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

Name: Problem 26, Chapter 9: Geometry
Uploaded: 2020-06-09T03:18:06-08:00
Duration: 7 min 3 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 and interior angles equal. This consistent symmetry allows for the derivation of specialized formulas to calculate properties such as area and perimeter using its equal side lengths and angles.

Area Calculation Formula for Regular Polygons

The area of a regular polygon can typically be found using the formula A = (1/2) × apothem × perimeter, where the apothem is the perpendicular distance from the center to a side. This formula leverages the uniform structure of the polygon by effectively decomposing it into congruent isosceles triangles.

Apothem

The apothem of a regular polygon is the distance from its center to the midpoint of one of its sides, and it is a key component in calculating the area. It plays a crucial role in breaking down the polygon into triangles, making the area calculation more straightforward through the relation of the apothem to the perimeter.

Rounding and Approximation

Rounding numerical results, such as rounding to the nearest tenth, is an essential practice in applied mathematics to present an answer with a desired precision. This process involves approximating a computed value to a specified decimal place while maintaining its accuracy for practical use.

Transcript

00:01 Okay, so for this problem, we are given a hexagon with six sides, and we're told that the apathem of this hexagon, or we can see in the diagram, that the apathem is equal to 14 inches, and we're told to find the area of this hexagon to given that information.

00:24 So we know that the area of any polygon, i'll write out this equation, the area of any polygon is equal to one half, the apathem times the perimeter.

00:40 And we're already given the apathem, and so our job is to find the perimeter and then plug all this back into that area equation.

00:49 So how do we find the perimeter? we know that the perimeter is equal to the number of size.

00:55 Times the length of each side.

00:58 So our goal to solving this equation is to first find the central angle and use that information with some trig to find this side length value and then the rest will come from there.

01:15 So let's start off by finding this the central angle.

01:19 Central angle of a hexagon is equal to 360 degrees divided by by n, which is 6.

01:29 Again, the formula for any polygon to find the central angle is 360 degrees divided by n.

01:37 And n is 6 because it's a hexagon.

01:40 So this is equal to 60 degrees.

01:45 That's our central angle.

01:47 So let's visualize this.

01:49 Say we have, say this point right here is the center point of the hexagon.

01:54 And then we have two adjacent vertices.

01:59 Right here and right here.

02:01 If we draw a line from these adjacent vertices to each other, from one vertex to the other, we get one side length of this hexagon.

02:14 And then if we draw a line from each vertex to the center point, we form this triangle in which this angle right here is equal to the central angle.

02:28 And what we can do is if we drop an apothex, down from the center point straight down to the midpoint of the side, which it forms a 90 degree angle...

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