A trapezoid is a four-sided figure with at least one pair of parallel sides. These parallel sides are known as the bases of the trapezoid. The height (or altitude) of a trapezoid is the perpendicular distance between these two bases.
Formula for the Area of a Trapezoid
The area of a trapezoid can be calculated using the following formula:
$A = frac{1}{2} (b_1 + b_2) h$
where:
- $A$ is the area of the trapezoid.
- $b_1$ and $b_2$ are the lengths of the two parallel sides (bases).
- $h$ is the height of the trapezoid.
Solving for the Height
To find the height of the trapezoid, we need to rearrange the area formula to solve for $h$. Here’s how you can do it:
- Start with the area formula:
$A = frac{1}{2} (b_1 + b_2) h$
- Multiply both sides of the equation by 2 to eliminate the fraction:
$2A = (b_1 + b_2) h$
- Divide both sides by the sum of the bases $(b_1 + b_2)$ to isolate $h$:
$h = frac{2A}{b_1 + b_2}$
Now, you have a formula to find the height of the trapezoid:
$h = frac{2A}{b_1 + b_2}$
Example Problem
Let’s go through an example to make this clearer. Suppose you have a trapezoid with bases $b_1 = 8$ units and $b_2 = 5$ units, and the area $A = 39$ square units. We can find the height as follows:
- Plug the values into the formula:
$h = frac{2 times 39}{8 + 5}$
- Simplify the expression:
$h = frac{78}{13}$
- Calculate the height:
$h = 6$
So, the height of the trapezoid is 6 units.
Conclusion
Finding the height of a trapezoid involves knowing its area and the lengths of its two bases. By rearranging the area formula, you can solve for the height using the equation $h = frac{2A}{b_1 + b_2}$. This method is straightforward and can be applied to any trapezoid as long as you have the necessary measurements.
3. CK-12 Foundation – Trapezoid Area