Calculate the Area of Deltoid ABCD with Height 5 and Base 8

Q: What's the difference between a deltoid and a regular quadrilateral?

A deltoid (or kite) has two pairs of adjacent equal sides, and its diagonals are perpendicular. This special property lets us use the simple formula: $ \text{Area} = \frac{1}{2} \times d_1 \times d_2 $.

Q: Why do we divide by 2 in the deltoid area formula?

The diagonals of a deltoid divide it into 4 right triangles. When you multiply the full diagonals together, you're counting the area twice, so dividing by 2 gives the correct area.

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

Diagonals connect opposite vertices and cross inside the shape. In this problem, AC = 5 and BD = 8 are the diagonals because they connect opposite corners of the deltoid.

Q: Can I use this formula for any quadrilateral?

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

Q: What if the diagonals aren't given directly?

You might need to use the Pythagorean theorem or coordinate geometry to find diagonal lengths first. Look for right triangles formed by the diagonals and sides.

Area of Deltoid with Diagonal Method

Given the deltoid ABCD

Find the area

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the area of the kite

00:03 We'll use the formula to calculate the area of a kite

00:07 (diagonal multiplied by diagonal) divided by 2

00:13 We'll substitute the appropriate values according to the given data and solve for the area

00:25 Divide 8 by 2

00:31 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

Find the area

Step-by-step solution

To solve the problem of finding the area of the deltoid (kite) ABCD, we will follow these steps:

Step 1: Identify the given diagonal lengths. Here, $AC = 5$ cm and $BD = 8$ cm.
Step 2: Use the formula for the area of a kite or deltoid: $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ where $d_1$ and $d_2$ are the lengths of the diagonals.
Step 3: Plug in the given values into the formula to calculate the area.

Now, let's calculate:
- The length of diagonal $AC = 5$ cm.
- The length of diagonal $BD = 8$ cm.

Applying the formula:

$\text{Area} = \frac{1}{2} \times 5 \times 8 = \frac{1}{2} \times 40 = 20$

Therefore, the area of the deltoid is $20$ cm².

Final Answer

$20$ cm².

Key Points to Remember

Essential concepts to master this topic

Deltoid Formula: Area equals half the product of diagonal lengths
Calculation: Multiply diagonals first: $5 \times 8 = 40$ , then divide by 2
Check: Verify diagonals are perpendicular and formula gives $\frac{1}{2} \times 40 = 20$ ✓

Common Mistakes

Avoid these frequent errors

Using wrong formula for area calculation
Don't use triangle area formula (base × height ÷ 2) for deltoids = wrong answer like 10 cm²! The deltoid formula specifically requires diagonal lengths, not base and height measurements. Always use Area = ½ × d₁ × d₂ for kites and deltoids.

Practice Quiz

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Indicate the correct answer

The next quadrilateral is:

FAQ

Everything you need to know about this question

What's the difference between a deltoid and a regular quadrilateral?

A deltoid (or kite) has two pairs of adjacent equal sides, and its diagonals are perpendicular. This special property lets us use the simple formula: $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ .

Why do we divide by 2 in the deltoid area formula?

The diagonals of a deltoid divide it into 4 right triangles. When you multiply the full diagonals together, you're counting the area twice, so dividing by 2 gives the correct area.

How do I know which measurements are the diagonals?

Diagonals connect opposite vertices and cross inside the shape. In this problem, AC = 5 and BD = 8 are the diagonals because they connect opposite corners of the deltoid.

Can I use this formula for any quadrilateral?

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

What if the diagonals aren't given directly?

You might need to use the Pythagorean theorem or coordinate geometry to find diagonal lengths first. Look for right triangles formed by the diagonals and sides.

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