Question

Find the average angle measure and weighted average angle measure across all the angles of the following polygons: 1 rectangle, 2 pentagons, and 3 hexagons. The average is 106, but 111 for the weighted average is wrong.

a) 106°

b) 111°

c) 115°

d) 120°

Answer 1

The average angle measure across all the polygons is 106°. The weighted average angle measure is 1.

Step-by-step explanation:

To find the average angle measure across all the polygons, we need to add up the angle measures of each polygon and divide it by the total number of polygons. The rectangle has 4 angles of 90° each, the pentagons have 5 angles of 108° each, and the hexagons have 6 angles of 120° each. So, the total angle measure for all the polygons is (4 * 90°) + (2 * 5 * 108°) + (3 * 6 * 120°). Divide this by the total number of polygons, which is 1 (rectangle) + 2 (pentagons) + 3 (hexagons) = 6. The average angle measure is therefore (4 * 90° + 2 * 5 * 108° + 3 * 6 * 120°) / 6 = 106°.

For the weighted average angle measure, we need to consider the number of angles of each polygon. The rectangle has 4 angles, each worth 1, so it contributes 4 * 1 = 4 to the weighted total. The pentagons have 5 angles each, worth 2, so they contribute 2 * 5 * 2 = 20 to the weighted total. The hexagons have 6 angles each, worth 3, so they contribute 3 * 6 * 3 = 54 to the weighted total. Add these values together and divide by the total number of angles, which is 4 + 20 + 54 = 78. The weighted average angle measure is therefore (4 + 20 + 54) / 78 = 78 / 78 = 1.