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Learning Targets
I can calculate the area of a complicated shape by breaking it into shapes whose area I know how to calculate.
Notes
Notes
Let’s estimate the areas of weird shapes.
We can find the area of some complex polygons by surrounding them with a simple polygon like a rectangle. For example, this octagon is contained in a rectangle.
The rectangle is 20 units long and 16 units wide, so its area is 320 square units. To get the area of the octagon, we need to subtract the areas of the four right triangles in the corners. These triangles are each 8 units long and 5 units wide, so they each have an area of 20 square units. The area of the octagon is
320 − (4 ⋅ 20) or 240 square units.
We can estimate the area of irregular shapes by approximating them with a polygon and finding the area of the polygon. For example, here is a satellite picture of Lake Tahoe with some one-dimensional measurements around the lake.
The area of the rectangle is 160 square miles, and the area of the triangle is 17.5 square miles for a total of 177.5 square miles. We recognize that this is an approximation, and not likely the exact area of the lake.
Remember...
Area of a Rectangle = b ⋅ h or A = l ⋅ w
Area or a Triangle = (b ⋅ h ) / 2 or (1/2) (b ⋅ h)
*Notice that the formula for rectangles and triangles can be written in two different ways. Which one you use is personal preference.*
Activities
Activities
6.2 House Floorplan
6.2 House Floorplan
Here is a floor plan of a house. Approximate lengths of the walls are given.
What is the approximate area of the home, including the balcony? Explain or show your reasoning.
Notice you can:
decomposing the floorplan into various rectangles and triangles OR
composing the floorplan with other shapes to create a large rectangle
6.3 Area of Nevada
6.3 Area of Nevada
Estimate the area of Nevada in square miles. Explain or show your reasoning.
Notice:
The shape of the state looks like a rectangle with a corner cut off and a bite taken out.
The shape of the state could be decomposed into a rectangle and a triangle.
Add to Your Notes
Add to Your Notes
You can estimated areas of both large and small things in the world by approximating them with polygons. We can find the area of any polygon by decomposing it into triangles and rectangles and using formulas we know to find the area. In practice, it is important to be strategic when composing and decomposing, taking advantage of measurements that are known and avoiding measurements that are unknown or difficult to calculate.
"What things are important to think about when asked to find the area of a figure?"
"What things do we know help us find area of any figure?"
It is important to consider the shape of the region, how polygons are helpful, and the ways polygons can be decomposed, rearranged or enclosed to find the area of the region.
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