Practice
Regular Polygons Practice
Fifty original questions on regular-polygon definitions, angles, apothems, perimeters, central triangles, and areas.
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Question 1
Explanation
Regularity requires both equal sides and equal angles; radius and apothem have different endpoints.
- Reasoning
Identify the endpoint and right-angle condition.
Question 2
Explanation
Regularity requires both equal sides and equal angles; radius and apothem have different endpoints.
- Reasoning
Identify the endpoint and right-angle condition.
Question 3
Explanation
Regularity requires both equal sides and equal angles; radius and apothem have different endpoints.
- Reasoning
Identify the endpoint and right-angle condition.
Question 4
Explanation
Regularity requires both equal sides and equal angles; radius and apothem have different endpoints.
- Reasoning
Identify the endpoint and right-angle condition.
Question 5
Explanation
Use \((n-2)180^\circ=(5-2)180^\circ=540^\circ\).
- Reasoning
Do not divide by n when the question asks for the sum.
Question 6
Explanation
Use \((n-2)180^\circ=(6-2)180^\circ=720^\circ\).
- Reasoning
Do not divide by n when the question asks for the sum.
Question 7
Explanation
Use \((n-2)180^\circ=(7-2)180^\circ=900^\circ\).
- Reasoning
Do not divide by n when the question asks for the sum.
Question 8
Explanation
Use \((n-2)180^\circ=(8-2)180^\circ=1080^\circ\).
- Reasoning
Do not divide by n when the question asks for the sum.
Question 9
Explanation
Use \((n-2)180^\circ=(9-2)180^\circ=1260^\circ\).
- Reasoning
Do not divide by n when the question asks for the sum.
Question 10
Explanation
Use \((n-2)180^\circ=(10-2)180^\circ=1440^\circ\).
- Reasoning
Do not divide by n when the question asks for the sum.
Question 11
Explanation
Use \((n-2)180^\circ=(12-2)180^\circ=1800^\circ\).
- Reasoning
Do not divide by n when the question asks for the sum.
Question 12
Explanation
Use \((n-2)180^\circ=(15-2)180^\circ=2340^\circ\).
- Reasoning
Do not divide by n when the question asks for the sum.
Question 13
Explanation
Divide the interior sum by n: \(\frac{(3-2)180^\circ}{3}=60^\circ\).
- Reasoning
Use regularity before dividing the sum equally.
Question 14
Explanation
Divide the interior sum by n: \(\frac{(4-2)180^\circ}{4}=90^\circ\).
- Reasoning
Use regularity before dividing the sum equally.
Question 15
Explanation
Divide the interior sum by n: \(\frac{(5-2)180^\circ}{5}=108^\circ\).
- Reasoning
Use regularity before dividing the sum equally.
Question 16
Explanation
Divide the interior sum by n: \(\frac{(6-2)180^\circ}{6}=120^\circ\).
- Reasoning
Use regularity before dividing the sum equally.
Question 17
Explanation
Divide the interior sum by n: \(\frac{(8-2)180^\circ}{8}=135^\circ\).
- Reasoning
Use regularity before dividing the sum equally.
Question 18
Explanation
Divide the interior sum by n: \(\frac{(9-2)180^\circ}{9}=140^\circ\).
- Reasoning
Use regularity before dividing the sum equally.
Question 19
Explanation
Divide the interior sum by n: \(\frac{(10-2)180^\circ}{10}=144^\circ\).
- Reasoning
Use regularity before dividing the sum equally.
Question 20
Explanation
Divide the interior sum by n: \(\frac{(12-2)180^\circ}{12}=150^\circ\).
- Reasoning
Use regularity before dividing the sum equally.
Question 21
Explanation
Both quantities equal \(360^\circ/n=360^\circ/5=72^\circ\).
- Reasoning
Central and regular exterior angles share the same 360/n measure.
Question 22
Explanation
Both quantities equal \(360^\circ/n=360^\circ/6=60^\circ\).
- Reasoning
Central and regular exterior angles share the same 360/n measure.
