Areas of Parallelograms, Triangles, and Trapezoids

We do not know a formula for finding the area of the given pentagon. However, we can divide it into polygons for which we know how to find the areas. For example, let's draw a segment connecting B and D.

We can see that the upper polygon is a triangle. We are told that BD and AE are parallel, which means that ABDE is a trapezoid.

We can find the area of the pentagon by adding the area of the trapezoid and the area of the triangle. A = A_1 + A_2 Let's write the formulas for the area of each polygon.

Area of a Trapezoid	Area of a Triangle
A_1=1/2h(b_1+b_2)	A_2=1/2bh

Finding A_1

The height of the trapezoid is 3 and its bases are 3 and 7. Let's use these values to find its area.

Finding A_2

In △ BCD, we can see that the height corresponding to BD is 2, so we will use BD=7 as the base of the triangle.

Finding A

Finally, we add the two areas to get the area of the pentagon. A = 15+7 ⇒ A = 22 The area of the pentagon is 22 square units.

We need to find the perpendicular distance between AB and CD. We know the area of the whole figure, but we do not know a formula to find it. Let's divide the figure into polygons for which we know how to find the area. For example, let's connect vertices A and D with a segment.

This segment divides the figure into two polygons, ABCD and DEFA. But there is more! Since AB∥DC and DE∥FA, these two polygons are both trapezoids!

Notice that x is the height of trapezoid ABCD. We know the lengths of the bases of this trapezoid. If we knew its area, we could use the formula for the area to find the height. A = 1/2h(b_1+b_2) A = ? h = ? b_1 = 7 b_2 = 4 We do not know the area of either of the trapezoids, but we are given the area of the whole figure — we know that the areas of the trapezoids add up to 17. \begin{gathered} A_\text{Total}=A_{ABCD}+A_{DEFA}\\ \Updownarrow\\ 17=A_{ABCD}+A_{DEFA} \end{gathered} If we find the area of trapezoid DEFA, we will be able to find the area of ABCD, and from there, the height of ABCD. Let's find the area of DEFA.

Finding A_(DEFA)

The bases of trapezoid DEFA are DE and FA and notice that EF is the height. Therefore, let's substitute the corresponding lengths to find the area of this trapezoid.

The area of trapezoid DEFA is 6 square units.

Finding A_(ABCD)

Let's substitute 17 for A_\text{Total} and 6 for A_(DEFA) into the equation for the area of the whole figure. Then, we will solve the equation for A_(ABCD).

The area of trapezoid ABCD is 11 square units.

Finding x

Now that we know the area of trapezoid ABCD, we can substitute it along with the lengths of the bases into the formula for the area and solve it for the height, x.

We found that the height of the trapezoid, x, is 2 units.

Property	Justification
The opposite sides are congruent	Parallelogram Opposite Sides Theorem
The opposite angles are congruent	Parallelogram Opposite Angles Theorem
The diagonals bisect each other	Parallelogram Diagonals Theorem

Property	Justification
The diagonals are congruent.	Isosceles Trapezoid Diagonals Theorem
Each pair of base angles is congruent.	Isosceles Trapezoid Base Angles Theorem

Length on the Whiteboard	Length on the Court
$d_{1} = 73$	$d_{1} = 273$
$d_{2} = 29$	$d_{2} = 229$

Length on the Whiteboard	Length on the Court
$b_{1} = 10.2$	$b_{1} = 20.4$
$b_{2} = 5.1$	$b_{2} = 10.2$

Tans Forming the Trapezoid	Dimensions	$A = \frac{1}{2} h (b_{1} + b_{2})$	Area $(cm^{2})$
$7$ and $3$	$b_{1} = 8$ $b_{2} = 4$ $h = 2$	$A_{7, 3} = \frac{1}{2} \cdot 2 (8 + 4)$	$12$
$1,$ $2,$ and $4$	$b_{1} = 12$ $b_{2} = 8$ $h = 4$	$A_{1, 2, 4} = \frac{1}{2} \cdot 4 (12 + 8)$	$40$
$5$ and $6$	$b_{1} = 28$ $b_{2} = 8$ $h = 8$	$A_{5, 6} = \frac{1}{2} \cdot 8 (28 + 8)$	$12$

	18 Theory slides
	8 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Areas of Parallelograms, Triangles, and Trapezoids

Catch-Up and Review

Proof

Proof for Right Triangles

Proof for Non-Right Triangles

Hint

Solution

Hint

Solution

Hint

Solution

Extra

Hint

Solution

Hint

Solution

Finding A_1

Finding A_2

Finding A

Finding A_(DEFA)

Finding A_(ABCD)

Finding x

Finding A_(△)

Finding A_r

Finding the Area of the Composite Figure

Areas of Parallelograms, Triangles, and Trapezoids

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