Introduction:
The Total Surface Area (TSA) and Curved Surface Area (CSA) are important measurements used to describe the surface area of three-dimensional objects. In this case, we will consider two spheres with the same radius and determine the ratio between their TSA and CSA.

Understanding the TSA and CSA of a Sphere:
Before we proceed, let's briefly understand the concepts of TSA and CSA for a sphere.
- The TSA of a sphere is the sum of the areas of all its surfaces, including its curved surface area and the areas of its two hemispherical caps.
- The CSA of a sphere refers only to the curved surface area, excluding the areas of the hemispherical caps.

Calculating the TSA of a Sphere:
The TSA of a sphere can be calculated using the formula:
TSA = 4πr²,
where r represents the radius of the sphere.

Calculating the CSA of a Sphere:
The CSA of a sphere can be calculated using the formula:
CSA = 2πr²,
where r represents the radius of the sphere.

Ratio of TSA:
To find the ratio of the TSA between two spheres with the same radius, we can substitute the radius (r) into the formula for TSA and calculate the ratio.

Let's assume the radius of both spheres is 'r'.

For the first sphere, the TSA = 4πr².
For the second sphere, the TSA = 4πr².

By dividing the TSA of the first sphere by the TSA of the second sphere, we get:
TSA1 / TSA2 = (4πr²) / (4πr²).

Simplifying the expression, we find that the ratio of the TSA between two spheres with the same radius is:
TSA1 / TSA2 = 1.

Ratio of CSA:
To find the ratio of the CSA between two spheres with the same radius, we can substitute the radius (r) into the formula for CSA and calculate the ratio.

Let's assume the radius of both spheres is 'r'.

For the first sphere, the CSA = 2πr².
For the second sphere, the CSA = 2πr².

By dividing the CSA of the first sphere by the CSA of the second sphere, we get:
CSA1 / CSA2 = (2πr²) / (2πr²).

Simplifying the expression, we find that the ratio of the CSA between two spheres with the same radius is:
CSA1 / CSA2 = 1.

Conclusion:
In conclusion, the ratio of the TSA and CSA for two spheres with the same radius is always 1. This means that the TSA and CSA of the two spheres will be equal, regardless of the size of the radius.