Calculate the Area of a Deltoid: Given Height 9 and Base 15

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

A deltoid (also called a kite) is a quadrilateral with two pairs of adjacent sides that are equal. Unlike rectangles or parallelograms, its diagonals are perpendicular and one bisects the other at right angles.

Q: Why do we use half the product of the diagonals?

The deltoid can be divided into four right triangles by its diagonals. Each triangle has area = $ \frac{1}{2} \times \text{base} \times \text{height} $, and when you add all four, you get $ \frac{1}{2} \times d_1 \times d_2 $.

Q: Do the diagonals have to be perpendicular for this formula to work?

Yes, absolutely! This formula $ \frac{1}{2} \times d_1 \times d_2 $ only works when the diagonals are perpendicular (meet at 90°). For deltoids, this is always true by definition.

Q: What if I mixed up which measurement is which diagonal?

Great news - it doesn't matter! Since we're multiplying the diagonals together, $ 9 \times 15 $ gives the same result as $ 15 \times 9 $. Multiplication is commutative!

Q: How can I remember this formula for tests?

Think: "Half the diagonals' product" - just like finding the area of a rectangle (length × width), but since the diagonals create triangular pieces, we divide by 2.

Deltoid Area with Diagonal Measurements

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 will use the formula to calculate the area of a kite

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

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

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 this problem, we'll follow these steps:

Identify the given information: The lengths of diagonals are 9 cm and 15 cm.
Apply the appropriate formula for the area of a deltoid.
Perform the necessary calculations.

Now, let's work through each step:
Step 1: We are given that diagonal $d_1 = 9$ cm and diagonal $d_2 = 15$ cm.
Step 2: We'll use the formula for the area of a deltoid: $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ .
Step 3: Plugging in our values, we get: $\text{Area} = \frac{1}{2} \times 9 \times 15 = \frac{1}{2} \times 135 = 67.5 \text{ cm}^2$

Therefore, the solution to the problem is $67.5$ cm².

Final Answer

$67.5$ cm².

Key Points to Remember

Essential concepts to master this topic

Formula: Area equals one-half times diagonal one times diagonal two
Technique: Multiply diagonals first: $9 \times 15 = 135$ , then divide by 2
Check: Verify using $\frac{1}{2} \times 135 = 67.5$ cm² matches answer ✓

Common Mistakes

Avoid these frequent errors

Adding diagonals instead of multiplying them
Don't add the diagonals like 9 + 15 = 24, then divide by 2 = 12 cm²! This completely ignores how area works in 2D shapes. Always multiply the diagonals first, then divide the product by 2.

Practice Quiz

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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 (also called a kite) is a quadrilateral with two pairs of adjacent sides that are equal. Unlike rectangles or parallelograms, its diagonals are perpendicular and one bisects the other at right angles.

Why do we use half the product of the diagonals?

The deltoid can be divided into four right triangles by its diagonals. Each triangle has area = $\frac{1}{2} \times \text{base} \times \text{height}$ , and when you add all four, you get $\frac{1}{2} \times d_1 \times d_2$ .

Do the diagonals have to be perpendicular for this formula to work?

Yes, absolutely! This formula $\frac{1}{2} \times d_1 \times d_2$ only works when the diagonals are perpendicular (meet at 90°). For deltoids, this is always true by definition.

What if I mixed up which measurement is which diagonal?

Great news - it doesn't matter! Since we're multiplying the diagonals together, $9 \times 15$ gives the same result as $15 \times 9$ . Multiplication is commutative!

How can I remember this formula for tests?

Think: "Half the diagonals' product" - just like finding the area of a rectangle (length × width), but since the diagonals create triangular pieces, we divide by 2.

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