diagonals. In this problem, it turns out both diagonals equal 8 cm, but that's just a special case. The formula works for any deltoid regardless of diagonal lengths.

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Calculate Side Length DB in Deltoid with Area 32 cm² and Height 8

Deltoid Area Formula with Diagonal Calculation

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

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

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

How do I identify which measurements are the diagonals?

What if I'm given the area and need to find the other diagonal?

Can deltoids have diagonals of different lengths?

🌟 Unlock Your Math Potential

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