Introduction
Determining the area of a trapezoid can be straightforward when certain elements are known. In this guide, we'll explore how to calculate the area of trapezoid ABCD, given the lengths of the bases AB and CD, and the angle ADC is 60 degrees.
The Formula for the Area of a Trapezoid
The area \( A \) of a trapezoid can be found using the formula:
\[ A = \frac{1}{2} \times (a + b) \times h \]
Where:
- \( a \) is the length of the top base AB.
- \( b \) is the length of the bottom base CD.
- \( h \) is the height of the trapezoid.
Understanding the Given Information
Given:
- \( AB = a \) (top base)
- \( CD = b \) (bottom base)
- \( \angle ADC = 60^\circ \)
The challenge is to find the height \( h \) using the given angle and the lengths of the bases.
Calculating the Height
The height \( h \) of the trapezoid can be calculated using trigonometry. From the right triangle formed by dropping a perpendicular from point D to line AB, we can use the sine function:
\[ h = b \sin(\theta) \]
Given \( \theta = 60^\circ \):
\[ h = b \sin(60^\circ) \]
Since \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \):
\[ h = b \times \frac{\sqrt{3}}{2} \]
Step-by-Step Calculation
Let's work through an example to illustrate the process.
Example:
Suppose we have a trapezoid with \( AB = 5 \) units, \( CD = 10 \) units, and \( \angle ADC = 60^\circ \). We want to find the area of the trapezoid.
Step 1: Identify the Given Values
Given:
- Top base \( AB = 5 \) units
- Bottom base \( CD = 10 \) units
- Angle \( \angle ADC = 60^\circ \)
Step 2: Calculate the Height
Using the height formula:
\[ h = 10 \times \frac{\sqrt{3}}{2} \]
\[ h = 10 \times 0.866 \]
\[ h \approx 8.66 \]
Step 3: Use the Formula to Find the Area
Using the area formula \( A = \frac{1}{2} \times (a + b) \times h \):
\[ A = \frac{1}{2} \times (5 + 10) \times 8.66 \]
\[ A = \frac{1}{2} \times 15 \times 8.66 \]
\[ A = 7.5 \times 8.66 \]
\[ A \approx 64.95 \]
Final Value
For trapezoid ABCD with bases \( AB = 5 \) units and \( CD = 10 \) units, and angle \( \angle ADC = 60^\circ \), the area is approximately \( 64.95 \) square units.
