Calculate Trapezoid Height: Finding AE Given Area 45 cm² and Bases 4 cm, 6 cm

Q: Why is there a 1/2 in the trapezoid area formula?

The 1/2 factor comes from averaging the two parallel bases. Think of it as finding the area of a rectangle with width equal to the average of the two bases: $ \frac{b_1 + b_2}{2} $ × height.

Q: How do I know which sides are the bases?

The bases are the parallel sides of the trapezoid. In this problem, BC = 4 cm and AD = 6 cm are clearly labeled as the bases. The height is always perpendicular to these parallel sides.

Q: What does AE represent in this trapezoid?

AE is the height of the trapezoid - the perpendicular distance between the two parallel bases. You can see in the diagram that AE forms a right angle with the base, shown by the small square symbol.

Q: Can I solve this if the bases were different lengths?

Yes! The trapezoid formula works for any two parallel bases. Just substitute the given base lengths and area into $ A = \frac{1}{2}(b_1 + b_2)h $ and solve for height.

Trapezoid Area Formula with Height Calculation

The area of trapezoid ABCD is equal to 45 cm².

The base of the trapezoid BC is equal to 4 cm.

The base of the trapezoid AD is equal to 6 cm.

Calculate the length of AE.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:13 Let's find the height AE.

00:16 We'll use the trapezoid area formula.

00:19 Add the bases, multiply by the height, and then divide by two.

00:28 Now, substitute the values from the data and solve for the height.

00:58 Let's isolate height AE to find its value.

01:11 And that's how we solve the question!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The area of trapezoid ABCD is equal to 45 cm².

The base of the trapezoid BC is equal to 4 cm.

The base of the trapezoid AD is equal to 6 cm.

Calculate the length of AE.

Step-by-step solution

The problem can be solved using the formula for the area of a trapezoid:

$S = \frac{1}{2} \times (b_1 + b_2) \times h$

Where $b_1$ and $b_2$ are the lengths of the two parallel sides (bases) of the trapezoid, and $h$ is the height. For this trapezoid, the data given is:

Area, $S = 45 \, \text{cm}^2$
Base 1, $b_1 = 4 \, \text{cm}$
Base 2, $b_2 = 6 \, \text{cm}$

Substitute these values into the area formula:

$45 = \frac{1}{2} \times (4 + 6) \times h$

Simplify the formula:

$45 = 5 \times h$

Divide both sides of the equation by 5 to solve for $h$ :

$h = \frac{45}{5} = 9$

Thus, the length of AE, or the height of the trapezoid, is $\mathbf{9 \, \text{cm}}$ .

Therefore, the solution to the problem is that the length of AE is $9 \, \text{cm}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Formula: Area = (1/2) × (base₁ + base₂) × height
Technique: Substitute known values: 45 = (1/2) × (4 + 6) × h
Check: Verify: (1/2) × 10 × 9 = 45 cm² ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply by 1/2 in the trapezoid formula
Don't use Area = (base₁ + base₂) × height = wrong result twice as large! This confuses trapezoid formula with rectangle formula and gives incorrect height values. Always include the 1/2 factor in the trapezoid area formula.

Practice Quiz

Test your knowledge with interactive questions

Calculate the area of the trapezoid.

FAQ

Everything you need to know about this question

Why is there a 1/2 in the trapezoid area formula?

The 1/2 factor comes from averaging the two parallel bases. Think of it as finding the area of a rectangle with width equal to the average of the two bases: $\frac{b_1 + b_2}{2}$ × height.

How do I know which sides are the bases?

The bases are the parallel sides of the trapezoid. In this problem, BC = 4 cm and AD = 6 cm are clearly labeled as the bases. The height is always perpendicular to these parallel sides.

What does AE represent in this trapezoid?

AE is the height of the trapezoid - the perpendicular distance between the two parallel bases. You can see in the diagram that AE forms a right angle with the base, shown by the small square symbol.

Can I solve this if the bases were different lengths?

Yes! The trapezoid formula works for any two parallel bases. Just substitute the given base lengths and area into $A = \frac{1}{2}(b_1 + b_2)h$ and solve for height.

How do I check my answer is reasonable?

Think about it logically: with bases of 4 cm and 6 cm (average = 5 cm), and area = 45 cm², the height should be $\frac{45 ÷ 5}{1/2} = 9$ cm. This seems reasonable for the diagram!

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