Exploring the Area of a Parallelogram, Triangle, Trapezoid, and Rhombus

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 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

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

d_{1} = 73

d_{1} = 273

d_{2} = 29

d_{2} = 229

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

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 $6$	$b_{1} = 28$ $b_{2} = 8$ $h = 8$	$A_{5, 6} = \frac{1}{2} \cdot 8 (28 + 8)$	$12$

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

What is the area of the letter A?

There is no formula that directly gives us the area of this letter A, but we can try breaking it down into some parts that we can find the area of. For example, this letter A can be made by cutting a triangle and small trapezoid out from a large trapezoid.

The area of the letter is the area of the large trapezoid minus the areas of the cut pieces. Unfortunately, we do not know the heights of the small pieces, so we cannot find their areas. However, we can also form the letter A by cutting a large triangle out of the original large trapezoid, then adding back in the middle trapezoid.

We do have all the information to find the area of each of these pieces. Let's calculate the area of the letter A by finding the area of the large trapezoid minus the area of the triangle plus the area of the middle trapezoid. A_(Letter A) = A_1 - A_2 + A_3 Let's find each area one by one.

Finding the Area of the Large Trapezoid A1

The large trapezoid has bases that are 80 and 10 centimeters long and is 61 centimeters high.

The area of a trapezoid is half the height times the sum of the lengths of the bases. Let's substitute these values into the formula and solve for the area of the large trapezoid A_1.

The area of the large trapezoid is 2745 square centimeters.

Finding the Area of the Triangle A_2

The triangle cutout has a base that is 20 centimeters long and is 51 centimeters high.

The area of a triangle is half the product of the base and height. Let's find the area of the triangle A_2!

The area of the triangle cutout is 510 square centimeters.

Finding the Area of the Middle Trapezoid A_3

The middle trapezoid has bases that are 15 and 8 centimeters long and is 20 centimeters high.

Let's substitute these values into the formula for the area of a trapezoid and simplify to find the area of the middle trapezoid A_3.

The area of the middle trapezoid is 230 square centimeters.

Finding the Area of the Letter A

Finally, we can substitute the areas we found into the equation we wrote for the area of the letter A. A_(Letter A) = A_1 - A_2 + A_3 Now let's find the total area of the letter. A_(Letter A)= 2745 - 510 + 230 ⇓ A_(Letter A) = 2465 The area of the letter A is 2465 square centimeters.

	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 the Area of the Large Trapezoid A1

Finding the Area of the Triangle A_2

Finding the Area of the Middle Trapezoid A_3

Finding the Area of the Letter A

Areas of Parallelograms, Triangles, and Trapezoids

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