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If the interior angles of a pentagon are in the ratio 1:2:3:4:5, find the smallest angle.
36°
40°
45°
50°
36°
Sum of ratios is 1+2+3+4+5=15. Since the sum of interior angles is 540°, the smallest part is 15540=36°.
A pentagon has interior angles in the ratio 1:2:3:4:5.
(n−2)×180°
Sum of ratios is 1+2+3+4+5=15. Since the sum of interior angles is 540°, the smallest part is 15540=36°.
Students often confuse the sum of interior angles (540°) with the sum of exterior angles (360°), leading to incorrect proportionality constants.
Calculate the sum of interior angles
For a polygon with n=5 sides, the sum of interior angles is given by (n−2)×180°.
(5−2)×180°=3×180°=540°
Set up the ratio equation
Let the angles be x,2x,3x,4x,5x. The sum of these parts equals the total sum calculated.
x+2x+3x+4x+5x=540°
Solve for x
Sum the coefficients to get 15x=540°, then divide by 15.
x=15540°=36°
A is correct because the smallest angle corresponds to 1 part of the total 15 parts, resulting in 36°.
This ratio method applies to any polygon; for a hexagon, use n=6 to get a total of 720° and proceed similarly.
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