The problem asks to find the length of the base of a square cake tin that has the same area as the base of a round cake tin.

Given:

• The round cake tin has a radius of $r = 10$ cm.
• The area of the base of the round cake tin is the area of a circle, $A_{\text{circle}}$.
• The area of the base of the square cake tin is the area of a square, $A_{\text{square}}$.
• The areas of the two bases are equal: $A_{\text{circle}} = A_{\text{square}}$.

Step-by-step Solution:

1. Area of the circle:
   The area of the base of the round cake tin (a circle) is given by the formula: $A_{\text{circle}} = \pi r^2$ Substituting $r = 10$ cm: $A_{\text{circle}} = \pi \times 10^2 = 100\pi \ \text{cm}^2$

2. Area of the square:
   Let the side length of the square be $l$. The area of the base of the square cake tin is: $A_{\text{square}} = l^2$

3. Equating the areas:
   Since the two areas are equal: $100\pi = l^2$ Solving for $l$: $l = \sqrt{100\pi}$

4. Approximation:
   Using $\pi \approx 3.1416$: $l = \sqrt{100 \times 3.1416} = \sqrt{314.16} \approx 17.73 \ \text{cm}$

Final Answer:

The length of the base of the square cake tin is approximately 17.73 cm.

Would you like further explanation on this? Here are some related questions for practice:

1. If the radius of the round cake tin was doubled, what would be the new side length of the square cake tin?
2. How would you find the area if the shape of the cake tin was a triangle instead of a square?
3. What is the circumference of the round cake tin in this case?
4. If the area of the square cake tin were doubled, what would the new side length be?
5. How does the area of the square cake tin compare to the perimeter of the round cake tin?

Tip: Always double-check units in geometric problems to ensure consistency when performing calculations.