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Area of Polygons (Using Decomposition)
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 We'll learn to find the area of polygons by decomposing shapes.
 We will also learn how to find the area of polygons displayed on a coordinate plane.
 And we will see how this knowledge helps people create video games, build houses and even skateboard ramps!

Discussion Questions

Before VideoWhat is area?ANSWER

Area is a measure of the space inside a 2D object.

I can measure the sides of the rectangle, and then find the area using the formula length times width.

I can break the shape into two rectangles. I can find the area of each rectangle and add the areas together to find the area of the bigger shape.

I can draw lines to break a rectangle into other rectangles, squares, triangles, and other polygons.

The total area of all of the shapes is equal to the area of the original rectangle.


After VideoHow is the area of a right triangle related to the area of a rectangle?ANSWER

The area of a right triangle is half the area of a rectangle with the same base and height.

I can break a parallelogram into two identical right triangles and a rectangle. When I combine the two triangles it forms a new rectangle. The area of this set of shapes is the same as the area of a rectangle with a length and width the same as the length and height of the parallelogram.

I can divide a polygon into rectangles and right triangles. Then I can find the area of each of the smaller shapes and add them together to find the total area.

First I plot each point. The coordinates tell me the position of the point on the xaxis and the yaxis. After I plot each point, I connect the points to make a polygon. I need to connect the points with straight lines and make a closed shape.

I can find the area in the same way, by dividing the shape into rectangles and triangles and finding the total area of those shapes.



Vocabulary

Area
DEFINE
The measure of the space enclosed by the sides of a shape.

Decompose (a shape)
DEFINE
Breaking a shape into smaller shapes.

Polygon
DEFINE
A closed shape with straight sides.

Rectangle
DEFINE
A closed shape with 4 straight sides with 4 right angles.

Triangle
DEFINE
A closed shape with 3 straight sides.

Right triangle
DEFINE
A triangle where one angle measures 90 degrees.

Parallelogram
DEFINE
A closed shape with 4 straight sides. Two pairs of sides are parallel.

Kite
DEFINE
A closed shape with 4 straight sides. Two pairs of adjacent (next to each other) sides have the same length.

Trapezoid
DEFINE
A closed shape with 4 straight sides. At least one pair of sides are parallel.

