What is the formula for the surface area of a rectangular prism?

The surface area of a rectangular prism is given by the formula SA = 2(lw + lh + wh), where l is length, w is width, and h is height.

How to Find Surface Area: Step-by-Step Guide with Visual Formulas

What is Surface Area?

Surface Area is the total area of all the exterior faces of a 3D object added together.

•Volume is how much water or space fits inside the box (cm³).
•Surface Area is how much wrapping paper covers the outside of the box (cm²).

1. Rectangular Prism Surface Area Formula

A rectangular prism (like a cereal box) has 6 rectangular faces that come in 3 matching pairs:

Surface Area = 2(l × w) + 2(l × h) + 2(w × h)

Where:

•l = Length
•w = Width
•h = Height

Worked Example 1: Standard Rectangular Prism

Problem:

A cardboard gift box measures length = 8 in, width = 5 in, and height = 3 in.

What is the total surface area of the box?

Step 1: Calculate the Top & Bottom Faces (2 faces)

A_{top/bottom} = 2 × (8 × 5) = 2 × 40 = 80 sq in

Step 2: Calculate the Front & Back Faces (2 faces)

A_{front/back} = 2 × (8 × 3) = 2 × 24 = 48 sq in

Step 3: Calculate the Left & Right Sides (2 faces)

A_{left/right} = 2 × (5 × 3) = 2 × 15 = 30 sq in

Step 4: Add all 6 faces together

SA = 80 + 48 + 30 = 158 in²

The total surface area of the box is 158 square inches.

Interactive Practice Check

Test Your Understanding

Find the surface area of a box with length = 6 cm, width = 4 cm, and height = 5 cm. Step 1: What is the combined area of just the **top and bottom** faces ($2 × (6 × 4)$)?

2. Cylinder Surface Area Formula

Think of a soup can. If you peel off the label and pop off the top and bottom lids, you get:

Two circular lids: Area of each circle is \pi r², so two circles = 2\pi r².
One unrolled rectangular label: The height is h, and the length across is the circumference of the can, 2\pi r. Area = 2\pi rh.

Total Cylinder Surface Area = 2\pi r² + 2\pi rh

Worked Example 2: Cylinder Surface Area

Problem:

A cylinder container has a radius of r = 4 cm and a height of h = 10 cm.

Find the total surface area (use \pi ≈ 3.14).

Step 1: Area of the 2 circular bases

2\pi r² = 2 × 3.14 × (4)² = 2 × 3.14 × 16 = 100.48 cm²

Step 2: Area of the curved body (lateral surface)

2\pi rh = 2 × 3.14 × 4 × 10 = 251.2 cm²

Step 3: Add both parts together

SA = 100.48 + 251.2 = 351.68 cm²

The total surface area is 351.68 cm² (or 112\pi cm² in exact form).

Common Surface Area Formulas Quick Sheet

3D Shape	Formula	Note
Cube	SA = 6s²	All 6 faces are identical squares
Rectangular Prism	SA = 2(lw + lh + wh)	3 pairs of matching rectangles
Cylinder	SA = 2\pi r² + 2\pi rh	2 circular bases + 1 curved rectangle
Sphere	SA = 4\pi r²	Exactly equal to 4 circular cross-sections

⚠️WATCH OUT

Watch Out for Diameter vs. Radius!

If a problem gives you a cylinder with a diameter of 10 cm, remember to divide by 2 first: r = 5 cm. Squaring the diameter by mistake will make your answer 4 times too large!

How to Find Surface Area: Step-by-Step Guide with Visual Formulas

The 4-Step Solving Process

Identify the 3D Shape and Its Faces

Calculate the Area of Each Face Pair

Add All Face Areas Together

For Cylinders: Add Bases and Lateral Area