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Base Of A Trapezoid

A trapezoid is a unique quadrilateral with one pair of parallel sides, known as its bases. Understanding these bases is key to calculating a trapezoid's area. This lesson will show you how to use the area formula, along with some simple algebra, to find the length of a missing base.

What Are the Bases of a Trapezoid?

The bases of a trapezoid are its two parallel sides. In any trapezoid, you will find exactly one pair of sides that run alongside each other like railroad tracks, never getting closer or farther apart. These are the bases. The other two sides, which are not parallel, are called the legs.

We typically label the bases as $b_{1}$ (base one) and $b_{2}$ (base two). It doesn't matter which parallel side you label as $b_{1}$ or $b_{2}$ , as the formulas we use will work either way. The key properties to remember are:

The bases are always parallel to each other.
The bases are not usually the same length. (If they were, the shape would be a parallelogram).
The orientation of the trapezoid doesn't change which sides are the bases. A base can be at the top, bottom, or on the side, as long as it's parallel to another side.

The perpendicular distance between these two bases is called the height ( $h$ ). It's crucial not to confuse the height with the length of a slanted leg. The height must form a right angle ( $90^{\circ}$ ) with both bases.

How Is the Area Formula Used to Find a Base?

The relationship between a trapezoid's area, its height, and its bases is described by the area formula. This formula is the most important tool we have for finding the length of a missing base when we know the other dimensions.

The standard formula for the area of a trapezoid is:

A = \frac{1}{2} (b_{1} + b_{2}) h

Let's break down what each variable means:

$A$ is the Area, which is the amount of space inside the trapezoid.
$b_{1}$ and $b_{2}$ are the lengths of the two parallel bases.
$h$ is the height, the perpendicular distance between the bases.

This formula tells us to add the lengths of the bases, multiply by the height, and then take half of the result. However, what if we already know the area ( $A$ ), the height ( $h$ ), and one base (say, $b_{1}$ ), but need to find the other base ( $b_{2}$ )? We can use algebra to work backward from the answer, which involves rearranging the formula to solve for the variable we need.

How Do You Rearrange the Trapezoid Area Formula?

Rearranging a formula, also known as solving for a variable, is a fundamental algebra skill. Our goal is to get the variable for the missing base (let's say $b_{2}$ ) all by itself on one side of the equals sign. We can do this by performing inverse operations in a specific order.

Let's start with our original formula:

A = \frac{1}{2} (b_{1} + b_{2}) h

Here is the step-by-step process to isolate $b_{2}$ :

Eliminate the fraction: The first step is to get rid of the $\frac{1}{2}$ . We do this by multiplying both sides of the equation by $2$ .
$2 \cdot A = 2 \cdot \frac{1}{2} (b_{1} + b_{2}) h$
This simplifies to: $2 A = (b_{1} + b_{2}) h$
Isolate the parentheses: The sum of the bases $(b_{1} + b_{2})$ is being multiplied by the height $h$ . The inverse operation of multiplication is division, so we divide both sides by $h$ .
$\frac{2 A}{h} = \frac{(b_{1} + b_{2}) h}{h}$
This simplifies to: $\frac{2 A}{h} = b_{1} + b_{2}$
Isolate the final variable: Now, $b_{2}$ is being added to $b_{1}$ . To get $b_{2}$ by itself, we subtract $b_{1}$ from both sides.
$\frac{2 A}{h} - b_{1} = b_{1} + b_{2} - b_{1}$
This leaves us with our final, rearranged formula:

b_{2} = \frac{2 A}{h} - b_{1}

Because it doesn't matter which base is which, the formula to find $b_{1}$ is identical:

b_{1} = \frac{2 A}{h} - b_{2}

Using one of these rearranged formulas makes solving for a missing base much faster.

Can We Walk Through an Example of Finding a Base?

Absolutely. Let's apply the formula we just derived to a practical problem. Following a clear, step-by-step process is the best way to ensure you get the correct answer every time.

Example 1

A trapezoid has a total area of $90$ square inches. Its height is $6$ inches and one of its bases measures $12$ inches. What is the length of the other base?

Step 1: Identify your known values.
From the problem, we know:
Area $A = 90 {in}^{2}$
Height $h = 6 in$
One base $b_{1} = 12 in$
We need to find the other base, $b_{2}$ .

Step 2: Choose the correct formula.
Since we need to find $b_{2}$ , we will use the rearranged formula:
$b_{2} = \frac{2 A}{h} - b_{1}$

Step 3: Substitute the known values into the formula.
$b_{2} = \frac{2 (90)}{6} - 12$

Step 4: Solve the equation using the order of operations (PEMDAS/BODMAS).
First, handle the multiplication in the numerator:
$b_{2} = \frac{180}{6} - 12$ Next, perform the division:
$b_{2} = 30 - 12$ Finally, do the subtraction:
$b_{2} = 18$

Step 5: State the final answer with units.
The length of the other base is $18$ inches.

Let's Try Another Example with a Check

Practice makes perfect. Let's work through another problem. This time, after we find the answer, we will plug it back into the original area formula to verify that it's correct.

Example 2

The area of a trapezoidal park is $5000$ square meters. The distance between its two parallel roads (the bases) is $50$ meters. If the shorter road is $80$ meters long, what is the length of the longer road?

Step 1: Identify your known values.
Area $A = 5000 m^{2}$
Height $h = 50 m$
One base $b_{1} = 80 m$
We are looking for the other base, $b_{2}$ .

Step 2: Use the rearranged formula.
$b_{2} = \frac{2 A}{h} - b_{1}$

Step 3: Substitute the values.
$b_{2} = \frac{2 (5000)}{50} - 80$

Step 4: Solve the equation.
Multiply the numerator:
$b_{2} = \frac{10000}{50} - 80$ Divide:
$b_{2} = 200 - 80$ Subtract:
$b_{2} = 120$

The length of the longer road is $120$ meters.

Step 5: Check your answer.
Let's plug our new base back into the original area formula: $A = \frac{1}{2} (b_{1} + b_{2}) h$ .
$A = \frac{1}{2} (80 + 120) (50)$ $A = \frac{1}{2} (200) (50)$ $A = (100) (50)$ $A = 5000$ This matches the area given in the problem, so our answer is correct!

Key formulas for base of a trapezoid by Algebra911.

What if You Know the Median Instead of the Area?

Sometimes, a problem might give you the median of a trapezoid instead of its area. The median (also called the midsegment) is a line segment that connects the midpoints of the non-parallel legs. It runs parallel to the bases, and its length is exactly the average of the lengths of the two bases.

The formula for the median $M$ is:

M = \frac{b_{1} + b_{2}}{2}

If you know the median and one base, you can easily find the other base by rearranging this simpler formula. Let's solve for $b_{2}$ :

Multiply both sides by 2: $2 M = b_{1} + b_{2}$
Subtract $b_{1}$ from both sides: $2 M - b_{1} = b_{2}$

This gives us a new tool for finding a missing base.

Example 3

A trapezoid has a median that is $25$ cm long. One of its bases measures $18$ cm. Find the length of the second base.

Step 1: Identify your knowns.
Median $M = 25 cm$
Base $b_{1} = 18 cm$

Step 2: Choose the correct formula.
We'll use the rearranged median formula: $b_{2} = 2 M - b_{1}$ .

Step 3: Substitute and solve.
$b_{2} = 2 (25) - 18$ $b_{2} = 50 - 18$ $b_{2} = 32$

The length of the second base is $32$ cm.

What Are Some Common Mistakes When Finding a Trapezoid's Base?

When working with trapezoid formulas, a few common errors can trip students up. Being aware of these can help you avoid them.

Forgetting to Multiply by 2: The most frequent mistake is forgetting to multiply the area $A$ by $2$ as the very first step. Students often calculate $\frac{A}{h} - b_{1}$ , which will give the wrong answer. Remember, the formula is $\frac{2 A}{h} - b_{1}$ .
Confusing Height and Leg Length: Always use the perpendicular height (the one that forms a right angle with the base), not the length of a slanted side (a leg). If the height isn't given directly, you may need to find it using other methods.
Incorrect Order of Operations: Make sure you follow the order of operations. In the formula $\frac{2 A}{h} - b_{1}$ , you must perform the division $\frac{2 A}{h}$ *before* you subtract $b_{1}$ .
Unit Mismatch: Ensure all your measurements are in the same units before you begin. If the area is in square feet and the height is in inches, you must convert one of them before plugging them into the formula.

Quick Reference: Key Formulas

Here is a quick summary of the essential definitions and formulas covered in this lesson for easy reference.

Concept	Formula	Description
Bases ( $b_{1}, b_{2}$ )	N/A	The two parallel sides of a trapezoid.
Area ( $A$ )	$A = \frac{1}{2} (b_{1} + b_{2}) h$	Calculates the total space inside the trapezoid.
Finding a Base (from Area)	$b_{2} = \frac{2 A}{h} - b_{1}$	Use this when you know the area, height, and one base.
Median ( $M$ )	$M = \frac{b_{1} + b_{2}}{2}$	The length of the midsegment, which is the average of the bases.
Finding a Base (from Median)	$b_{2} = 2 M - b_{1}$	Use this when you know the median and one base.

Frequently Asked Questions

Can a trapezoid have more than two bases?

No, a trapezoid is a quadrilateral defined by having exactly one pair of parallel sides. These two parallel sides are its only bases.

Does it matter which base I call b1 and which I call b2?

No, it does not matter at all. Since the bases are added together in the formula ( $b_{1} + b_{2}$ ), the order is irrelevant. You can assign either parallel side to be $b_{1}$ or $b_{2}$ and get the same result.

What if I'm given the lengths of the slanted sides (legs) instead of the height?

The lengths of the legs cannot be used directly in the area formula to find a base. You must know the perpendicular height. In more advanced problems, you might use the Pythagorean theorem on a right triangle formed by the height, a leg, and a portion of the base to find the height first.

Can the two bases of a trapezoid be equal in length?

If the two parallel bases were equal in length, the shape would be a parallelogram, not a trapezoid. The definition of a trapezoid requires only one pair of parallel sides, and they are typically of different lengths.

Is the base always the bottom side of the trapezoid?

No, the term 'base' in geometry refers to the parallel sides, not the orientation of the shape. A trapezoid can be rotated in any direction, and the two sides that are parallel to each other are always considered the bases.

How is the median of a trapezoid related to its bases?

The median's length is the average of the lengths of the two bases. You can find it with the formula $M = (b_{1} + b_{2}) / 2$ . This also means that the area can be found by simply multiplying the median by the height ( $A = M \cdot h$ ).

What happens if I forget to multiply the Area by 2 when solving for a base?

Forgetting to multiply the area by 2 is a very common mistake. This will cause your result for the term $\frac{2 A}{h}$ to be half of what it should be, leading to an incorrect final answer for the missing base.