Surface Area of a Cuboid Notes

Published on: September 29, 2026

This Surface Area of a Cuboid Calculator finds the total area of the outside of a cuboid from its length, width and height, and shows the working line by line. A cuboid has six rectangular faces that match in opposite pairs, so only three different face areas are worked out, added together, and then doubled. The answer is written in square units. Following the same pattern every time is what makes the surface area of a cuboid easy to check and repeat.

Cuboid labeled l, w, h

In the diagram, the dimensions are labeled \(l\), \(w\) and \(h\).

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the length, the width and the height into the formula.
  4. Work out the area of each pair of faces.
  5. Add the three face areas inside the brackets.
  6. Double that total to get the surface area.

Formula:

\(SA = 2(lw + lh + wh)\)
See Example 1 Hide Example 1

Problem

\(l = 8, w = 5, h = 3\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the length, the width and the height into the formula.
  4. Work out the area of each pair of faces.
  5. Add the three face areas inside the brackets.
  6. Double that total to get the surface area.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the surface area of a cuboid whose length is \(l = 8\), whose width is \(w = 5\) and whose height is \(h = 3\).

\(l = 8,\; w = 5,\; h = 3\)

Step 2

Step 2 - Write the formula.

In this problem: A cuboid has six faces that match in opposite pairs, so the three different face areas are added together and that total is doubled.

\(SA = 2(lw + lh + wh)\)

Step 3

Step 3 - Substitute the length, the width and the height into the formula.

In this problem: Replacing \(l\), \(w\) and \(h\) with \(8\), \(5\) and \(3\) in \(SA = 2(lw + lh + wh)\) gives \(SA = 2(8 \times 5 + 8 \times 3 + 5 \times 3)\).

\(SA = 2(8 \times 5 + 8 \times 3 + 5 \times 3)\)

Step 4

Step 4 - Work out the area of each pair of faces.

In this problem: Inside the brackets the three face areas are still written as multiplications, so each one is worked out in turn.

\(SA = 2(8 \times 5 + 8 \times 3 + 5 \times 3)\)

Step 4a

Step 4a - Work out the area of each pair of faces.

In this problem: Multiplying the length and the width gives \(8 \times 5 = 40\).

\(SA = 2(40 + 8 \times 3 + 5 \times 3)\)

Step 4b

Step 4b - Work out the area of each pair of faces.

In this problem: Multiplying the length and the height gives \(8 \times 3 = 24\).

\(SA = 2(40 + 24 + 5 \times 3)\)

Step 4c

Step 4c - Work out the area of each pair of faces.

In this problem: Multiplying the width and the height gives \(5 \times 3 = 15\).

\(SA = 2(40 + 24 + 15)\)

Step 5

Step 5 - Add the three face areas inside the brackets.

In this problem: Adding the three face areas gives \(40 + 24 + 15 = 79\).

\(SA = 2(79)\)

Step 6

Step 6 - Double that total to get the surface area.

In this problem: Doubling that total covers the matching opposite faces and gives \(2 \times 79 = 158\).

\(SA = 2(79) = 158\)

Final Answer

\(SA = 158\)
See Example 2 Hide Example 2

Problem

\(l = 2, w = 2.5, h = 4\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the length, the width and the height into the formula.
  4. Work out the area of each pair of faces.
  5. Add the three face areas inside the brackets.
  6. Double that total to get the surface area.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the surface area of a cuboid whose length is \(l = 2\), whose width is \(w = 2.5\) and whose height is \(h = 4\).

\(l = 2,\; w = 2.5,\; h = 4\)

Step 2

Step 2 - Write the formula.

In this problem: A cuboid has six faces that match in opposite pairs, so the three different face areas are added together and that total is doubled.

\(SA = 2(lw + lh + wh)\)

Step 3

Step 3 - Substitute the length, the width and the height into the formula.

In this problem: Replacing \(l\), \(w\) and \(h\) with \(2\), \(2.5\) and \(4\) in \(SA = 2(lw + lh + wh)\) gives \(SA = 2(2 \times 2.5 + 2 \times 4 + 2.5 \times 4)\).

\(SA = 2(2 \times 2.5 + 2 \times 4 + 2.5 \times 4)\)

Step 4

Step 4 - Work out the area of each pair of faces.

In this problem: Inside the brackets the three face areas are still written as multiplications, so each one is worked out in turn.

\(SA = 2(2 \times 2.5 + 2 \times 4 + 2.5 \times 4)\)

Step 4a

Step 4a - Work out the area of each pair of faces.

In this problem: Multiplying the length and the width gives \(2 \times 2.5 = 5\).

\(SA = 2(5 + 2 \times 4 + 2.5 \times 4)\)

Step 4b

Step 4b - Work out the area of each pair of faces.

In this problem: Multiplying the length and the height gives \(2 \times 4 = 8\).

\(SA = 2(5 + 8 + 2.5 \times 4)\)

Step 4c

Step 4c - Work out the area of each pair of faces.

In this problem: Multiplying the width and the height gives \(2.5 \times 4 = 10\).

\(SA = 2(5 + 8 + 10)\)

Step 5

Step 5 - Add the three face areas inside the brackets.

