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Example Questions
Example Question #11 : Geometry
Archimedes High School has an unusual track in that it is shaped like a regular hexagon, as above. Each side of the hexagon measures 264 feet.
Alvin runs at a steady speed of seven miles an hour for twelve minutes, starting at point A and working his way clockwise. When he is finished, which of the following points is he closest to?
Point D
Point E
Point B
Point C
Point F
Point E
Alvin runs at a rate of seven miles an hour for twelve minutes, or hours. The distance he runs is equal to his rate multiplied by his time, so, setting in this formula:
miles.
One mile comprises 5,280 feet, so this is equal to
feet
Since each side of the track measures 264 feet, this means that Alvin runs
sidelengths.
,
which means that Alvin runs around the track four complete times, plus four more sides of the track. Alvin stops when he is at Point E.
Example Question #2 : Hexagons
A circle with circumference is inscribed in a regular hexagon. Give the perimeter of the hexagon.
None of these
Below is the figure referenced; note that the hexagon is divided by its diameters, and that an apothem—a perpendicular bisector from the center to one side—has been drawn.
The circle has circumference ; its radius, which coincides with the apothem of the hexagon, is the circumference divided by :
The hexagon is divided into six equilateral triangles. One, , is divided by an apothem of the hexagon - a radius of the circle - into two 30-60-90 triangles, one of which is . Since has length 30, and it is a long leg of , then short leg has length
is the midpoint of , one of the six congruent sides of the hexagon, so
;
this makes the perimeter of the hexagon six times this, or
.
Example Question #1 : Hexagons
How many diagonals are there in a regular hexagon?
A diagonal is a line segment joining two non-adjacent vertices of a polygon. A regular hexagon has six sides and six vertices. One vertex has three diagonals, so a hexagon would have three diagonals times six vertices, or 18 diagonals. Divide this number by 2 to account for duplicate diagonals between two vertices. The formula for the number of vertices in a polygon is:
where .
Example Question #12 : Geometry
How many diagonals are there in a regular hexagon?
10
9
3
6
18
9
A diagonal connects two non-consecutive vertices of a polygon. A hexagon has six sides. There are 3 diagonals from a single vertex, and there are 6 vertices on a hexagon, which suggests there would be 18 diagonals in a hexagon. However, we must divide by two as half of the diagonals are common to the same vertices. Thus there are 9 unique diagonals in a hexagon. The formula for the number of diagonals of a polygon is:
where n = the number of sides in the polygon.
Thus a pentagon thas 5 diagonals. An octagon has 20 diagonals.
Example Question #2 : Hexagons
Hexagon is a regular hexagon with sides of length 10. is the midpoint of . To the nearest tenth, give the length of the segment .
Below is the referenced hexagon, with some additional segments constructed.
Note that the segments and have been constructed. Along with , they form right triangle with hypotenuse .
is the midpoint of , so
.
has been divided by drawing the perpendicular from to the segment and calling the point of intersection . is a 30-60-90 triangle with hypotenuse , short leg , and long leg , so by the 30-60-90 Triangle Theorem,
and
For the same reason, , so
By the Pythagorean Theorem,
when rounded to the nearest tenth.
Example Question #2 : Hexagons
The provided image represents a track in the shape of a regular hexagon with perimeter one fourth of a mile.
Teresa starts at Point A and runs clockwise until she gets halfway between Point E and Point F. How far does she run, in feet?
One mile is equal to 5,280 feet; one fourth of a mile is equal to
Each of the six congruent sides measures one sixth of this, or
Teresa runs clockwise from Point A to halfway between Point E and Point F, so she runs along four and one half sides, for a total of
Example Question #1 : How To Find An Angle In A Hexagon
210
190
200
180
170
Example Question #1 : Hexagons
If a triangle has 180 degrees, what is the sum of the interior angles of a regular octagon?
The sum of the interior angles of a polygon is given by where = number of sides of the polygon. An octagon has 8 sides, so the formula becomes
Example Question #1 : Hexagons
Find the sum of all the inner angles in a hexagon.
To solve, simply use the formula to find the total degrees inside a polygon, where n is the number of vertices.
In this particular case, a hexagon means a shape with six sides and thus six vertices.
Thus,
Example Question #22 : Geometry
An equilateral triangle with side length has one of its vertices at the center of a regular hexagon, and the side opposite that vertex is one of the sides of the hexagon. What is the hexagon's area?
Because it can be split into two triangles, the area of an equilateral triangle can be expressed as...
With that in mind, the equilateral triangle in question has area of .
Now consider that a regular hexagon can be split into six congruent equilateral triangles with a vertex at the center and the side opposite the center as one of the hexagon's sides (a handy way of finding a hexagon's area if you can't use the regular polygon formula requiring an apothem.) Knowing that, our answer is .
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