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.

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

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

Given the following trapezoid:

Calculate the area of the trapezoid ABCD.

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!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Trapeze questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime