Surface Area of a Rectangular Prism: Express in Terms of Variable 'a' (5×1×a)

Q: Why do we multiply by 2 in the surface area formula?

Because every rectangular prism has 6 faces arranged in 3 pairs! Each pair consists of two identical rectangles on opposite sides. So we calculate the area of each unique face once, then multiply by 2.

Q: How do I remember which dimensions to multiply together?

Think of the three different face types: front/back faces (length × height), left/right faces (width × height), and top/bottom faces (length × width). Each pair uses two different dimensions.

Q: What if I get the terms in a different order like 10 + 12a instead of 12a + 10?

Both expressions are mathematically equivalent! Addition is commutative, so $ 10 + 12a = 12a + 10 $. Choose whichever form the problem asks for.

Q: Can I use this formula even when one dimension is a variable?

Absolutely! The surface area formula works with any dimensions - numbers, variables, or expressions. Just substitute carefully and combine like terms at the end.

Surface Area Formula with Variable Dimensions

Express the surface area of the rectangular prism below in terms of a.

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the surface area of the box

00:03 We'll use the formula for calculating rectangle area (side times side)

00:13 Let's calculate each face

00:22 Since the box has 6 faces

00:32 To calculate the surface area, we'll multiply each area by 2 and sum

01:01 Let's group the factors

01:07 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Express the surface area of the rectangular prism below in terms of a.

Step-by-step solution

To solve this problem, we must find the surface area of the rectangular prism with given dimensions $1$ , $5$ , and $a$ .

The formula for the surface area $S$ of a rectangular prism with length $l$ , width $w$ , and height $h$ is:

$S = 2(lw + lh + wh)$

For this prism, let's identify the dimensions:

Length ( $l$ ) = 1
Width ( $w$ ) = 5
Height ( $h$ ) = a

Now, substitute these dimensions into the surface area formula:

$S = 2(1 \times 5 + 1 \times a + 5 \times a)$

Simplify the expression inside the parentheses:

$S = 2(5 + a + 5a)$

Combine the terms:

$S = 2(5 + 6a)$

Multiply through by 2:

$S = 10 + 12a$

Thus, the surface area of the rectangular prism expressed in terms of $a$ is $12a + 10$ .

Final Answer

12a+10

Key Points to Remember

Essential concepts to master this topic

Formula: Surface area equals 2(lw + lh + wh) for all rectangular prisms
Technique: Substitute dimensions: 2(1×5 + 1×a + 5×a) = 2(5 + 6a)
Check: Verify by counting faces: two 5×1, two 1×a, two 5×a rectangles ✓

Common Mistakes

Avoid these frequent errors

Forgetting to double each face area
Don't calculate just lw + lh + wh = 5 + a + 5a = 5 + 6a! This gives only half the surface area because each rectangular face appears twice on opposite sides. Always multiply by 2: 2(5 + 6a) = 10 + 12a.

Practice Quiz

Test your knowledge with interactive questions

A cuboid is shown below:

What is the surface area of the cuboid?

FAQ

Everything you need to know about this question

Why do we multiply by 2 in the surface area formula?

Because every rectangular prism has 6 faces arranged in 3 pairs! Each pair consists of two identical rectangles on opposite sides. So we calculate the area of each unique face once, then multiply by 2.

How do I remember which dimensions to multiply together?

Think of the three different face types: front/back faces (length × height), left/right faces (width × height), and top/bottom faces (length × width). Each pair uses two different dimensions.

What if I get the terms in a different order like 10 + 12a instead of 12a + 10?

Both expressions are mathematically equivalent! Addition is commutative, so $10 + 12a = 12a + 10$ . Choose whichever form the problem asks for.

Can I use this formula even when one dimension is a variable?

Absolutely! The surface area formula works with any dimensions - numbers, variables, or expressions. Just substitute carefully and combine like terms at the end.

How can I double-check my algebra?

Try substituting a simple value for a, like a = 1. Calculate the surface area both ways: using your formula and by adding up all 6 face areas individually. They should match!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Cuboids questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime