This is an inscribed figures question testing the relationship between a circle's radius and an inscribed square's diagonal. Choice B (32) is correct — the diameter of the circle (8 inches) equals the diagonal of the inscribed square. Using the relationship diagonal = side × √2: 8 = s√2 → s = 8/√2 = 4√2. Area = s² = (4√2)² = 16 × 2 = 32 square inches. Choice A (16) uses the radius as the side length: 4² = 16 — confusing radius with the square's side. Choice C (16π) gives the area of the circle, not the inscribed square. Choice D (64) uses the diameter as the side length: 8² = 64 — the diameter is the diagonal, not the side. Pro tip: When a square is inscribed in a circle, the circle's diameter is the square's diagonal (it connects opposite corners through the center). From diagonal to side: s = d/√2 = d√2/2. Or use s² + s² = d² (Pythagorean theorem on the half-square): 2s² = 64 → s² = 32. The area is s², so no further calculation needed!

This is a perimeter and area question testing the relationship between area, dimensions, and perimeter. Choice C (38) is correct — find the width: Width = Area ÷ Length = 84 ÷ 12 = 7 inches. Perimeter = 2(length + width) = 2(12 + 7) = 2(19) = 38 inches. Choice A (7) stops after finding the width, reporting the intermediate step rather than the perimeter. Choice B (19) adds length + width = 12 + 7 = 19, but forgets to multiply by 2 — computing half the perimeter. Choice D (96) likely comes from multiplying area × length: 84 × 12 ÷ ... or adding area + length: 84 + 12 = 96. Pro tip: Finding a missing dimension from area is just the first step — remember to plug both dimensions into P = 2(l + w) to get the perimeter. The factor of 2 is easy to forget.

This question asks for the number of sides in a regular polygon with an interior angle sum of 1080 degrees. The formula for the sum is 180(n-2) degrees, so set this equal to 1080 and solve for n. Thus, 180(n-2)=1080, divide both sides by 180 to get n-2=6, so n=8. This means it's an octagon. For verification, a regular octagon has each angle 135 degrees, and 135*8=1080. Choice A, 6, gives sum 720 degrees, perhaps from 180(n-3) or similar error.

This question asks for the number of sides in a regular polygon where each exterior angle measures 45 degrees. The sum of exterior angles for any polygon is always 360 degrees, and in a regular polygon, each exterior angle is 360/n degrees. Setting 360/n = 45, we solve for n=360/45=8. This means the polygon is an octagon. A key point is that interior and exterior angles sum to 180 degrees each, so the interior would be 135 degrees, but the question focuses on exterior. Choice A, 6, might be from confusing with a hexagon's 60-degree exterior angles, as 360/6=60.

Given that the sum of interior angles is 1080°, we need to find the number of sides. Using the formula 180(n-2) = 1080, we solve for n: 180(n-2) = 1080, so n-2 = 6, therefore n = 8. The polygon has 8 sides (octagon).

This question asks for the sum of interior angles of a decagon (10 sides). The formula for the sum of interior angles is 180(n-2)° where n is the number of sides. Substituting n = 10: 180(10-2) = 180(8) = 1440°. Choice A would result from incorrectly using 180(6) for an 8-sided polygon.

This question asks for each interior angle in a regular triangle (3 sides). For a regular polygon, each interior angle equals 180(n-2)/n degrees. Substituting n = 3: 180(3-2)/3 = 180(1)/3 = 180/3 = 60°. An equilateral triangle has all angles equal to 60°.

This question asks for the sum of interior angles of an octagon (8 sides). The formula for the sum of interior angles is 180(n-2)° where n is the number of sides. Substituting n = 8: 180(8-2) = 180(6) = 1080°. Choice B would result from using a nonagon (9 sides) with sum 180(7) = 1260°.

This question asks how many sides a regular polygon has when each interior angle is 150°. For a regular polygon, each interior angle equals 180(n-2)/n degrees. Setting 180(n-2)/n = 150 and solving: 180n - 360 = 150n, so 30n = 360, giving n = 12. A regular dodecagon has interior angles of 150° each.

The question asks for the measure of each interior angle in a regular pentagon, which has 5 sides. The formula for the sum of interior angles is 180(n-2) degrees, and for regular, each is divided by n. For n=5, sum=180*(5-2)=180*3=540 degrees, each=540/5=108 degrees. Exterior angles sum to 360 degrees, each 72 degrees, so interior=180-72=108 degrees confirms. Choice B, 120 degrees, might be from confusing with a hexagon's 120-degree angles.