Calculate Trapezoid Area with Equilateral Triangle and Parallelogram Properties

The tapezoid ABCD and the parallelogram ABED are shown below.

EBC is an equilateral triangle.

What is the area of the trapezoid?

Video Solution

Solution Steps

00:00 Find the area of the trapezoid

00:03 equilateral triangle

00:08 Let's draw the height in the triangle

00:11 The height in an equilateral triangle is also a median

00:17 Let's use the Pythagorean theorem in the small triangle we created

00:21 Let's substitute the side values to find the height

00:24 Let's isolate H

00:41 This is the size of height H

00:47 Opposite sides are equal in parallelograms

00:51 Let's use the formula for calculating trapezoid area

00:55 (Sum of bases(AB+DC) multiplied by height(H))divided by 2

01:00 The triangle and trapezoid share the same height H

01:12 And this is the solution to the problem

Step-by-Step Solution

To find the area of trapezoid $ABCD$ , we need to determine the height using $\triangle EBC$ , which is equilateral with side $BC = 3$ cm.

Step 1: Calculating height of $\triangle EBC$ .
For $\triangle EBC$ , the height ( $h_t$ ) is $h_t = \frac{\sqrt{3}}{2} \times 3 = \frac{3\sqrt{3}}{2}$ cm.
Step 2: Confirm equal base length.
The base $AB$ is considered equal to $ED$ in parallelogram $ABED$ , it is shared in the trapezoid.
Step 3: Use trapezoid area formula $A = \frac{1}{2} \times (b_1 + b_2) \times h$ .
Considering $b_1 = AB = 3$ cm (since $BE = BC$ ) and $b_2 = 9$ cm (given $DC$ ), with the height $h$ same as equilateral triangle $EBC$ , calculate:

The exact calculation becomes:

$A = \frac{1}{2} \times (3 + 9) \times \frac{3\sqrt{3}}{2} = \frac{1}{2} \times 12 \times \frac{3\sqrt{3}}{2} = 18\sqrt{3}$ square centimeters.

Approximating, $18\sqrt{3} \approx 27.3$ cm².

Therefore, the area of trapezoid $ABCD$ is $\boldsymbol{27.3}$ cm².