A regular hexagon (ABCDEF) shape of city center in Egypt with sides all of lenght 4 km and with center at 0 (letter). The interior angles of the hexagon are all equal. The interior angles of a polygon with n equal sides and n equal angles (regular polygon) add up to (n-2) x 180 degrees.
a.) Calculate the size of angle ABC.
b.) Given that OB=OC=BC, find the area of the triangle OBC
c.) Find the area of the whole city center.
(a) since there are 6 of them, 4*180/6 = 120
(b) Draw the diagram, and you will see lots o f 30,60,90-degree angles
(c) there are 6 equilateral triangles of side 4. You know how to find the area of an equilateral triangle, right?
a.) To calculate the size of angle ABC, we can use the formula for the sum of interior angles of a regular polygon:
Sum of interior angles = (n-2) x 180 degrees
For a regular hexagon, n (number of sides) is 6, so:
Sum of interior angles = (6-2) x 180 degrees
= 4 x 180 degrees
= 720 degrees
Since all interior angles of a regular hexagon are equal, we divide the sum by 6 to find the size of angle ABC:
720 degrees / 6 = 120 degrees
Therefore, the size of angle ABC is 120 degrees.
b.) Given that OB=OC=BC, we know that triangle OBC is an equilateral triangle. In an equilateral triangle, all three angles are equal to 60 degrees.
Hence, the size of angle OBC (or BOC) is 60 degrees.
To find the area of triangle OBC, we can use the formula for the area of an equilateral triangle:
Area = (side length^2 x √3) / 4
Since the side length is given as 4 km, we can substitute this value into the formula:
Area = (4^2 x √3) / 4
= (16 x √3) / 4
= 4√3 square km
Therefore, the area of triangle OBC is 4√3 square km.
c.) Since the city center is in the shape of a regular hexagon, we can divide it into six equilateral triangles, each with side length 4 km.
The area of one equilateral triangle is 4√3 square km, as we calculated in part b.
So, to find the area of the whole city center, we multiply the area of one triangle by 6:
Area of city center = 4√3 square km x 6
= 24√3 square km
Therefore, the area of the whole city center is 24√3 square km.
To find the answers to the questions, we can follow these steps:
a.) To calculate the size of angle ABC, we need to know the formula for finding the measure of each interior angle of a regular hexagon. The formula is:
Interior angle = (n - 2) x 180 / n
Since we have a regular hexagon (a polygon with 6 equal sides and angles), we can substitute n = 6 into the formula:
Interior angle = (6 - 2) x 180 / 6
= 4 x 180 / 6
= 120 degrees
Therefore, the size of angle ABC is 120 degrees.
b.) To find the area of triangle OBC, we need to know the length of one side (OB, OC, or BC). Since OB = OC = BC (as given in the question), we can choose any of these sides to calculate the area. Let's choose OB.
Using the formula for the area of an equilateral triangle (a special case of an isosceles triangle):
Area = (side length^2 * √3) / 4
Substituting the side length, which is 4 km, into the formula:
Area = (4^2 * √3) / 4
= (16 * √3) / 4
= 4√3 km²
Therefore, the area of triangle OBC is 4√3 km².
c.) To find the area of the whole city center, we can calculate the area of the regular hexagon. The area of a regular polygon can be found using the formula:
Area = (n * side length^2) / (4 * tan(π/n))
Substituting n = 6 and the side length = 4 km into the formula:
Area = (6 * 4^2) / (4 * tan(π/6))
= (6 * 16) / (4 * (√3 / 3)
= 96 / (√3 / 3)
= 96 * (3 / √3)
= 96 * (√3 / 1)
= 96√3 km²
Therefore, the area of the whole city center is 96√3 km².