What are the measures of each interior and exterior angles of regular quadrilateral MNOP?
1 answer:
Answer:
A regular polygon is a polygon that is equilateral and equiangular.
The only regular quadrilateral is the square.
We know that all the interior angles of a square are 90° each, so we are fine with that.
Now, for the exterior angles.
If we fix our position in the vertex, we can do from one side, to the same side, a full rotation of 360°.
Now, if we only do an external rotation, this means, from one side to other side, we will do a rotation of 360° minus the interior angle:
this is:
360° - 90° = 270°
Below is an image as a guide of this.
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Answer:
The length of one side of the octagon is 7.65 cm
Step-by-step explanation:
The parameters given are;
A regular octagon inscribed in a circle of radius, r, of 10 cm.
The length of each side is found from the isosceles triangle formed by the radius and one side of the octagon
The sum of interior angles in a polygon, ∑θ = 180 × (n - 2)
Where;
n = The number of sides of the polygon
θ = The interior angle of the polygon
For the octagon, we have;
n = 8, therefore;
∑θ = 180 × (8 - 2) = 1080
Given that there are eight equal angles in a regular octagon, we have;
∑θ = 8 × θ
= 1080
θ = 1080/8 = 135°
The sum of angles at the center of the circle = 360
Therefore, the angle at the center (tip angle) of the isosceles triangle formed by the radius and one side of the octagon = 360/8 = 45°
The base angles of the isosceles triangle is therefore, (180 - 45)/2 = 67.5° = θ/2
The length of the base of the isosceles triangle formed by the radius and one side of the octagon = The length of one side of the octagon
From trigonometric ratios, the length of the base of the isosceles triangle is therefore;
2 × r × cos(θ/2) = 2×10 × cos(67.5°) = 7.65 cm
The length of the base of the isosceles triangle = 7.65 cm = The length of one side of the octagon.