Calculate Deltoid Area: Finding Area with Height 5 and Base 8

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

A deltoid (or kite) is a quadrilateral with two pairs of adjacent sides that are equal. Unlike a rhombus where all sides are equal, or a rectangle where opposite sides are equal, a deltoid has a unique "kite" shape with perpendicular diagonals.

Q: How do I identify the diagonals in a deltoid from a diagram?

The diagonals are the lines connecting opposite vertices. In deltoid ABCD, the diagonals are AC and BD. They always intersect at right angles, which is why we can use the diagonal formula for area.

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

When two perpendicular lines intersect, they create a rectangle with area = length × width. The deltoid takes up exactly half of that rectangle, so we divide by 2 to get the actual deltoid area.

Q: Can I use this same formula for other quadrilaterals?

This $ \frac{1}{2} \times d_1 \times d_2 $ formula works for any quadrilateral with perpendicular diagonals, including rhombi, squares, and deltoids. But it won't work for rectangles or parallelograms unless their diagonals are perpendicular.

Q: What if I'm given different measurements like side lengths?

If you only have side lengths, you'll need additional information like angles or one diagonal length. The diagonal method is the most direct way to find deltoid area when diagonal measurements are provided.

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'll use the formula to calculate the kite's area

00:07 (diagonal times diagonal) divided by 2

00:14 We'll substitute the appropriate values according to the given data and solve for the area

00:25 Divide 8 by 2

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

Given the deltoid ABCD

Find the area

Step-by-step solution

To solve for the area of deltoid $ABCD$ , we employ the formula for the area of a deltoid given its diagonals:

The formula is $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ , where $d_1$ and $d_2$ are the diagonals.
From the problem, the length of the first diagonal ( $d_1$ ) is 5 units and the second diagonal ( $d_2$ ) is 8 units.

Now, substituting these values into the formula:

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

Thus, the area of the deltoid is $\mathbf{20 \, \text{cm}^2}$ .

The correct choice from the given options is choice 2.

Final Answer

$20$ 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: Multiply diagonals first: 5 × 8 = 40, then divide by 2
Check: Verify diagonals are perpendicular and formula gives $20 \, \text{cm}^2$ ✓

Common Mistakes

Avoid these frequent errors

Confusing deltoid with rhombus or using side lengths instead of diagonals
Don't use side lengths or base × height formulas = wrong calculation! A deltoid has two pairs of adjacent equal sides, but its area specifically needs the diagonal measurements. Always identify the perpendicular diagonals and use $\frac{1}{2} \times d_1 \times d_2$ .

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 quadrilaterals?

A deltoid (or kite) is a quadrilateral with two pairs of adjacent sides that are equal. Unlike a rhombus where all sides are equal, or a rectangle where opposite sides are equal, a deltoid has a unique "kite" shape with perpendicular diagonals.

How do I identify the diagonals in a deltoid from a diagram?

The diagonals are the lines connecting opposite vertices. In deltoid ABCD, the diagonals are AC and BD. They always intersect at right angles, which is why we can use the diagonal formula for area.

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

When two perpendicular lines intersect, they create a rectangle with area = length × width. The deltoid takes up exactly half of that rectangle, so we divide by 2 to get the actual deltoid area.

Can I use this same formula for other quadrilaterals?

This $\frac{1}{2} \times d_1 \times d_2$ formula works for any quadrilateral with perpendicular diagonals, including rhombi, squares, and deltoids. But it won't work for rectangles or parallelograms unless their diagonals are perpendicular.

What if I'm given different measurements like side lengths?

If you only have side lengths, you'll need additional information like angles or one diagonal length. The diagonal method is the most direct way to find deltoid area when diagonal measurements are provided.

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