Calculate Deltoid Area: Finding Area with Height 5 and Base 16

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

A deltoid is a type of kite - a quadrilateral with two pairs of adjacent equal sides. Unlike rectangles or parallelograms, deltoids have perpendicular diagonals that intersect at right angles.

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

Look for the perpendicular lines in the diagram! In this problem, the vertical line (height 5) and horizontal line (base 16) cross at 90° - these are your diagonals $ d_1 $ and $ d_2 $.

Q: Why can't I just use base times height like for rectangles?

Because a deltoid isn't a rectangle! The formula $ \text{base} \times \text{height} $ works for rectangles, but deltoids are kite-shaped and need the special kite formula: $ \frac{1}{2} \times d_1 \times d_2 $.

Q: What if the diagonals aren't perpendicular?

Then it's not a true deltoid or kite! For the area formula $ \frac{1}{2} \times d_1 \times d_2 $ to work, the diagonals must be perpendicular. Always check the diagram for that 90° angle.

Q: How can I double-check my area calculation?

Substitute back: $ \frac{1}{2} \times 16 \times 5 = \frac{80}{2} = 40 $. Also, think logically - the area should be less than a rectangle with the same dimensions (16 × 5 = 80), so 40 makes sense!

Deltoid Area with Kite Formula

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 kite's area

00:07 (diagonal times diagonal) divided by 2

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

00:26 We'll divide 16 by 2

00:37 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 find the area of the deltoid ABCD, we use the external height formula for deltoids:

Given:
- Height ( $h$ ) = $16$ cm
- Segment related to base ( $b$ ) = $5$ cm

The area of the deltoid can be calculated by:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

Plugging in our values, we have:

$\text{Area} = \frac{1}{2} \times 5 \times 16$

Calculating the result:

$\text{Area} = \frac{1}{2} \times 80 = 40$ cm $^2$

Therefore, the area of deltoid ABCD is $40$ cm $^2$ .

Final Answer

$40$ cm².

Key Points to Remember

Essential concepts to master this topic

Formula: Area = $\frac{1}{2} \times d_1 \times d_2$ for deltoid diagonals
Technique: Identify diagonals as base 16 and height 5
Check: $\frac{1}{2} \times 16 \times 5 = 40$ cm² ✓

Common Mistakes

Avoid these frequent errors

Confusing deltoid with triangle area formula
Don't use $\frac{1}{2} \times \text{base} \times \text{height}$ thinking of it as a triangle = wrong result! This treats it as a simple triangle instead of a kite shape. Always use the kite formula $\frac{1}{2} \times d_1 \times d_2$ where the diagonals are perpendicular.

Practice Quiz

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Indicate the correct answer

The next quadrilateral is:

FAQ

Everything you need to know about this question

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

A deltoid is a type of kite - a quadrilateral with two pairs of adjacent equal sides. Unlike rectangles or parallelograms, deltoids have perpendicular diagonals that intersect at right angles.

How do I identify which measurements are the diagonals?

Look for the perpendicular lines in the diagram! In this problem, the vertical line (height 5) and horizontal line (base 16) cross at 90° - these are your diagonals $d_1$ and $d_2$ .

Why can't I just use base times height like for rectangles?

Because a deltoid isn't a rectangle! The formula $\text{base} \times \text{height}$ works for rectangles, but deltoids are kite-shaped and need the special kite formula: $\frac{1}{2} \times d_1 \times d_2$ .

What if the diagonals aren't perpendicular?

Then it's not a true deltoid or kite! For the area formula $\frac{1}{2} \times d_1 \times d_2$ to work, the diagonals must be perpendicular. Always check the diagram for that 90° angle.

How can I double-check my area calculation?

Substitute back: $\frac{1}{2} \times 16 \times 5 = \frac{80}{2} = 40$ . Also, think logically - the area should be less than a rectangle with the same dimensions (16 × 5 = 80), so 40 makes sense!

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