Question 23
Explanation
Both quantities equal \(360^\circ/n=360^\circ/8=45^\circ\).
- Reasoning
Central and regular exterior angles share the same 360/n measure.
Question 24
Explanation
Both quantities equal \(360^\circ/n=360^\circ/9=40^\circ\).
- Reasoning
Central and regular exterior angles share the same 360/n measure.
Question 25
Explanation
Both quantities equal \(360^\circ/n=360^\circ/10=36^\circ\).
- Reasoning
Central and regular exterior angles share the same 360/n measure.
Question 26
Explanation
Both quantities equal \(360^\circ/n=360^\circ/12=30^\circ\).
- Reasoning
Central and regular exterior angles share the same 360/n measure.
Question 27
Explanation
The exterior angles total 360 degrees, so \(n=360/72=5\).
- Reasoning
Divide 360 by one regular exterior or central angle.
Question 28
Explanation
The exterior angles total 360 degrees, so \(n=360/60=6\).
- Reasoning
Divide 360 by one regular exterior or central angle.
Question 29
Explanation
The exterior angles total 360 degrees, so \(n=360/45=8\).
- Reasoning
Divide 360 by one regular exterior or central angle.
Question 30
Explanation
The exterior angles total 360 degrees, so \(n=360/40=9\).
- Reasoning
Divide 360 by one regular exterior or central angle.
Question 31
Explanation
The exterior angles total 360 degrees, so \(n=360/36=10\).
- Reasoning
Divide 360 by one regular exterior or central angle.
Question 32
Explanation
The exterior angles total 360 degrees, so \(n=360/30=12\).
- Reasoning
Divide 360 by one regular exterior or central angle.
Question 33
Explanation
A regular hexagon central triangle is equilateral. Splitting it gives a 30-60-90 triangle with half-side 1, so \(a=1\sqrt3\).
- Reasoning
Use half the side as the short leg of the central 30-60-90 triangle.
Question 34
Explanation
A regular hexagon central triangle is equilateral. Splitting it gives a 30-60-90 triangle with half-side 2, so \(a=2\sqrt3\).
- Reasoning
Use half the side as the short leg of the central 30-60-90 triangle.
Question 35
Explanation
A regular hexagon central triangle is equilateral. Splitting it gives a 30-60-90 triangle with half-side 3, so \(a=3\sqrt3\).
- Reasoning
Use half the side as the short leg of the central 30-60-90 triangle.
Question 36
Explanation
A regular hexagon central triangle is equilateral. Splitting it gives a 30-60-90 triangle with half-side 4, so \(a=4\sqrt3\).
- Reasoning
Use half the side as the short leg of the central 30-60-90 triangle.
Question 37
Explanation
A regular hexagon central triangle is equilateral. Splitting it gives a 30-60-90 triangle with half-side 5, so \(a=5\sqrt3\).
- Reasoning
Use half the side as the short leg of the central 30-60-90 triangle.
Question 38
Explanation
A regular hexagon central triangle is equilateral. Splitting it gives a 30-60-90 triangle with half-side 6, so \(a=6\sqrt3\).
- Reasoning
Use half the side as the short leg of the central 30-60-90 triangle.
Question 39
Explanation
Perimeter is \(p=6(8)=48\). Then \(A=\tfrac12ap=\tfrac12(5)(48)=120\).
- Reasoning
Find the full perimeter before using one-half apothem times perimeter.
Question 40
Explanation
Perimeter is \(p=5(10)=50\). Then \(A=\tfrac12ap=\tfrac12(7)(50)=175\).
- Reasoning
Find the full perimeter before using one-half apothem times perimeter.
Question 41
Explanation
Perimeter is \(p=8(6)=48\). Then \(A=\tfrac12ap=\tfrac12(9)(48)=216\).
- Reasoning
Find the full perimeter before using one-half apothem times perimeter.
Question 42
Explanation
Perimeter is \(p=10(5)=50\). Then \(A=\tfrac12ap=\tfrac12(12)(50)=300\).
