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Calculate Sum of Interior and Exterior Angles for a 16-Sided Polygon
Mathematics
Grade 8 (Junior High School)
Question Content
In a 16-sided polygon, what is the sum of a) the interior angles? b) the exterior angles?
Correct Answer
a) 2520 degrees; b) 360 degrees
Detailed Solution Steps
1
Step 1: Solve for the sum of interior angles. Use the formula for the sum of interior angles of an n-sided polygon: Sum = (n - 2) × 180°, where n is the number of sides. Here, n = 16.
2
Step 2: Substitute n = 16 into the formula: (16 - 2) × 180° = 14 × 180° = 2520°. This is the sum of the interior angles.
3
Step 3: Solve for the sum of exterior angles. Recall the polygon exterior angle theorem: the sum of the exterior angles of any convex polygon (regardless of the number of sides) is always 360°. So for a 16-sided polygon, the sum of exterior angles is 360°.
Knowledge Points Involved
1
Sum of Interior Angles of a Polygon
The formula (n - 2) × 180° calculates the total measure of all interior angles in an n-sided polygon. It comes from dividing the polygon into (n - 2) non-overlapping triangles, each with an interior angle sum of 180°. This applies to all simple, convex polygons.
2
Polygon Exterior Angle Theorem
This theorem states that the sum of the exterior angles of any convex polygon, when one exterior angle is taken at each vertex, is always 360°. This rule holds true no matter how many sides the polygon has, making it a constant value for all convex polygons.
3
Convex Polygon Definition
A convex polygon is a polygon where all interior angles are less than 180°, and all vertices point outward. The interior and exterior angle formulas for polygons are standardly applied to convex polygons, which is the default type of polygon in basic geometry problems.
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