# ACT Math : Other Polygons

## Example Questions

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### Example Question #1 : How To Find The Perimeter Of A Polygon

All segments of the polygon meet at right angles (90 degrees). The length of segment is 10. The length of segment is 8. The length of segment is 3. The length of segment is 2.

Find the perimeter of the polygon.

Explanation:

The perimeter of the polygon is 46. Think of this polygon as a rectangle with two of its corners "flipped" inwards. This "flipping" changes the area of the rectangle, but not its perimeter; therefore, the top and bottom sides of the original rectangle would be 12 units long . The left and right sides would be 11 units long . Adding all four sides, we find that the perimeter of the recangle (and therefore, of this polygon) is 46.

### Example Question #2 : How To Find The Perimeter Of A Polygon

In the figure below, each pair of intersecting line segments forms a right angle, and all the lengths are given in feet. What is the perimeter, in feet, of the figure?

Explanation:

Fill in the missing sides by thinking about the entire figure as a big rectangle. In the figure below, the large rectangle is outlined in blue and the missing numbers are supplied in red.

### Example Question #3 : How To Find The Perimeter Of A Polygon

What is the perimeter of a regular,  polygon with a side length of ?

Explanation:

To find the perimeter of an -sided polygon with a side length of , simply multiply the side length by the number of sides:

### Example Question #1 : Other Polygons

Polygon  is a regular nine-sided polygon, or nonagon, with perimeter 500. Which choice comes closest to the length of diagonal ?

Explanation:

Congruent sides , and , and the diagonal  form an isosceles trapezoid.

and . being angles of a nine-sided regular polygon, have measure

The other two angles are supplementary to these:

The length of one side of the nonagon is one-ninth of 500, so

The trapezoid formed is below (figure NOT drawn to scale):

Altitudes  and  to the base have been drawn, so

This makes 140 the best choice.

### Example Question #2 : Other Polygons

Polygon  is a regular seven-sided polygon, or heptagon, with perimeter 500. Which choice comes closest to the length of diagonal ?

Explanation:

Congruent sides , and , and the diagonal  form an isosceles trapezoid.

and . being angles of a seven-sided regular polygon, have measure

The other two angles are supplementary to these:

The length of one side is one-seventh of 500, so

The trapezoid formed is below (figure NOT drawn to scale):

Altitudes  and  to the base have been drawn, so

This makes 160 the best choice.

### Example Question #3 : Other Polygons

Polygon  is a regular seven-sided polygon, or heptagon, with perimeter 500. Which choice comes closest to the length of diagonal ?

Explanation:

Congruent sides  and  and the diagonal  form an isosceles triangle.

, being an angle of a seven-sided regular polygon, has measure

The other two are congruent, and each has measure

The length of one side is one-seventh of 500, or

can be found using the Law of Sines:

Of the given choices, 130 comes closest.

### Example Question #4 : Other Polygons

Polygon  is a regular nine-sided polygon, or nonagon, with perimeter 500. Which choice comes closest to the length of diagonal ?

Explanation:

Congruent sides  and  and the diagonal  form an isosceles triangle.

, being an angle of a nine-sided regular polygon, has measure

The other two are congruent, and each has measure

The length of one side is one-ninth of 500, or

can be found using the Law of Sines:

Of the given choices, 105 comes closest.

### Example Question #5 : Other Polygons

What is the side length of a 12-sided polygon with a perimeter of

Explanation:

To find the side length, , of an -sided polygon with a perimeter of .

Use the formula:

### Example Question #1 : How To Find An Angle In A Polygon

In the diagram below, what is  equal to?

Explanation:

The figure given is a hexagon with an embedded triangle. The fact that it is embedded in a triangle is mainly to throw you off, as it has little to no consequence on the correct answer. Of the available answer choices, you must choose a relationship that would give the value of Tangent describes the relationship between an angle and the opposite and adjacent sides of that angle. Or in other words, tan = opposite side/adjacent side. However, when solving for an angle, we must use the inverse function. Therefore, if we know the opposite and adjacent sides are, we can use the inverse of the tangent, or arctangent (tan-1), of  to find .

Thus,

### Example Question #2 : How To Find An Angle In A Polygon

What is the value of angle  in the figure above?