Area
DEFINE

Reading Material
Download as PDF Download PDF View as Separate PageWHAT IS THE AREA OF POLYGONS?A polygon is a closed shape with straight sides. You can use the area of a rectangle to find the area of other polygons. Decompose (break apart) polygons into other shapes to find the area.
To better understand the area of polygons…
WHAT IS THE AREA OF POLYGONS?. A polygon is a closed shape with straight sides. You can use the area of a rectangle to find the area of other polygons. Decompose (break apart) polygons into other shapes to find the area. To better understand the area of polygons…LET’S BREAK IT DOWN!
Review the area of a rectangle.
Area is the space inside a 2D shape. We measure area in square units. You know that the area of a rectangle is its length multiplied by its width. For example, a rectangle that is 6 cm wide and 5 cm long has area 5 × 6 = 30 square centimeters. You can use the area of a rectangle to find the area of other polygons.
Review the area of a rectangle. Area is the space inside a 2D shape. We measure area in square units. You know that the area of a rectangle is its length multiplied by its width. For example, a rectangle that is 6 cm wide and 5 cm long has area 5 × 6 = 30 square centimeters. You can use the area of a rectangle to find the area of other polygons.Find the area of a triangle on a skateboard ramp.
The side of a skateboard ramp is in the shape of a right triangle. A right triangle has one angle that is 90 degrees. You can make a right triangle by taking a rectangle and dividing it along the diagonal. One rectangle makes 2 right triangles with equal area. A 10 cm by 5 cm rectangle has area 50 cm2. Divide the rectangle along to diagonal to make 2 right triangles that have base and height 10 cm and 5 cm. The area is half of 50, or 25 cm2. The area of a triangle is [ggfrac]1/2[/ggfrac] × base × height. The triangle of the skateboard ramp has base 100 cm and height 30 cm. The area is [ggfrac]1/2[/ggfrac]×100×30 = 1,500 cm2.
Find the area of a triangle on a skateboard ramp. The side of a skateboard ramp is in the shape of a right triangle. A right triangle has one angle that is 90 degrees. You can make a right triangle by taking a rectangle and dividing it along the diagonal. One rectangle makes 2 right triangles with equal area. A 10 cm by 5 cm rectangle has area 50 cm2. Divide the rectangle along to diagonal to make 2 right triangles that have base and height 10 cm and 5 cm. The area is half of 50, or 25 cm2. The area of a triangle is [ggfrac]1/2[/ggfrac] × base × height. The triangle of the skateboard ramp has base 100 cm and height 30 cm. The area is [ggfrac]1/2[/ggfrac]×100×30 = 1,500 cm2.Find the area of polygons in blueprints.
A blueprint is a 2D plan that shows the walls of a building. A blueprint you have has a room in the shape of a trapezoid and a room in the shape of a parallelogram. You need to find the area of these rooms to estimate how much flooring to buy. You can decompose polygons into triangles and rectangles to find their area. The trapezoid shape can be divided into one rectangle and one right triangle. The rectangle is 6 meters by 3 meters, so the area is 6 × 3 = 18 m2. The triangle has base 4 meters and height 3 meters, so the area is [ggfrac]1/2[/ggfrac]×4×3 = 6 m2. The total area is 18 + 6 = 24 m2. The parallelogram can be divided into one rectangle and two right triangles. The rectangle has area 5 × 3 = 15 m2. Each triangle has area [ggfrac]1/2[/ggfrac]×3×2 = 3 m2. So, the total area is 15 + 3 + 3 = 21 m2.
Find the area of polygons in blueprints. A blueprint is a 2D plan that shows the walls of a building. A blueprint you have has a room in the shape of a trapezoid and a room in the shape of a parallelogram. You need to find the area of these rooms to estimate how much flooring to buy. You can decompose polygons into triangles and rectangles to find their area. The trapezoid shape can be divided into one rectangle and one right triangle. The rectangle is 6 meters by 3 meters, so the area is 6 × 3 = 18 m2. The triangle has base 4 meters and height 3 meters, so the area is [ggfrac]1/2[/ggfrac]×4×3 = 6 m2. The total area is 18 + 6 = 24 m2. The parallelogram can be divided into one rectangle and two right triangles. The rectangle has area 5 × 3 = 15 m2. Each triangle has area [ggfrac]1/2[/ggfrac]×3×2 = 3 m2. So, the total area is 15 + 3 + 3 = 21 m2.Draw polygons on a coordinate plane.
You can plot points on a coordinate plane. A coordinate plane has two axes: the xaxis and the yaxis. A coordinate pair tells you where to plot a point on each axis. The point (8, 5) has xcoordinate 8 and ycoordinate 5. So, you should plot the point 8 units right on the xaxis and 5 units up the yaxis. You can plot points to help you design a video game. To draw a hexagon, you need to plot each point of the shape. To plot the point (5, 2), for example, go right on the xaxis 5 units then go up 2 units. After plotting all of the points, you can draw lines to connect the dots and make the shape.
Draw polygons on a coordinate plane. You can plot points on a coordinate plane. A coordinate plane has two axes: the xaxis and the yaxis. A coordinate pair tells you where to plot a point on each axis. The point (8, 5) has xcoordinate 8 and ycoordinate 5. So, you should plot the point 8 units right on the xaxis and 5 units up the yaxis. You can plot points to help you design a video game. To draw a hexagon, you need to plot each point of the shape. To plot the point (5, 2), for example, go right on the xaxis 5 units then go up 2 units. After plotting all of the points, you can draw lines to connect the dots and make the shape.Find the area of shapes on a coordinate plane.
You can find the area of shapes you plot on a coordinate plane. You have a hexagon plotted on the coordinate plane. It has 6 straight sides, and you can divide this hexagon into one rectangle and four right triangles. The area of the rectangle is 5 × 2 = 8 square units. All four triangles have the same area: [ggfrac]1/2[/ggfrac]×2×2 = 2. The total area is 8 + 2 + 2 + 2 + 2 = 16 square units.
Find the area of shapes on a coordinate plane. You can find the area of shapes you plot on a coordinate plane. You have a hexagon plotted on the coordinate plane. It has 6 straight sides, and you can divide this hexagon into one rectangle and four right triangles. The area of the rectangle is 5 × 2 = 8 square units. All four triangles have the same area: [ggfrac]1/2[/ggfrac]×2×2 = 2. The total area is 8 + 2 + 2 + 2 + 2 = 16 square units.There are many careers that use area.
Wildlife ecologists are scientists who study animals and how they interact in the wild. It is important to know how much area animals have, to ensure they have what they need. Aerospace engineers calculate the area of designs they make, like the shapes needed to build an airplane. Mural artists use area to plan how to construct their paintings.
There are many careers that use area. Wildlife ecologists are scientists who study animals and how they interact in the wild. It is important to know how much area animals have, to ensure they have what they need. Aerospace engineers calculate the area of designs they make, like the shapes needed to build an airplane. Mural artists use area to plan how to construct their paintings. 
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Science & Math$_{/yr}

Science Only$_{/yr}
A nonright triangle has base length 6 cm and height 7 cm. What is the area?
A parallelogram has length 10 m and height 4 m. What is the area?
A trapezoid has height 10 cm, bottom length 8 cm and top length 4 cm. What is the area of the shape?
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