How Changes in Dimensions Affect Area & Perimeter

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Tweaking just one dimension in a figure can make an important difference in its perimeter and area. In this lesson, we'll look at rectangles, pentagons, and circles to explore how a change in one dimension can affect a figure's area and perimeter.

Defining Dimension, Area, and Perimeter

One of the first things we learn in math is how to calculate the area and perimeter of a figure. For example, we can find the area of a rectangle by multiplying its length times its width. We can find the perimeter of a circle by multiplying its diameter by π. So, what happens if we tweak one of the dimensions? How does that affect the calculation? In this lesson, we'll explore how the individual dimensions affect the overall perimeter and area, and how small changes at some points can make a huge difference in the results.

First of all, some definitions. A dimension is one of the measurable characteristics that determine the shape of the figure. It might be the length of one of the sides of a polygon (a figure with straight sides) or the radius of a circle. It might be the height of a triangle or of a three-dimensional figure, such as a cube or pyramid. Whether it's length, width, height, or something else, a dimension provides a good starting point for examining a figure's other properties.

Perimeter is a measure of the distance along the outside edge of the figure. We can find a a figure's perimeter by combining its outside dimensions. For example, the perimeter of a rectangle is found by doubling the length, doubling the width, and then adding the two together. You can find the perimeter of a regular octagon (8-sided figure with equal sides) by multiplying the length of one of the sides by 8.

The area of a figure is the measure of how large its surface is. Area is given in square units, such as square inches or square miles, and it's usually determined by calculations that involve multiplying the figure's dimensions. For example, the area of a circle is determined by multiplying its radius (distance from center to the outside edge) by the radius again and then by π (approximately 3.14). The area of a square is found by multiplying the length of one of its sides times that length again. The surface area of a cube is found by multiplying the area of one of its faces (a square) times 6. In each case, the result is a measure of the size of the figure's surface.

Effect on Perimeter When Dimensions Change

So, what happens when dimensions start changing? You can find out by looking at how the equation uses that dimension. For example, say you have an irregular pentagon (5-sided figure with straight sides that aren't equal length), and you want to know how the perimeter will change if you chop an inch off of one of the sides.

Well, the formula is simple: perimeter P = S1 + S2 + S3 + S4 + S5, where each of the S terms represents the length of one of the sides. If I make S1 an inch shorter, how will that affect P? A brief look at the formula tells us that P will also become an inch shorter.

What if I reduce the length of a rectangle by one inch? The formula for the perimeter of a rectangle is P = 2L + 2W. Notice that our length L term is being multiplied by 2, so the perimeter will lose two inches when we make L one inch shorter.

Okay, now let's go back to our pentagon, but let's now make it a regular pentagon, where all the sides are the same length L. The equation is now P = 5L, which means, if we nip an inch off of L, we're reducing the perimeter of that pentagon by five inches! Looking at the perimeter formula, you can see how a change in one dimension affects the perimeter of the figure.

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