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Formula

Formula

An expression used to calculate a desired result, such as a formula to find volume or a formula to count combinations. Formulas can also be equations involving numbers and/or variables, such as Euler's formula.

 

 

See also

Explicit formula, recursive formula

Key Formula

A=πr2A = \pi r^2
Where:
  • AA = The area of the circle (the quantity you are solving for)
  • π\pi = The constant pi, approximately 3.14159
  • rr = The radius of the circle

Worked Example

Problem: Use the formula for the area of a circle to find the area when the radius is 5 cm.
Step 1: Write down the formula for the area of a circle.
A=πr2A = \pi r^2
Step 2: Substitute the known value of the radius into the formula.
A=π(5)2A = \pi (5)^2
Step 3: Evaluate the exponent first.
A=π⋅25A = \pi \cdot 25
Step 4: Multiply by pi to get the final result.
A=25π≈78.54 cm2A = 25\pi \approx 78.54 \text{ cm}^2
Answer: The area of the circle is 25π≈78.5425\pi \approx 78.54 cm².

Another Example

This example uses an algebraic formula (the quadratic formula) rather than a geometric one, showing that formulas apply across many branches of math and can yield multiple answers.

Problem: Use the quadratic formula to solve the equation 2x2+3x−5=02x^2 + 3x - 5 = 0.
Step 1: Identify the coefficients from the standard form ax2+bx+c=0ax^2 + bx + c = 0. Here a=2a = 2, b=3b = 3, and c=−5c = -5.
a=2,b=3,c=−5a = 2,\quad b = 3,\quad c = -5
Step 2: Write the quadratic formula.
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Step 3: Substitute the values of aa, bb, and cc into the formula.
x=−3±(3)2−4(2)(−5)2(2)=−3±9+404x = \frac{-3 \pm \sqrt{(3)^2 - 4(2)(-5)}}{2(2)} = \frac{-3 \pm \sqrt{9 + 40}}{4}
Step 4: Simplify under the square root and compute both solutions.
x=−3±494=−3±74x = \frac{-3 \pm \sqrt{49}}{4} = \frac{-3 \pm 7}{4}
Step 5: Calculate each solution separately.
x=−3+74=1orx=−3−74=−104=−52x = \frac{-3 + 7}{4} = 1 \qquad \text{or} \qquad x = \frac{-3 - 7}{4} = -\frac{10}{4} = -\frac{5}{2}
Answer: The solutions are x=1x = 1 and x=−52x = -\dfrac{5}{2}.

Frequently Asked Questions

What is the difference between a formula and an equation?
An equation is any mathematical statement that two expressions are equal, such as 3x+1=73x + 1 = 7. A formula is a specific type of equation (or expression) designed to compute a particular quantity every time you substitute values into it, like A=lwA = lw for the area of a rectangle. Every formula that contains an equals sign is an equation, but not every equation is a formula.
What is the difference between a formula and an expression?
An expression is a combination of numbers, variables, and operations that represents a value, such as 2x+52x + 5. A formula typically includes an equals sign and defines how to calculate a specific result from given inputs. For instance, d=rtd = rt is a formula because it tells you exactly how to find distance from rate and time, whereas rtrt alone is just an expression.
How do you use a formula in math?
To use a formula, identify which quantity you want to find, determine the known values, and substitute those values into the appropriate positions in the formula. Then simplify using the order of operations. If the unknown variable is not already isolated on one side, you may need to rearrange the formula before substituting.

Formula vs. Equation

FormulaEquation
DefinitionA rule that expresses how to calculate a specific quantity from given inputsA statement that two expressions are equal
PurposeRepeatedly compute a result by substituting valuesState a relationship or pose a problem to solve
ExampleV=lwhV = lwh (volume of a rectangular prism)3x+1=73x + 1 = 7 (solve for xx)
Contains an equals sign?Usually, but some formulas are written as expressionsAlways
Reusable?Yes — designed to be used with many different input valuesNot necessarily — may describe a single specific relationship

Why It Matters

Formulas appear in nearly every math course you will take, from the area and perimeter formulas in geometry to the compound interest formula in finance and the kinematic equations in physics. Learning to identify the right formula, substitute values correctly, and rearrange it when necessary is one of the most transferable skills in mathematics. Standardized tests like the SAT and ACT provide formula sheets, but understanding how each formula works — not just memorizing it — is what lets you apply them under pressure.

Common Mistakes

Mistake: Substituting values into the wrong variables, especially when a formula has several similar-looking letters.
Correction: Before substituting, write out what each variable represents and match it to the corresponding given value. For example, in d=rtd = rt, confirm that rr is the rate and tt is the time — not the other way around.
Mistake: Forgetting to follow the order of operations after substituting into a formula.
Correction: Always handle exponents and roots before multiplication, and multiplication before addition. For instance, in A=πr2A = \pi r^2, you must square rr first and then multiply by π\pi, not multiply π\pi and rr first and then square the result.

Related Terms