https://file.aiquickdraw.com/mathbot/file/1725593692515gqoolsbc.jpg

The image shows a cuboid with the dimensions \( 8 \, \text{cm} \), \( 6 \, \text{cm} \), and \( 12 \, \text{cm} \), and several questions that are listed next to it. The questions ask to calculate:

a. Perpendicular diagonal between points.

b. Longest diagonal of the cuboid.

c. Surface area.

d. Volume.

Let me go ahead and solve these problems.

### a. Perpendicular Diagonal (Diagonal of a Face)
Assume the question asks for the diagonal across a face of the cuboid, such as the diagonal between the sides 6 cm and 8 cm. The diagonal of a rectangle can be calculated using the Pythagorean theorem:
\[
d = \sqrt{l^2 + w^2}
\]
Where \( l = 8 \, \text{cm} \) and \( w = 6 \, \text{cm} \).

### b. Longest Diagonal (Body Diagonal)
For the longest diagonal, which is the body diagonal of the cuboid, we use the 3D Pythagorean theorem:
\[
d = \sqrt{l^2 + w^2 + h^2}
\]
Where \( l = 12 \, \text{cm} \), \( w = 8 \, \text{cm} \), and \( h = 6 \, \text{cm} \).

### c. Surface Area
The surface area of the cuboid is the sum of the areas of all six faces:
\[
A = 2(lw + lh + wh)
\]
Where \( l = 12 \, \text{cm} \), \( w = 8 \, \text{cm} \),

The image shows a cuboid with dimensions 8 cm, 6 cm, and 12 cm, asking for:
a. Perpendicular diagonal between points.
b. Longest diagonal of the cuboid.
c. Surface area.
d. Volume.

Solve geometry problems involving a cuboid with dimensions 8 cm, 6 cm, and 12 cm. Learn to calculate the perpendicular diagonal, longest diagonal, surface area, and volume using Pythagorean theorem and key geometric formulas. Suitable for Grades 7-9 students.

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