What is the measure of each angle of a regular decagon? If necessary, round to the nearest tenth.
The Correct Answer and Explanation is:
To find the measure of each interior angle of a regular decagon, we follow this method:
Step-by-step Calculation:
A regular decagon has 10 equal sides and 10 equal angles.
Step 1: Use the formula for the sum of interior angles of a polygon:
Sum of interior angles=(n−2)×180∘\text{Sum of interior angles} = (n – 2) \times 180^\circSum of interior angles=(n−2)×180∘
Where n is the number of sides. For a decagon:(10−2)×180∘=8×180∘=1440∘(10 – 2) \times 180^\circ = 8 \times 180^\circ = 1440^\circ(10−2)×180∘=8×180∘=1440∘
Step 2: Divide the sum by the number of angles to find each interior angle:
Each interior angle=1440∘10=144∘\text{Each interior angle} = \frac{1440^\circ}{10} = 144^\circEach interior angle=101440∘=144∘
So, each interior angle of a regular decagon measures 144 degrees.
Full Explanation
A regular polygon is a shape where all sides and all angles are equal. In the case of a regular decagon, there are 10 sides and 10 equal angles. To find how large each angle is, we begin by finding the total measure of all the interior angles combined.
The general rule for the sum of interior angles in any polygon is to multiply 180 degrees by the number of triangles that can be drawn within the shape. This number is always two less than the number of sides. So, for a decagon which has 10 sides, we subtract 2 and get 8. We then multiply 8 by 180 to get the total angle sum of 1440 degrees.
Since all 10 angles are equal in a regular decagon, we divide this sum by 10. Doing so gives 144 degrees for each angle.
This makes sense geometrically as well. In a regular polygon, each interior angle must be less than 180 degrees so the shape can close in on itself and form a complete two-dimensional figure. An angle of 144 degrees fits this requirement and ensures all angles add up properly to 1440 degrees.
Therefore, the measure of each angle of a regular decagon is 144 degrees, with no rounding needed.
