Deltoid Diagonal Calculation: Finding AC When Area = 24cm² and DB = 4

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

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 use the diagonal formula for area.

Q: Why do we multiply the diagonals and divide by 2?

The diagonals of a deltoid are perpendicular and divide the shape into 4 right triangles. The area formula $ \frac{1}{2} \times d_1 \times d_2 $ comes from adding up these triangular areas.

Q: How do I know which diagonal is which in the formula?

It doesn't matter! Since we're multiplying the diagonals, $ d_1 \times d_2 = d_2 \times d_1 $. You can call either diagonal d₁ or d₂ and get the same answer.

Q: What if I'm given the area and need to find a diagonal like in this problem?

Substitute what you know into $ S = \frac{1}{2} \times d_1 \times d_2 $, then solve for the unknown diagonal. In this case: 24 = ½ × AC × 4, so AC = 12 cm.

Q: Can this formula work for any kite-shaped figure?

Yes! Any quadrilateral with perpendicular diagonals uses this area formula. This includes squares, rhombuses, and all deltoids/kites.

Deltoid Area Formula with Diagonal Calculation

Look at the deltoid ABCD below.

Diagonal DB = 4

The area of the deltoid is 24 cm².

Calculate the diagonal AC.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate diagonal AC

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

00:06 (diagonal times diagonal) divided by 2

00:15 We'll substitute appropriate values according to the given data and solve for AC

00:26 Divide 4 by 2

00:31 Isolate AC

00:39 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the deltoid ABCD below.

Diagonal DB = 4

The area of the deltoid is 24 cm².

Calculate the diagonal AC.

Step-by-step solution

To find the length of diagonal AC of the deltoid, follow these steps:

We know the formula for the area of a deltoid (kite) given by $S = \frac{1}{2} \times d_1 \times d_2$ . Here, $S = 24 \, \text{cm}^2$ , $d_1 = AC$ , and $d_2 = BD = 4 \, \text{cm}$ .
Substitute the known values into the formula: $24 = \frac{1}{2} \times AC \times 4$ .
Simplify the equation: $24 = 2 \times AC$ because $\frac{1}{2} \times 4 = 2$ .
Solve for $AC$ by dividing both sides by 2: $AC = \frac{24}{2}$ .
This simplifies to $AC = 12$

Therefore, the length of diagonal AC is $12 \, \text{cm}$ .

Final Answer

12 cm

Key Points to Remember

Essential concepts to master this topic

Formula: Deltoid area equals half the product of diagonal lengths
Technique: Use $S = \frac{1}{2} \times d_1 \times d_2$ where S = 24, d₂ = 4
Check: Verify $\frac{1}{2} \times 12 \times 4 = 24$ cm² ✓

Common Mistakes

Avoid these frequent errors

Using wrong area formula for deltoids
Don't use base × height formula from rectangles = wrong answer! Deltoids aren't rectangles, so this gives completely incorrect results. Always use the diagonal formula: area = ½ × diagonal₁ × diagonal₂.

Practice Quiz

Test your knowledge with interactive questions

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?

FAQ

Everything you need to know about this question

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

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 use the diagonal formula for area.

Why do we multiply the diagonals and divide by 2?

The diagonals of a deltoid are perpendicular and divide the shape into 4 right triangles. The area formula $\frac{1}{2} \times d_1 \times d_2$ comes from adding up these triangular areas.

How do I know which diagonal is which in the formula?

It doesn't matter! Since we're multiplying the diagonals, $d_1 \times d_2 = d_2 \times d_1$ . You can call either diagonal d₁ or d₂ and get the same answer.

What if I'm given the area and need to find a diagonal like in this problem?

Substitute what you know into $S = \frac{1}{2} \times d_1 \times d_2$ , then solve for the unknown diagonal. In this case: 24 = ½ × AC × 4, so AC = 12 cm.

Can this formula work for any kite-shaped figure?

Yes! Any quadrilateral with perpendicular diagonals uses this area formula. This includes squares, rhombuses, and all deltoids/kites.

How can I check if my diagonal calculation is reasonable?

Ask yourself: does the calculated diagonal make sense compared to the given diagonal? In our problem, AC = 12 cm and DB = 4 cm seems reasonable for a deltoid with area 24 cm².

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Deltoid Diagonal Calculation: Finding AC When Area = 24cm² and DB = 4

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 shapes?

Why do we multiply the diagonals and divide by 2?

How do I know which diagonal is which in the formula?

What if I'm given the area and need to find a diagonal like in this problem?

Can this formula work for any kite-shaped figure?

How can I check if my diagonal calculation is reasonable?

🌟 Unlock Your Math Potential

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