a²+b²+c² Formula with Examples

a² + b² + c² Formula helps us find the total of squares of three numbers—a, b, and c—in a quick and smart way. Instead of solving a², b², and c² separately, we can use one single formula to save time. The formula is a² + b² + c² = (a + b + c)² – 2ab – 2bc – 2ca.

This is one of the important formulas in algebra. We also call it the formula of a² + b² + c². It is made by expanding the square of (a + b + c). Many students also see a different formula called a² + b² = c², which is used in triangles. Keep reading to learn how to use the a^2 + b^2 + c^2 formula with easy-to-understand solved examples.

Read More: What is Factor?

What is a2+b2+c2 Formula?

a² + b² + c² formula is a basic rule in algebra that helps us find the sum of the squares of three numbers. This means if we have any three values like x, y, z or a, b, c, and we want to add their squares, we can use the formula a^2+b^2+c^2. In mathematics, a square means multiplying a number by itself. So a² means a × a, b² means b × b, and so on. Instead of solving each square one by one, the formula helps us do it in a faster way.

The formula of a2+b2+c2 is part of algebraic identities, which are special rules used to solve math problems. In algebra, we generally use letters like a, b, or c in place of numbers. These letters can stand for any value. 

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a^2+b^2+c^2 Formula Derivation

As we learned, the a² + b² + c² formula is an important identity in algebra. It helps us find the total of squares of three terms without calculating each square one by one. To understand the a^2+b^2+c^2 formula derivation, let us start by looking at the square of a bracket.

We know that when we square a number or an expression, we multiply it by itself. So if we have (a + b + c)², it means we are doing: (a + b + c) × (a + b + c). Now, we will open this bracket step by step using the distributive rule, which means we will multiply each term in the first bracket with every term in the second bracket.

• So first, let’s multiply a with (a + b + c):

• a × a = a²

• a × b = ab

• a × c = ac

• Now take b and multiply it with (a + b + c):

• b × a = ba (which is the same as ab)

• b × b = b²

• b × c = bc

• Now take c and multiply it with (a + b + c):

• c × a = ca (same as ac)

• c × b = cb (same as bc)

• c × c = c²

• Now, let’s put all these together: a² + ab + ac + ab + b² + bc + ac + bc + c²

• Now group the similar terms like: a² + b² + c² + ab + ab + ac + ac + bc + bc

• Now add the like terms: a² + b² + c² + 2ab + 2ac + 2bc

• So we get: (a + b + c) ² = a² + b² + c² + 2ab + 2ac + 2bc

This is the full expansion of the square of (a + b + c). But we want only a² + b² + c², so we need to move the other terms (the ones with 2ab, 2bc, and 2ca) to the other side of the equation.

• So we write: a² + b² + c² = (a + b + c) ² – 2ab – 2bc – 2ca

• This is the final form of the a² + b² + c² formula.

So, the a^2 + b^2 + c^2 identity comes from simply expanding the square of three terms added together. This formula is very helpful in solving algebra problems, and it also saves time because we don’t have to calculate each square and product separately.

Read More: What are the Factors of 36?

a² + b² = c² Formula Overview

The formula a² + b² = c² is one of the most well-known formulas in mathematics. It is called the Pythagorean Theorem and is used to find the sides of a right-angled triangle. In this formula, a and b are the lengths of the two smaller sides (the ones that make the right angle), and c is the longest side, called the hypotenuse, which is the side opposite the right angle.

a² + b² = c² formula tells us that when we square the two shorter sides of a right-angled triangle and add them, we get the square of the longest side. For example, if one side is 3 units and the other is 4 units, then a² = 9 and b² = 16. When we add them, we get 25. So, c² = 25, which means c = 5.

a² + b² + c² Formula vs a² + b² = c² Formula: Are they the Same?

Both a^2+b^2+c^2 and a²+b²=c² may look similar as they have squares in them, but they are used in different ways and in different topics of mathematics.

• The a² + b² + c² formula is used in algebra. It is a shortcut to find the sum of the squares of three numbers. We don’t use it for shapes or triangles.

• The full formula is: a² + b² + c² = (a + b + c)² – 2ab – 2bc – 2ca. This helps us solve algebraic problems faster, especially when we work with expressions.

On the other hand, a² + b² = c² is a very famous formula from geometry, known as the Pythagorean Theorem. It is used only for right-angled triangles. In this, a and b are the shorter sides, and c is the hypotenuse (the longest side). This formula tells us how the sides of a triangle are related to each other.

So, the formula of a² + b² + c² is for general math problems in algebra, and a² + b² = c² is only for finding the sides of a right-angled triangle.

Read More: What is Factorization Formula? 

a² + b² + c² Formula with Solved Examples

To understand how the a² + b² + c² formula works, it is important to look at some examples with solutions. These will help students learn how to use the formula of a2+b2+c2 while solving real maths problems.

Example 1: Find the value of a² + b² + c² if a + b + c = 8 and ab + bc + ca = 5.

Solution:

We are given:

• a + b + c = 8

• ab + bc + ca = 5

We will use the formula of a² + b² + c²: a² + b² + c² = (a + b + c) ² – 2(ab + bc + ca)

Now substitute the values:
= (8)² – 2(5)
= 64 – 10
= 54

Answer: So, a² + b² + c² = 54.

Example 2: Find the value of a² + b² + c² if a + b + c = –6, 1/a + 1/b + 1/c = 1, and abc = –3.

Solution:

We are given:

• a + b + c = –6

• 1/a + 1/b + 1/c = 1

• abc = –3

Step 1: Multiply the last two values: abc × (1/a + 1/b + 1/c) = –3 × 1 = –3

So, ab + bc + ca = –3

Now, use the a^2 + b^2 + c^2 formula: a² + b² + c² = (a + b + c)² – 2(ab + bc + ca)
= (–6)² – 2(–3)
= 36 + 6
= 42

Answer: So, a² + b² + c² = 42.

Example 3: Find the value of a² + b² + c² if a + b + c = 12 and ab + bc + ca = 20.

Solution:

Given:

• a + b + c = 12

• ab + bc + ca = 20

Use the formula of a2+b2+c2: a² + b² + c² = (a + b + c) ² – 2(ab + bc + ca)
= (12)² – 2(20)
= 144 – 40
= 104

Answer: So, a² + b² + c² = 104.

Also Read: Imperial system

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