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To solve this problem, we need to calculate the surface area of the cuboid (rectangular box). The surface area is the total area of all six faces of the cuboid. 

The cuboid has three distinct dimensions:
- Length (\(l\)) = 27 cm
- Width (\(w\)) = 6 cm
- Height (\(h\)) = 14 cm

The surface area \(A\) of a cuboid is given by the formula:
\[
A = 2(lw + lh + wh)
\]
Now let's calculate the surface area step by step:

1. \( lw = 27 \times 6 = 162 \, \text{cm}^2 \)
2. \( lh = 27 \times 14 = 378 \, \text{cm}^2 \)
3. \( wh = 6 \times 14 = 84 \, \text{cm}^2 \)

Now, summing up the three areas:
\[
lw + lh + wh = 162 + 378 + 84 = 624 \, \text{cm}^2
\]

Finally, multiplying by 2 to account for both pairs of identical faces:
\[
A = 2 \times 624 = 1248 \, \text{cm}^2
\]

So, Heidi needs **1248 cm²** of purple paper to cover the box.

Would you like more details or have any questions about this solution? 

Here are five related questions to expand on this concept:
1. How would the surface area change if the height of the box were doubled?
2. What is the formula to calculate the volume of this cuboid?
3. How much material would be saved if the box was only covered on 4 of its 6 sides?
4. What is the surface area of a cube with a side length of 6 cm?
5. Can you think of a real-world scenario where surface area calculations are important?

**Tip:** Always double-check units in surface area problems to ensure consistency throughout the calculations.

Heidi is covering the cuboid-shaped box below with purple paper. How much purple paper does she need to cover the box? Give your answer in cm².

Learn how to calculate the surface area of a cuboid with dimensions 27 cm x 6 cm x 14 cm. This detailed solution covers key geometric concepts and applies the surface area formula for cuboids. Perfect for students in Grades 5-7.

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