# SAT Math : Hexagons

## Example Questions

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### Example Question #1 : Hexagons

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 B

Point F

Point C

Point E

Point D

Point E

Explanation:

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

Explanation:

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 #2 : How To Find The Length Of The Diagonal Of A Hexagon

How many diagonals are there in a regular hexagon?

Explanation:

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 #3 : How To Find The Length Of The Diagonal Of A Hexagon

How many diagonals are there in a regular hexagon?

9

18

10

6

3

9

Explanation:

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 #3 : 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 .

Explanation:

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 #4 : 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?

Explanation:

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

170

210

190

200

180

190

Explanation:

### Example Question #11 : Hexagons

If a triangle has 180 degrees, what is the sum of the interior angles of a regular octagon?

Explanation:

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 #5 : Hexagons

Find the sum of all the inner angles in a hexagon.

Explanation:

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 #1 : How To Find The Area Of A Hexagon

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?