What would be the total area of all the triangles of Exercise 32, disregarding the overlaps?

Name: Problem 33, Chapter 12: Calculus with Applications
Uploaded: 2020-02-10T03:21:13-08:00
Duration: 5 min 31 s
Channel: John Robb
Description: What would be the total area of all the triangles of Exercise 32, disregarding the overlaps?

step 1 And hello calculus student. We are looking at chapter 12, section 4, problem number 33. We're to

What would be the total area of all the triangles of...

Key Concepts

Triangle Area Calculation

This concept involves determining the area of a triangle using fundamental geometric formulas. Typically, the area is calculated using the formula 1/2 × base × height, though other methods like Heron’s formula can also be employed when appropriate measurements are available. This calculation is central to understanding how to quantify the space enclosed by a triangle.

Additivity of Areas

Additivity of areas refers to the principle that the total area of a composite shape can be found by adding the areas of its individual components, as long as those components do not overlap. Even in cases where overlaps exist, if the problem specifies to disregard (or ignore) these overlapping sections, the total area is simply the sum of each component’s area, which is essential for solving such problems.

Handling Overlapping Regions

In geometry, overlapping regions occur when two or more shapes share common interior space. Normally, care is taken to ensure that the overlapping area is not counted more than once when calculating the area of the union. However, in scenarios where overlaps are to be disregarded, each figure’s area is treated independently and added together, simplifying the calculation by not subtracting the shared areas.

Transcript

00:01 And hello calculus student.

00:03 We are looking at chapter 12, section 4, problem number 33.

00:09 We're to find a sum of areas formed from equilateral triangles where the first side of each of the triangle has a side of two.

00:21 And so we'll go like this for our first.

00:25 And then each side is formed by connecting the midpoints of the previous.

00:32 So i can form a second triangle by finding the midpoints of my first equilateral triangle.

00:41 And not great straight lines, but we can see they would have a length of one.

00:48 And if we did the same thing again, we're going to have another equilateral triangle with a length of 0 .5.

00:58 So we could write that out.

01:01 We could say this.

01:02 So the sides of our first equilateral triangle is going to be two.

01:12 Our second equal lateral triangle, each side will be a length of one.

01:18 Our third, each side will have a length of one half, etc.

01:30 And so we want to remember that the area of an equal lateral.

01:36 Triangle.

01:38 So area of our equilateral triangles in general, equal lateral triangle.

01:51 We know from geometry is the square root of three over four times the length of a side squared.

02:05 Therefore, we can say the sum of our areas, write it like that, is going to equal so our first triangle is going to be square root of three over four times the side squared, so times two squared.

02:26 Then our second is going to be the square root of three over four times one squared...

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