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

Name: Problem 7, Chapter 9: Geometry
Uploaded: 2020-06-08T17:14:15-08:00
Duration: 6 min 57 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 uniformity simplifies many calculations, such as finding the area or perimeter, because formulas can be applied consistently to each of its congruent parts.

Area Calculation for Regular Polygons

The area of a regular polygon can be calculated by decomposing the polygon into a set of congruent isosceles triangles. By computing the area of one triangle—using the apothem as the height—and then multiplying by the total number of sides, the overall area can be determined efficiently with the formula A = 1/2 × perimeter × apothem.

Apothem

The apothem is the perpendicular distance from the center of a regular polygon to any of its sides. It serves as the height of the isosceles triangles formed within the polygon and is essential in calculating the area, as it directly influences the size of the triangles.

Rounding

Rounding is the process of adjusting a numerical result to a specified level of precision, such as to the nearest tenth, which is often necessary in practical applications to provide a clear, concise, and usable answer without excessive decimal detail.

Transcript

00:01 Okay, so for this problem, we're given a polygon that we can see has seven sides.

00:06 So i'm going to make a note that number of sides is equal to seven.

00:12 Now on the diagram that they give, you can see that each side is three centimeters long.

00:19 And we're told using this information to find the area of polygon.

00:25 So our first step is going to be to find the central angle of this polygon, which, sorry, i'll write i'll be as ca.

00:35 So c .a.

00:39 Central angle is going to be equal to 360 degrees divided by 7.

00:47 So you get approximately, i'll approximate it, to the 10th.

00:55 So 51 .4 degrees is your central angle.

01:03 Now, if we look at, for instance, if we have the center point of the polygon, and two vertices that are next to each other.

01:15 And we draw a line between them.

01:17 This is the length of one side.

01:19 So we know that this is 3 centimeters.

01:22 And if we connect the center point to each vertices, you can see we create a central angle.

01:31 So we know that this is equal to about 51 .4 degrees.

01:40 But what happens is if we drop down an apatham from the center point to the midpoint of the side, we bisect that angle.

01:51 So we have two new triangles, each are right triangles, that now have an angle towards the top that is equal to 560 .4 divided by 2 because it's bisected.

02:06 So this is approximately equal to 25 .7 degrees.

02:17 And what i'm going to do is i'm just going to redraw this one of these right triangles bigger.

02:24 So you can see better.

02:34 Okay.

02:36 So we know that this line right here, this one that i'm redrawing, is the apatham...

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