Answer:
50 cm²
Step-by-step explanation:
Given that the regular polygon is a square, there are multiple ways you can jump directly to the answer. Perhaps the simplest is to use the formula for the area of a rhombus:
A = 1/2(d1)(d2)
where d1 and d2 are the lengths of the diagonals. Here, we see that half the diagonal is 5 cm, so the area is ...
A = (1/2)(10 cm)(10 cm) = 50 cm² . . . . area of the polygon
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Alternate solution
If you want to use the radius and the number of sides in a formula, you can consider the area of each triangle formed by radii and a side. That triangle has area ...
A = 1/2r²sin(α)
where r is the radius and α is the central angle. For an n-sided polygon, the area is the sum of n of these triangles, and the central angle is 360°/n. Then the polygon area is ...
A = n/2·r²·sin(360°/n)
For n = 4 and r = 5 cm, the area is ...
A = (4/2)(5 cm)²(sin(360°/4)) = 2(5 cm)²(1) = 50 cm² . . . . area of square
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Additional comment
The formula is somewhat different if you start with the length of the apothem. One way to find the area is using the above formula and the relation between the radius and apothem:
r = a·sec(180°/n)
Another formula uses the apothem directly:
A = n·a²·tan(180°/n)
