Calculating the Surface Area of a Cuboid Given the Sides and Diagonal

Understanding the geometric properties of a cuboid can be crucial for numerous applications, including mathematics and engineering. In this article, we will explore how to calculate the total surface area of a cuboid when given the sum of its sides and the length of its diagonal. This is a useful problem that involves the interplay between volume and surface area in three-dimensional geometry.

Given Information

Consider a cuboid with sides denoted as p, q, r. The sum of these sides is given as 45 cm, and the length of the space diagonal is 21 cm. Let’s denote the dimensions of the cuboid as a, b, c, where a, b, and c correspond to p, q, r.

Step-by-Step Calculation

To find the total surface area of the cuboid, we can use the known relationships between the sides, the diagonal, and the surface area. We will follow these steps:

Step 1: Express the Known Information

Given the sum of the three sides:

a b c 45 cm The length of the space diagonal L 21 cm

From the properties of a cuboid, the length of the space diagonal L is given by:

[ L^2 a^2 b^2 c^2 ]

Given L 21 cm, we can write:

[ 21^2 a^2 b^2 c^2 ]

Simplifying this, we get:

[ 441 a^2 b^2 c^2 ]

Step 2: Calculate the Total Surface Area

The total surface area A of a cuboid is given by:

[ A 2(ab bc ca) ]

Given that:

[ a^2 b^2 c^2 441 ]

and knowing that:

[ abc 45 ]

We can substitute these into the formula for surface area:

[ A 2abc 2 times 45 ]

Calculating this, we get:

[ A 90 times 2 180 ]

However, to ensure accuracy, let's verify if this matches the direct formula derived from the problem statement:

[ A 2(ab bc ca) frac{L^2 (a^2 b^2 c^2 - (a^2 b^2 c^2)) 2abc(a b c)}{a b c} ]

Simplifying further:

[ A 2(ab bc ca) frac{21^2 times 441 2 times 45 times 45}{45} ]

And thus:

[ A 1584 text{ square centimeters} ]

Conclusion

The total surface area of the cuboid, when the sum of its sides is 45 cm and the length of the space diagonal is 21 cm, is 1584 square centimeters. This calculation highlights the relationship between the sum of the sides, the diagonal, and the total surface area of a cuboid, making it a useful and practical problem for geometry and spatial understanding.

Related Keywords

cuboid, surface area, diagonal, sum of sides