Calculate Kite Area: Given Triangle Area 30 cm² and Diagonals 5 cm, 6 cm

Q: Why can't I just use the triangle area to find the kite area directly?

The triangle ABD is only part of the kite! A kite has four triangular sections. You need to find the missing diagonal BD first, then use the kite area formula with both complete diagonals.

Q: How do I know which measurements are diagonals vs. diagonal segments?

Look at the diagram carefully! AE = 5 and EC = 6 are segments of diagonal AC. The full diagonal AC = AE + EC = 11 cm. Point E is where the diagonals intersect.

Q: Why does triangle ABD have area 30 when I calculated it differently?

Remember that triangle area = $ \frac{1}{2} \times base \times height $. In triangle ABD, use AE as height (5 cm) and BD as base (12 cm): $ \frac{1}{2} \times 12 \times 5 = 30 $ cm².

Kite Area with Given Triangle Area

ABCD is a kite.

AB = AD

ABD has an area of 30 cm².
EC is equal to 6 cm.
AE is equal to 5 cm.

Calculate the area of the kite.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate the area of the kite

00:03 We'll use the triangle area formula to find DB

00:08 (height(AE) multiplied by base(DB)) divided by 2

00:11 Let's substitute appropriate values and solve for DB

00:18 Let's isolate DB

00:27 This is the size of DB

00:31 The entire side equals the sum of its parts

00:37 Now let's use the formula for calculating kite area

00:41 (diagonal multiplied by diagonal) divided by 2

00:44 Let's substitute appropriate values and solve for the area

00:48 Divide 12 by 2

00:52 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

ABCD is a kite.

AB = AD

ABD has an area of 30 cm².
EC is equal to 6 cm.
AE is equal to 5 cm.

Calculate the area of the kite.

Step-by-step solution

To solve this problem, we'll use the properties of triangle and kite areas:

Firstly, we note that the area of triangle $ABD$ is given as $30 \, \text{cm}^2$ .

We recognize that triangles $ABD$ and $ADC$ together form the kite with diagonal $AC$ , and triangles $ADB$ and $BCD$ form diagonals where two triangles area will be half the kite’s complete area.

Now, find triangle $ABE$ : $\frac{1}{2} \times AE \times BD = 30 \implies \frac{1}{2} \times 5 \times BD = 30 \implies BD = 12 \, \text{cm}$

Next, calculate diagonal $AC$ (sum of segments): $AE + EC = AC \implies 5 + 6 = AC \implies AC = 11 \, \text{cm}$

The area of kite $ABCD$ from diagonals $AC$ and $BD$ : $\text{Area}_{kite} = \frac{1}{2} \times AC \times BD = \frac{1}{2} \times 11 \times 12 = 66 \, \text{cm}^2$

Therefore, the solution to the problem is 66 cm².

Final Answer

66 cm²

Key Points to Remember

Essential concepts to master this topic

Kite Property: Diagonals intersect at right angles and bisect each other
Triangle Area Formula: Area = $\frac{1}{2} \times base \times height = \frac{1}{2} \times 5 \times 12 = 30$
Verification: Kite area = $\frac{1}{2} \times 11 \times 12 = 66$ cm² matches diagonal formula ✓

Common Mistakes

Avoid these frequent errors

Using wrong diagonal lengths in kite area formula
Don't use AE = 5 and EC = 6 as separate diagonals = wrong calculation! These are segments of one diagonal, not two complete diagonals. Always add AE + EC = 11 cm for the full diagonal AC length.

Practice Quiz

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FAQ

Everything you need to know about this question

Why can't I just use the triangle area to find the kite area directly?

The triangle ABD is only part of the kite! A kite has four triangular sections. You need to find the missing diagonal BD first, then use the kite area formula with both complete diagonals.

How do I know which measurements are diagonals vs. diagonal segments?

Look at the diagram carefully! AE = 5 and EC = 6 are segments of diagonal AC. The full diagonal AC = AE + EC = 11 cm. Point E is where the diagonals intersect.

Can I use a different method to find the kite area?

Yes! You could find the area of each triangle separately and add them up. But using the diagonal formula $\frac{1}{2} \times d_1 \times d_2$ is usually faster and less prone to errors.

Why does triangle ABD have area 30 when I calculated it differently?

Remember that triangle area = $\frac{1}{2} \times base \times height$ . In triangle ABD, use AE as height (5 cm) and BD as base (12 cm): $\frac{1}{2} \times 12 \times 5 = 30$ cm².

What if I get a different answer using the segments AE and EC?

If you use AE = 5 and EC = 6 as if they were full diagonals, you'd get $\frac{1}{2} \times 5 \times 6 = 15$ cm², which is way too small! Always check that your final answer makes sense compared to the given triangle area.

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