Find the area of each regular polygon. Round to the nearest tenth. FIGURE CAN'T COPY.

Name: Problem 6, Chapter 9: Geometry
Uploaded: 2020-05-29T19:19:17-08:00
Duration: 5 min 44 s
Channel: Natalie Daly
Description: Find the area of each regular polygon. Round to the nearest tenth. FIGURE CAN'T COPY.

Find the area of each regular polygon. Round to the nearest...

Key Concepts

Regular Polygon

A regular polygon is a geometric figure with all sides of equal length and all interior angles congruent. This uniformity allows for the derivation of specific formulas to compute attributes like area and perimeter, based on the symmetry inherent in the shape.

Polygon Area Formula

The area of a regular polygon can be calculated using the formula: Area = 1/2 × apothem × perimeter. This formula leverages the consistent structure of a regular polygon where the apothem (a line from the center perpendicular to a side) and the perimeter (the sum of all side lengths) are key measurements.

Apothem

The apothem of a regular polygon is the perpendicular distance from the center of the polygon to one of its sides. It is crucial in area calculations, as it directly factors into the formula for the area, linking the internal symmetry of the polygon with its external measurements.

Perimeter

The perimeter of a regular polygon is the total length around the figure, calculated by multiplying the length of one side by the number of sides. This total boundary length is essential for determining the area when used in conjunction with the apothem.

Rounding

Rounding is a mathematical process used to reduce the number of digits in a number while keeping its value close to what it was originally. In many calculations, such as finding the area of a polygon, rounding to a specified precision, like the nearest tenth, ensures clarity and practicality in the result.

Transcript

00:01 Okay, so for this problem, we are given a diagram of a polygon, and that one of the lines in the polygon is drawn on the polygon, as seen here, is 10 inches long, and we're supposed to use this information to figure out the area of this polygon, which we can identify as a hexagon because it has six sides.

00:30 So as you learned in the chapter, there's this helpful formula that we can use to solve this problem, which is area is of any polygon.

00:40 So the area of this hexagon will be equal to one half of the epithem times the perimeter.

00:49 So first we're going to find the area, sorry, first we're going to find the length of the ophthalm, and then we're going to find the length of the perimeter.

00:59 So let's turn attention back to this diagram.

01:04 One useful property of a hexagon is that if you draw just another line from vertex to the center point, you can create this equilateral triangle, and you can drop an apothem between these two lines right down the middle to cut this equilateral triangle and it makes this 90 degree angle with the side length and here you can see that this part right here will be equal to one half the length of a side.

01:44 So what i'm going to do is i'm going to take that half of the triangle.

01:49 I'm going to blow it up so you can see it larger.

01:55 So what we did was we took half that equilateral triangle.

01:59 You can see that this is still 10 inches.

02:01 But now we know from our properties of equilateral triangles that this angle right here is 60 degrees.

02:09 This is 90.

02:11 And because we bisected the angle, this is now 30 degrees.

02:18 So we have a 60, 30, 90 degree triangle.

02:24 We know that this is the one -half length of a side.

02:31 This is the aft.

02:31 Them.

02:32 I'm just copying down what we discovered in that earlier diagram.