1. If a square has a radius of twelve, would its apothem be 6sqrt2? What will its area be? How would I solve it?

2. If triangle has a radius of 12, would its apothem be 7sqrt3? How would I find the area?

3. How would I find the area of a hexagon if it's apothem is 7sqrt3?

if the perimeter is p and the apothem is a, then the area is 1/2 ap

#1. correct: 12/√2
#2. hint: the triangle's altitude is 18.
#3. side is 12

1. First, we need to clarify that squares do not have radii. Radii are used to describe circles, not squares. However, if we assume you mean the diagonal of the square instead, we can solve the problem.

To find the length of the diagonal of a square, we can use the Pythagorean theorem: diagonal^2 = side^2 + side^2. Since all sides of a square are equal, we can replace "side" with "s". Therefore, diagonal^2 = s^2 + s^2 = 2s^2.

Given the diagonal length of 12, we can use the Pythagorean theorem to solve for the length of the side. 12^2 = 2s^2. Rearranging the equation, we get 144 = 2s^2. Dividing both sides by 2, we have s^2 = 72. Taking the square root of both sides, we find s = √(72) which simplifies to 6√2.

Next, let's calculate the area of the square. The area of a square is given by the formula A = side^2. Substituting the value of the side we found earlier, we have A = (6√2)^2 = 36 * 2 = 72.

Therefore, if a square has a diagonal length of 12, the length of each side would be 6√2, and the area would be 72.

2. Similar to the first question, triangles do not have radii. However, if you mean the radius of the circumcircle of the triangle, we can proceed with the question. The circumradius (not apothem) of a triangle is the distance from the center of the circumcircle to any vertex of the triangle.

To find the area of a triangle when given the circumradius, you can use the formula A = (r^2) * sin(A) * sin(B) * sin(C), where r is the radius (in your case, 12), and A, B, C are the angles of the triangle.

To find the angles of the triangle, you can use the Law of Sines, which states that sin(A) / a = sin(B) / b = sin(C) / c, where a, b, c are the sides opposite the respective angles.

Once you have the values of the angles A, B, C, you can substitute them into the area formula along with the radius.

3. To find the area of a hexagon when given the apothem (the distance from the center to any side), you can use the formula A = (3 * apothem^2 * √3)/2.

In your case, if the apothem is given as 7√3, we can substitute the value into the formula:

A = (3 * (7√3)^2 * √3)/2
= (3 * 49 * 3 * √3)/2
= (441 * √3)/2

Therefore, the area of the hexagon with an apothem of 7√3 would be (441 * √3)/2.