- Reasoning
Find the full perimeter before using one-half apothem times perimeter.
Question 43
Explanation
Perimeter is \(p=12(4)=48\). Then \(A=\tfrac12ap=\tfrac12(14)(48)=336\).
- Reasoning
Find the full perimeter before using one-half apothem times perimeter.
Question 44
Explanation
Perimeter is \(p=6(12)=72\). Then \(A=\tfrac12ap=\tfrac12(15)(72)=540\).
- Reasoning
Find the full perimeter before using one-half apothem times perimeter.
Question 45
Explanation
The side is \(96/8=12\). Area is \(\tfrac12(14)(96)=672\).
- Reasoning
Use n to recover side, but use the full given perimeter in area.
Question 46
Explanation
The half-central angle is \(180^\circ/5=36^\circ\). In the central right triangle, \(\cos36^\circ=a/10\).
- Reasoning
Use half the central angle and place apothem adjacent to it.
Question 47
Explanation
The expression gives the sum for all interior angles; regularity allows that sum to be divided equally among n angles.
- Reasoning
Distinguish a total from one equal share.
Question 48
Explanation
The formula sums the areas of all n central triangles, so it requires the entire boundary length.
- Reasoning
Expand perimeter from all congruent sides.
Question 49
Explanation
The corresponding exterior angle is \(180-150=30^\circ\). Thus \(n=360/30=12\), and the central angle is also 30 degrees.
- Reasoning
Convert interior to exterior, then use 360 divided by one exterior angle.
Question 50
Explanation
A regular hexagon side equals its radius, so \(s=10\), \(p=60\), and \(a=5\sqrt3\). Therefore \(A=\tfrac12(5\sqrt3)(60)=150\sqrt3\).
- Reasoning
Use the equilateral central triangles before applying the area formula.
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Questions to review
No mistakes this time. Excellent work.
- Question 1Regular-polygon vocabularyEasy
- Question 2Regular-polygon vocabularyEasy
- Question 3Regular-polygon vocabularyEasy
- Question 4Regular-polygon vocabularyEasy
- Question 5Interior-angle sumEasy
- Question 6Interior-angle sumEasy
- Question 7Interior-angle sumEasy
- Question 8Interior-angle sumEasy
- Question 9Interior-angle sumEasy
- Question 10Interior-angle sumEasy
- Question 11Interior-angle sumEasy
- Question 12Interior-angle sumEasy
- Question 13Each regular interior angleEasy
- Question 14Each regular interior angleEasy
- Question 15Each regular interior angleEasy
- Question 16Each regular interior angleMedium
- Question 17Each regular interior angleMedium
- Question 18Each regular interior angleMedium
- Question 19Each regular interior angleMedium
- Question 20Each regular interior angleMedium
- Question 21Central and exterior anglesMedium
- Question 22Central and exterior anglesMedium
- Question 23Central and exterior anglesMedium
- Question 24Central and exterior anglesMedium
- Question 25Central and exterior anglesMedium
- Question 26Central and exterior anglesMedium
- Question 27Number of sidesMedium
- Question 28Number of sidesMedium
- Question 29Number of sidesMedium
- Question 30Number of sidesMedium
- Question 31Number of sidesMedium
- Question 32Number of sidesMedium
- Question 33Apothem from a special triangleMedium
- Question 34Apothem from a special triangleMedium
- Question 35Apothem from a special triangleMedium
- Question 36Apothem from a special triangleMedium
- Question 37Apothem from a special triangleMedium
- Question 38Apothem from a special triangleMedium
- Question 39Regular-polygon areaMedium
- Question 40Regular-polygon areaMedium
- Question 41Regular-polygon areaHard
- Question 42Regular-polygon areaHard
- Question 43Regular-polygon areaHard
- Question 44Regular-polygon areaHard
- Question 45Perimeter and areaHard
- Question 46Trigonometric apothemHard
- Question 47Interior-angle error analysisHard
- Question 48Regular-area error analysisHard
- Question 49Angle synthesisHard
- Question 50Regular-hexagon synthesisHard