12 multiple-choice questions, progressively harder.
A hexagon is constructed by radius stepping in a circle of radius 272\sqrt727 cm. What is its exact area?
Solution
Correct answer: D
The squared radius is 28 square centimeters. Use the apothem and perimeter relationship for the justified regular hexagon.
An equilateral triangle obtained from alternate hexagon vertices has area 483 cm248\sqrt3\text{ cm}^2483 cm2. What radius must the original given circle have?
Correct answer: B
Use the triangle area in terms of the positive supporting-circle radius.
A constructed inscribed square has perimeter 36236\sqrt2362 cm. A regular hexagon is then constructed in the same circle. What is the hexagon perimeter?
Correct answer: C
The square side is 9 times the square root of 2 cm, so the circle radius is 9 cm. The radius-stepped hexagon has six 9 cm sides.
A hexagon is constructed in a given circle. Its radius is (2x+5)(2x+5)(2x+5) cm, and a side of its alternate-vertex triangle is (4x−1)3(4x-1)\sqrt3(4x−1)3 cm. Find x.
Correct answer: A
The alternate-vertex triangle side is the circle radius times the square root of 3. Cancel that positive common factor.
The radius is then 11 cm, so the required positive-length condition holds.
In a square construction, the equal helper circles at diameter endpoints A and C use opening AC. They meet at U and V. If UV=183UV=18\sqrt3UV=183 cm, what is the constructed square's area?
Let the original radius be r. Each helper radius is 2r. The right triangle AOU gives OU equal to r times the square root of 3. Reflection across AC places V the same distance below O.
A square and an equilateral triangle are constructed in the same circle. The square area is 72 cm272\text{ cm}^272 cm2. What is the triangle area?
The square area is twice the squared circle radius. Use that squared radius in the triangle relationship.
The perimeter of a constructed inscribed regular hexagon is 102 cm. What is the area of the square constructed in the same circle?
A hexagon side equals its supporting-circle radius. Divide its perimeter by six before using the square's area relationship.
In a constructed hexagon ABCDEF, triangle ACE is formed. If AC=93AC=9\sqrt3AC=93 cm, what is the altitude from E to AC?
The triangle is equilateral. Its altitude is half a side times the square root of 3.
The shorter distance from O to AC is only one third of this full altitude.
How many different equilateral triangles have all three vertices among the vertices of a given regular hexagon? Count each set of three vertices once.
Equal triangle sides on the same circle subtend equal minor central angles. The three equal gaps must each span two hexagon steps, because the six steps must be shared equally.
The choices are ACE and BDF. Starting at a different vertex of either triangle does not produce a new set.
In the square construction with diameter AC = 10 cm, helper circles at A and C use opening AC and meet at U above AC. What is the ratio of the area of triangle AUC to the area of square ABCD?
Triangle AUC is equilateral because both helper radii equal AC. Its area is 25 times the square root of 3 square centimeters. The square has diagonal 10 cm, so its area is 50 square centimeters.
A regular hexagon and its alternate-vertex triangle are constructed in a circle of radius 11\sqrt{11}11 cm. What is the area inside the hexagon but outside the triangle?
The triangle lies inside the convex hexagon because its sides join hexagon vertices. Subtract the triangle area from the hexagon area, using the common squared radius 11.
A circular sheet has a constructed inscribed square opening with side 666\sqrt666 cm. The same sheet could instead have a constructed inscribed equilateral triangle opening. What would the triangle opening's perimeter be?
Use the square side to recover the circle radius, then derive the triangle side from its different center angle.
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