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

Name: Problem 30, Chapter 9: Geometry
Uploaded: 2020-06-09T04:01:01-08:00
Duration: 6 min 21 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

Rounding

Rounding is the process of adjusting a numerical result to a specified degree of precision. In many problems, including those involving geometric calculations, results are often rounded to a certain number of decimal places to provide a practical, approximate answer, such as rounding to the nearest tenth.

Regular Polygon

A regular polygon is a geometric figure with all sides of equal length and all interior angles equal. Understanding its properties is fundamental because these equal properties allow for uniform formulas in calculating characteristics like the area and perimeter.

Area Calculation

The area of a regular polygon can be found using the formula that involves the apothem and the perimeter. This formula, A = 1/2 × (apothem) × (perimeter), is critical since it leverages the symmetry of a regular polygon to simplify the computation of its area.

Apothem

The apothem is the line segment from the center of a regular polygon perpendicular to one of its sides. It is essential in determining the area because it acts as the height for the triangular segments created when the polygon is divided, enabling the area calculation via the formula.

Perimeter

The perimeter of a polygon is the sum of the lengths of all its sides. In the context of regular polygons, since all sides are equal, the perimeter is calculated by multiplying the number of sides by the length of one side. This concept is integral when using the area formula that involves the perimeter.

Trigonometric Relationships

Trigonometric relationships are often employed to relate the side lengths and angles in a regular polygon, especially when the apothem is not directly given. By using trigonometric ratios and the central angles, one can calculate the necessary dimensions to apply in the area formula.

Transcript

00:01 Okay, so for this problem, we are shown an octagon, which we can identify because the polygon we're given has eight sides.

00:10 And we're shown that the apathem of this polygon is equal to two centimeters.

00:17 And using this information, we're asked to find the area of this polygon.

00:22 So our first step in finding the area will be to find the central angle of this polygon.

00:33 Which will be equal to 360 degrees, divide by n, which is the number of sides in the polygon.

00:42 So 360 degrees divided by 8, which is equal to 45 degrees.

00:59 Okay, and next we're going to visualize what a triangle with the central angle would look like.

01:07 So say that this is our symbol.

01:10 Point in that these two points are two rivets that are adjacent to each other.

01:18 If we drew a line from one vertex to the other, we would have side length.

01:24 And if we drew a line from each vertex to the center point, you can see that we formed triangle in which this angle right here is equal to 45 degrees central angle.

01:43 Now, because the information we're given is about the apathem.

01:47 We're going to want to draw an apathem down from this center point to the midpoint of the side length right here.

01:58 It creates the 90 degree angle with that side to form two bright triangles.

02:08 And what the apathom does is it also bisects the central angle.

02:13 So i'll draw one of these right triangles bigger.

02:23 And you can see that we have this new angle right here, which i'll call theta.

02:32 And because the absent bisects the central angle, theta is going to be equal to one half of the central angle, which i abbreviated central angle so theta is equal to 45 divide by 2, which is 22 .5 degrees.

03:05 And then we can also see from the diagram above here that after the apathemus dropped, the side length is divided in 2...

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