When we are given the diagonal of a cube, we can use some mathematical relationships to find the surface area. This article will guide you through the process of calculating the surface area of a cube when its diagonal is given as 24 cm. We will explore different methods and solutions to arrive at the surface area, which is 1152 square centimeters (cm2).
Introduction to Cube Diagonal and Surface Area
The cube, being a three-dimensional shape with six square faces, has several important properties. One of the most significant is its diagonal, which can be utilized to find the surface area. The diagonal of a cube, denoted by d, is the longest line segment connecting two opposite vertices of the cube. The relationship between the side length (a) and the diagonal of a cube is given by the formula:
d a√3
The surface area (A) of a cube, which is the total area of all its faces, is given by:
A 6a2
Calculating the Side Length from the Diagonal
To find the surface area of a cube when given its diagonal, we first need to determine the side length of the cube. Given that the diagonal of the cube is 24 cm, we can use the formula for the diagonal:
d a√3
Substituting the given value of d into the equation:
24 a√3
Solving for a, we get:
24/√3 a a 24/√3 ≈ 13.8564 cmCalculating the Surface Area Using the Side Length
Now that we know the side length of the cube, we can calculate the surface area using the surface area formula for a cube. Substituting the side length into the formula:
A 6a2
Substituting the value of a ≈ 13.8564 cm:
A 6(13.8564)2
A 1152 cm2
Thus, the surface area of the cube is 1152 square centimeters.
Alternative Calculation Method
Another method to find the surface area is to use the fact that the diagonal of a cube is related to the side length as follows:
d a√3
Given that the diagonal is 24 cm, we can solve for a as:
a 24/√3 8√3 cm
Using the surface area formula:
A 6a2
Substituting a 8√3 cm:
A 6(8√3)2
A 6 × 64 × 3
A 6 × 192
A 1152 cm2
Summary
By utilizing the mathematical relationships between the diagonal and side length of a cube, we can easily find the surface area. The given diagonal of 24 cm led us to calculate the side length as approximately 13.8564 cm, and substituting this into the surface area formula resulted in an area of 1152 square centimeters. Alternatively, we could use the fact that the side length is 8√3 cm, and the surface area would still be 1152 square centimeters.