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The problem shows a prism with a square cross-section \( ABCD \) and a volume of \( 960 \, \text{cm}^3 \). The length of the prism is \( 15 \, \text{cm} \). You are asked to calculate:

(i) The length of edge \( AB \) (side of the square) in cm.  
(ii) The total surface area of the prism in \( \text{cm}^2 \).

### Part (i): Length of Edge \( AB \)

We know that:
- The volume \( V \) of a prism is given by:
  \[
  V = \text{Area of the cross-section} \times \text{Length of the prism}
  \]
  Since the cross-section is a square, let \( AB = a \) (side of the square). The area of the square cross-section is \( a^2 \), and the length of the prism is \( 15 \, \text{cm} \). Therefore:
  \[
  960 = a^2 \times 15
  \]
  Solving for \( a \):
  \[
  a^2 = \frac{960}{15} = 64
  \]
  \[
  a = \sqrt{64} = 8 \, \text{cm}
  \]
Thus, the length of edge \( AB \) is **8 cm**.

### Part (ii): Total Surface Area

The total surface area of the prism consists of:
- Two square faces (\( ABCD \) and the opposite face) each with area \( a^2 \).
- Four rectangular faces, each with area \( a \times \text{Length of the prism} \).

Total surface area \( S \) is:
\[
S = 2 \times a^2 + 4 \times (a \times 15)
\]
Substitute \( a = 8 \):
\[
S = 2 \times 8^2 + 4 \times (8 \times 15)
\]
\[
S = 2 \times 64 + 4 \times 120 = 128 + 480 = 608 \, \text{cm}^2
\]
Thus, the total surface area of the prism is **608 cm²**.

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Would you like further details on any part of the solution or have any questions?

Here are 5 related questions for further practice:
1. What would the length of the edge be if the volume of the prism was doubled?
2. How would the surface area change if the length of the prism was reduced by half?
3. If the cross-section was a rectangle instead of a square, how would you calculate the dimensions?
4. What is the formula for the volume of a triangular prism?
5. Can the surface area be minimized by changing the shape of the cross-section?

**Tip**: The formula for the surface area of a prism depends on the shape of the cross-section, so always start by identifying that shape first.

The diagram shows a prism of volume 960 cm³. The cross-section ABCD is a square. The length of the prism is 15 cm. Calculate: (i) the length of the edge AB, in cm, (ii) the total surface area of the prism, in cm².

Solve a geometry problem involving a square prism with a given volume of 960 cm³ and a length of 15 cm. Learn how to calculate the edge length of the square cross-section and the total surface area. Suitable for students in Grades 8-10.

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