Calculate the Diagonal Length in a Deltoid: Given Area 32 cm² and AC = 8 cm

Kite Area Formula with Diagonal Calculations

Shown below is the deltoid ABCD.

The diagonal AC is 8 cm long.

The area of the deltoid is 32 cm².

Calculate the diagonal DB.

S=32S=32S=32888AAABBBCCCDDD

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Calculate diagonal DB
00:03 We'll use the formula for calculating the area of a kite
00:06 (diagonal times diagonal) divided by 2
00:10 We'll substitute appropriate values and solve for BD
00:17 Multiply by 2 to eliminate the fraction
00:26 Isolate BD
00:30 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Shown below is the deltoid ABCD.

The diagonal AC is 8 cm long.

The area of the deltoid is 32 cm².

Calculate the diagonal DB.

S=32S=32S=32888AAABBBCCCDDD

2

Step-by-step solution

First, we recall the formula for the area of a kite: multiply the lengths of the diagonals by each other and divide the product by 2.

We substitute the known data into the formula:

8DB2=32 \frac{8\cdot DB}{2}=32

We reduce the 8 and the 2:

4DB=32 4DB=32

Divide by 4

DB=8 DB=8

3

Final Answer

8 cm

Key Points to Remember

Essential concepts to master this topic
  • Area Formula: Kite area equals half the product of diagonal lengths
  • Substitution: 8DB2=32 \frac{8 \cdot DB}{2} = 32 becomes 4DB=32 4 \cdot DB = 32
  • Check: Verify 8×82=32 \frac{8 \times 8}{2} = 32 cm² ✓

Common Mistakes

Avoid these frequent errors
  • Using wrong area formula for kites
    Don't use base × height like for rectangles = wrong answer! Kites aren't rectangles, so this formula doesn't apply and gives completely incorrect results. Always use the diagonal formula: Area = ½ × diagonal₁ × diagonal₂.

Practice Quiz

Test your knowledge with interactive questions

Indicate the correct answer

The next quadrilateral is:

AAABBBCCCDDD

FAQ

Everything you need to know about this question

Is a deltoid the same as a kite?

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Yes! A deltoid is just another name for a kite. Both shapes have two pairs of adjacent sides that are equal in length, and their diagonals are perpendicular.

Why do we divide by 2 in the area formula?

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The diagonals of a kite divide it into four right triangles. When you multiply the full diagonals, you're calculating twice the actual area, so dividing by 2 gives the correct result.

What if both diagonals were unknown?

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You'd need additional information like the lengths of the sides or angles. With just the area, you can only find one diagonal if you know the other.

Do the diagonals have to be perpendicular?

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Yes! In a kite, the diagonals are always perpendicular (meet at 90°). This is a special property that makes the area formula work.

Could I get a decimal answer for the diagonal?

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Absolutely! If the area or known diagonal creates a decimal result, that's perfectly valid. Just make sure your calculation is accurate and round appropriately if needed.

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