In this problem: Adding the three face areas gives \(5 + 8 + 10 = 23\).

\(SA = 2(23)\)

Step 6

Step 6 - Double that total to get the surface area.

In this problem: Doubling that total covers the matching opposite faces and gives \(2 \times 23 = 46\).

\(SA = 2(23) = 46\)

Final Answer

\(SA = 46\)
See Example 3 Hide Example 3

Problem

\(l = 6, w = 4, h = 2\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the length, the width and the height into the formula.
  4. Work out the area of each pair of faces.
  5. Add the three face areas inside the brackets.
  6. Double that total to get the surface area.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the surface area of a cuboid whose length is \(l = 6\), whose width is \(w = 4\) and whose height is \(h = 2\).

\(l = 6,\; w = 4,\; h = 2\)

Step 2

Step 2 - Write the formula.

In this problem: A cuboid has six faces that match in opposite pairs, so the three different face areas are added together and that total is doubled.

\(SA = 2(lw + lh + wh)\)

Step 3

Step 3 - Substitute the length, the width and the height into the formula.

In this problem: Replacing \(l\), \(w\) and \(h\) with \(6\), \(4\) and \(2\) in \(SA = 2(lw + lh + wh)\) gives \(SA = 2(6 \times 4 + 6 \times 2 + 4 \times 2)\).

\(SA = 2(6 \times 4 + 6 \times 2 + 4 \times 2)\)

Step 4

Step 4 - Work out the area of each pair of faces.

In this problem: Inside the brackets the three face areas are still written as multiplications, so each one is worked out in turn.

\(SA = 2(6 \times 4 + 6 \times 2 + 4 \times 2)\)

Step 4a

Step 4a - Work out the area of each pair of faces.

In this problem: Multiplying the length and the width gives \(6 \times 4 = 24\).

\(SA = 2(24 + 6 \times 2 + 4 \times 2)\)

Step 4b

Step 4b - Work out the area of each pair of faces.

In this problem: Multiplying the length and the height gives \(6 \times 2 = 12\).

\(SA = 2(24 + 12 + 4 \times 2)\)

Step 4c

Step 4c - Work out the area of each pair of faces.

In this problem: Multiplying the width and the height gives \(4 \times 2 = 8\).

\(SA = 2(24 + 12 + 8)\)

Step 5

Step 5 - Add the three face areas inside the brackets.

In this problem: Adding the three face areas gives \(24 + 12 + 8 = 44\).

\(SA = 2(44)\)

Step 6

Step 6 - Double that total to get the surface area.

In this problem: Doubling that total covers the matching opposite faces and gives \(2 \times 44 = 88\).

\(SA = 2(44) = 88\)

Final Answer

\(SA = 88\)
See Example 4 Hide Example 4

Problem

\(l = 3, w = 1.5, h = 5\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the length, the width and the height into the formula.
  4. Work out the area of each pair of faces.
  5. Add the three face areas inside the brackets.
  6. Double that total to get the surface area.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the surface area of a cuboid whose length is \(l = 3\), whose width is \(w = 1.5\) and whose height is \(h = 5\).

\(l = 3,\; w = 1.5,\; h = 5\)

Step 2

Step 2 - Write the formula.

In this problem: A cuboid has six faces that match in opposite pairs, so the three different face areas are added together and that total is doubled.

\(SA = 2(lw + lh + wh)\)

Step 3

Step 3 - Substitute the length, the width and the height into the formula.

In this problem: Replacing \(l\), \(w\) and \(h\) with \(3\), \(1.5\) and \(5\) in \(SA = 2(lw + lh + wh)\) gives \(SA = 2(3 \times 1.5 + 3 \times 5 + 1.5 \times 5)\).

\(SA = 2(3 \times 1.5 + 3 \times 5 + 1.5 \times 5)\)

Step 4

Step 4 - Work out the area of each pair of faces.

In this problem: Inside the brackets the three face areas are still written as multiplications, so each one is worked out in turn.

\(SA = 2(3 \times 1.5 + 3 \times 5 + 1.5 \times 5)\)

Step 4a

Step 4a - Work out the area of each pair of faces.

In this problem: Multiplying the length and the width gives \(3 \times 1.5 = 4.5\).

\(SA = 2(4.5 + 3 \times 5 + 1.5 \times 5)\)

Step 4b

Step 4b - Work out the area of each pair of faces.

In this problem: Multiplying the length and the height gives \(3 \times 5 = 15\).

\(SA = 2(4.5 + 15 + 1.5 \times 5)\)

Step 4c

Step 4c - Work out the area of each pair of faces.

In this problem: Multiplying the width and the height gives \(1.5 \times 5 = 7.5\).

\(SA = 2(4.5 + 15 + 7.5)\)

Step 5

Step 5 - Add the three face areas inside the brackets.

In this problem: Adding the three face areas gives \(4.5 + 15 + 7.5 = 27\).

\(SA = 2(27)\)

Step 6

Step 6 - Double that total to get the surface area.

In this problem: Doubling that total covers the matching opposite faces and gives \(2 \times 27 = 54\).

\(SA = 2(27) = 54\)

Final Answer

\(SA = 54\)