Lesson 3: Using Formulas

New Concept

This lesson shows how to use algebraic formulas in real-world contexts. You'll apply formulas for volume and surface area, use the Pythagorean theorem to find unknown lengths, and practice rearranging equations to solve for a specific variable.

What’s next

Next, you'll work through interactive examples on volume and surface area, and then apply your skills to challenge problems using the Pythagorean theorem.

Property

• Volume is the amount of space contained within a three-dimensional object. It is measured in cubic units, such as cubic feet or cubic centimeters.

• Surface area is the sum of the areas of all the faces or surfaces that contain a solid. It is measured in square units.

Common Formulas:
Sphere: $V = \frac{4}{3}\pi r^3$, $S = 4\pi r^2$

Property

Taking a square root is the opposite of squaring a number. To solve an equation of the form $x^2 = k$ (where $k > 0$), we take the square root of both sides. Because a positive number has two square roots, the solution is written as:

Examples

• To solve the equation $x^2 = 81$, we take the square root of both sides. The solutions are $x = \pm\sqrt{81}$, which means $x = 9$ and $x = -9$.

Property

A right triangle contains a 90° angle. The side opposite the right angle is the hypotenuse ($c$), and the other two sides are the legs ($a$ and $b$).

The Pythagorean Theorem states that for any right triangle:

Property

To solve a formula for one variable, treat it as the unknown and all other variables as constants. Isolate the desired variable by applying inverse operations to both sides of the equation. Remember to follow the order of operations in reverse and do not combine unlike terms.

Examples

• To solve the perimeter formula $P = 2l + 2w$ for $l$, first subtract $2w$ from both sides: $P - 2w = 2l$. Then, divide by 2: $l = \frac{P - 2w}{2}$.

• To solve the interest formula $A = P + Prt$ for $r$, first subtract $P$: $A - P = Prt$. Then, divide by $Pt$ to isolate $r$: $r = \frac{A - P}{Pt}$.