Circular Sector Problems

Category: Plane Geometry

"Published in Vacaville, California, USA"

A sector in a circle of radius 25 ft has an area of 125 ft². Find the central angle of the sector.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

The area of a circular sector is given by the formula

where A is the area of a circular sector, R is the radius of a circular sector, and θ is the central angle of circular sector in radians. Radians is a unit less value of an angle.

Substitute the values of A and R in order to solve for the value of central angle, as follows

You can also express the value of central angle in degrees, as follows

or

Regular Polygon Problems

Category: Plane Geometry

"Published in Vacaville, California, USA"

Find the perimeter of a regular hexagon that is inscribed in a circle of radius 8 m.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

If you will draw the diagonals that passes through the center of a circle, there are six triangles in a regular hexagon as follows

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The center of a circle bisects the three diagonals of a regular hexagon. The bisected diagonals are equal to the radius of a circle which is 8 m. The vertex angle of each triangles can be calculated as follows

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The triangles in hexagon are isosceles triangles because the two sides of each triangles are congruent which is 8 m. If the two sides of an isosceles triangle are congruent, then its base angles are congruent also. We can calculate the base angle of an isosceles triangle as follows

Since the base angles of an isosceles triangle are the same as the vertex angle which is 60°, then the isosceles triangle is an equiangular or equilateral triangle. Hence, the sides of an hexagon is x = 8 m. 

Therefore, the perimeter of a regular hexagon is

Circle - Triangle Problems

Category: Trigonometry, Plane Geometry

"Published in Vacaville, California, USA"

Express the lengths a and b in the figure in terms of the trigonometric ratios of θ. 

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

The given figure consists of a circle and a triangle. One side of a triangle is equal to the radius of a circle which is 1. The other side of a triangle which is a is tangent to a circle at the y-axis. If a is perpendicular to the other side of a triangle and to the y-axis, then it follows that the given triangle is a right triangle.

Photo by Math Principles in Everyday Life

   
If a is perpendicular to y-axis, then it follows that a is parallel to x-axis. If b is a transversal line that passes thru the parallel lines a and the x-axis, then it follows that the alternating interior angles are congruent. In this case, the angle formed by lines b and the x-axis is congruent to the angle formed by lines a and b which is θ.

Since we know the side and the opposite angle of a right triangle, then we can get the values of a and b in terms of trigonometric functions of θ as follows