অটল সমীকরণের মাধ্যমে জ্যামিতির সূত্র

Literal equations appear everywhere in geometry. Equations describe relationships between different geometric objects, such as angles, volumes, and side lengths. We call them literal equations because they are built by taking a phrase written in words and turn it into an equation, written with variables instead of words. Formally, a literal equation is an equation containing more than one variable.

As a result of the connection between geometry and literal equations, I want to devote a bit of attention towards building literal equations used to describe geometric objects, and then using those equations, along with some algebra, to solve problems.

We're going to walk through a two-part example of solving literal equations. It's time to bundle up and make some snowballs!

Jenni and Mike are having a snowball fight. They start by assembling blocks of snow in the shapes of rectangular prisms, from which they will create snowballs. Jenni and Mike each make a block of snow with width $10$ centimeters, length $12$ centimeters, and height $2$ centimeters. What is the volume of one of the blocks of snow?

This involves constructing a literal equation, and the first step in doing that is defining some variables. Since "width" starts with a "w" we let $w$ define the width. In general it's a good rule of thumb to use the first letter of the word describing the geometric object as a variable name (unless this would cause you to repeat letters, in which case just pick something else). Using this convention we let $h$ be the "height," $\ell$ be the "length" (don't use $l,$ because it looks like a $1$), and $V$ be the "volume."

One of our formulas from geometry is that the volume of a rectangular prism equals the product of the length, width, and height. To build this equation, we use the "equals" sign to indicate how to divide the written expression into the two sides of the equation.

To reiterate, "the volume of a rectangular prism equals the product of the length, width, and height." The word "volume" appears to the left of the word "equals," so we start building our equation like

$$V = ?$$

The other side is "the product of the length, width, and height," so we multiply these variables together and that becomes our right-hand side.

$$V = \ell w h.$$

Ta-da! This is our literal equation, so now we switch gears to doing algebra. Remember, we are given $\ell,$ $w,$ and $h,$ and want to find the volume, $V.$ Our goal would be to solve for $V$ by adding, subtracting, multiplying, and dividing things from both sides of the equation, but there's no need, because $V$ is already isolated on one side of the equation. That means we can go ahead and start plugging in values. We know that $\ell = 12 \ \text{cm},$ $w = 10 \ \text{cm},$ and $h = 2 \ \text{cm}.$ Thus to find $V$ we can multiply those numbers together:

$$V = 12 \ \text{cm} \ \times 10 \ \text{cm} \ \times 2 \ \text{cm} = 240 \ \text{cm}^3.$$

Let's try another problem: if Jenni and Mike each use their blocks of snow to make snowballs with volume $8$ cubic centimeters, how many snowballs are made in total?

There are three quantities we care about: the total volume of snow, $S;$ the number of snowballs, $n;$ and the volume of each snowball, $w.$ But we calculated the volume earlier and called it $V,$ so why are we calling it $S$ now?

We use $S$ as the variable name because the volume we're dealing with now is not the volume we calculated earlier. The volume we calculated earlier was the volume of one block of snow, but we actually now care about the volume of both blocks of snow. In other words, we can add the volume of one block of snow to itself to get the total volume of snow:

$$S = V + V.$$

And we take the value of $V$ we found earlier: $240 \ \text{cm}^3,$ and plug it in:

$$S = 240 \ \text{cm}^3 + 240 \ \text{cm}^3 = 480 \ \text{cm}^3.$$

Now let's remind ourselves of what the other variables are. $n$ is the number of snowballs made, which is our "target" quantity to find, and $w$ is the volume of each snowball made. We're told that $w = 8 \ \text{cm}^3,$ so now we construct another literal equation to find $n.$

The key is that all of the snowballs made have the same volume, so the total volume of all the snowballs equals the number of snowballs times the volume of any single snowball. Now we turn this into a literal equation.

As before, split the description of the equation into two sides based on where the word "equals" appears. One side of the equation is "the total volume of all the snowballs," which we already said equals $S,$ so beginning to build the equation looks like

$$S = ?$$

What about the other side? We're told that the other side of the equation is described by "the number of snowballs times the volume of any single snowball." This is just the product of $n$ and $w,$ so our equation is

$$S = nw.$$

Now we have a complete literal equation, and have to solve it for $n.$ Although you can plug in the numbers and then solve, it's easier to solve for $n$ in terms of the other variables, and then plug in its values, so we'll do that. To isolate the $n,$ we divide both sides of the equation by $w.$

$$\frac{S}{w} = \frac{nw}{w}.$$

The $w$s on the right-hand side cancel out, so

$$\frac{S}{w} = n.$$

This is the equation we want, so we just need to plug in $S$ and $w.$ We know that $S = 480 \ \text{cm}^3,$ and $w = 8 \ \text{cm}^3,$ so substitute these in and divide:

$$\frac{480 \ \text{cm}^3}{8 \ \text{cm}^3} = n.$$

The units cancel, and $\frac{480}{8} = 60,$ so $n = 60.$ This means Jenni and Mike create a total of $60$ snowballs. They must have a long snowball fight coming up.

One of the most famous literal equations is the Pythagorean theorem:

In words, this states that in a right triangle, the sum of the squares of the lengths of the legs equals the square of the hypotenuse. This is a literal equation because it has three different variables: $a, b,$ and $c.$

Once we have a literal equation given to us, we can solve for one of the variables. At that point, we can ignore what geometric magic gave us the equation in the first place and just do algebra. For instance, if we want to solve for $c$ in terms of $a$ and $b,$ we can take the square root of both sides of $a^2 + b^2 = c^2$ and get

$$\sqrt{a^2 + b^2} = c.$$

Geometric formulas are used to help figure out things about the geometry of shapes and objects.

When we say "geometry," we mean things like length, width, radius, circumference, volume, surface area, etc.

Geometric formulas are most often written as literal equations—equations with more than one variable. For instance, if we were to write down the formula for area, we'd have three variables: length, width, and area. Given enough information about two of the variables, we can easily solve for the value of the third one.

Check out all those variables. This is definitely a literal equation.

Let's consider a problem about perimeter.

Janae is measuring the perimeter of her garden so she knows how much fence she needs to buy. Her garden is rectangular, and Janae finds that the length of both sides is $6.4$ meters and the width is $3.7$ meters.

Recall that the formula for perimeter of a rectangle is a literal equation—it has more than one variable. In fact, there are three variables in that equation. Think about it for a minute, then try to answer the question.

Janae's garden is $6.4$ meters long and $3.7$ meters wide, so we can figure out that $l = 6.4$ and $w = 3.7$. We can plug them into our equation and find:

$$\begin{align} P &= 2l + 2w\\ P &= 2(6.4) + 2(3.7)\\ P &= 12.8 + 7.4\\ P &= 20.2 \end{align}$$

Done!

Now, let's consider the formula for the area of a circle. Do you remember that formula? There are two variables in it.

A= area, r=radius

Note that π is not a variable. It has a fixed, known value that doesn't change ($3.1415$...). The two variables whose values are completely unknown and can vary by circle are area and radius.

Try out this problem:

Note: Normally when you take the square root of a term in order to "cancel" out a squared term on the other side of the equation, you need to consider both the positive and negative values of the answer (e.g., $+4.97$ and $-4.97$).

However, when you're working with measures like radius, the value cannot be negative. (You'd basically be in Narnia if you measured the radius of a circle and found it to be $-4.97\text{ cm}$. There aren't any rulers that can measure negative distances). So, we just stuck with the positive value here.

(Video) Literal Equations - Area Formula

by Brian McLogan

This video by Brian McLogan explains how to solve for $b$ in the area of a triangle formula.

This video uses the area formula to explain literal equations.

The formula for area of a triangle is:

$$A=(½)bh$$

We want to solve for $b$, so we want to get $b$ on its own.

If we divide both sides by $h$,

$$A/h = (½)b$$

Then, we want to multiple by the reciprocal, $2$,

$$2A/h=b$$

Lastly, let's rewrite with $b$ first,

$$b=2A/h$$