Find the measure of an interior angle of a regular hendecagon (11-sided polygon).
[tex]x=[?]^{\circ}[/tex]
Hint: Sum [tex]=(n-2) 180[/tex]
Round to the nearest tenth.
Answer (2)
Calculate the sum of the interior angles of the hendecagon using the formula S u m = ( n − 2 ) × 18 0 ∘ , where n = 11 , resulting in S u m = 162 0 ∘ .
Divide the sum of the interior angles by the number of sides to find the measure of one interior angle: x = 11 162 0 ∘ = 147.2727.. . ∘ .
Round the result to the nearest tenth: x ≈ 147. 3 ∘ .
The measure of an interior angle of a regular hendecagon is 147. 3 ∘ .
Explanation
Problem Analysis We are asked to find the measure of an interior angle of a regular hendecagon (11-sided polygon). We are given the formula for the sum of the interior angles of a polygon with n sides: S u m = ( n − 2 ) × 18 0 ∘ . Since the hendecagon is regular, all its interior angles are equal. Therefore, we can find the measure of one interior angle by dividing the sum of the interior angles by the number of sides, which is 11.
Calculate the Sum of Interior Angles First, we calculate the sum of the interior angles of the hendecagon using the formula: S u m = ( n − 2 ) × 18 0 ∘ where n = 11 .
S u m = ( 11 − 2 ) × 18 0 ∘ = 9 × 18 0 ∘ = 162 0 ∘ So, the sum of the interior angles of the hendecagon is 162 0 ∘ .
Calculate the Measure of One Interior Angle Next, we find the measure of one interior angle by dividing the sum of the interior angles by the number of sides: x = n S u m = 11 162 0 ∘ = 147.272727.. . ∘ where x is the measure of one interior angle.
Round to the Nearest Tenth Finally, we round the result to the nearest tenth: x ≈ 147. 3 ∘ Therefore, the measure of an interior angle of a regular hendecagon is approximately 147. 3 ∘ .
Examples
Understanding the angles in regular polygons like hendecagons is useful in architecture and design. For example, if you're designing a building with a hendecagonal base, knowing the interior angle helps ensure that the structure is stable and aesthetically pleasing. Also, in creating tessellations or patterns, knowing the angles of regular polygons is crucial for ensuring that the shapes fit together without gaps or overlaps.
The measure of an interior angle of a regular hendecagon is approximately 147.3 degrees. This is calculated by first determining the sum of its interior angles, which is 1620 degrees, and then dividing by the number of sides, 11. After rounding, we find that each interior angle measures about 147.3 degrees.
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