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Regular Polygon Area Help

— McGraw-Hill Professional
Updated on Oct 3, 2011

Trigonometry Overview - Finding the Interior Area of a Regular Polygon

Some of the following rules involve trigonometry. This branch of mathematics has an undeserved bad reputation among some students. According to various rumors, trigonometry is esoteric (this is not true), is inherently incomprehensible (also not true), was dreamed up by sadistic theoreticians with the intent of confusing people of ordinary intelligence (unlikely, but no one knows for sure), and is peppered with Greek symbology (well, yes). There are six trigonometric functions, also known as circular functions . They are the sine, cosine, tangent, cosecant, secant , and cotangent. All six of these functions produce specific numbers at their “outputs” when certain angular measures are fed into their “inputs.”

We won’t concern ourselves here with formal definitions of the circular functions, or how they are derived. All you need to know in order to use the following rules is how to use the sine and cosine function keys on a calculator. The sine of an angle is found by entering the angle’s measure in degrees or radians into a calculator, and then hitting the “sine” or “sin” function key. The cosine is found by entering the angle’s measure in degrees or radians and then hitting “cosine” or “cos.” Some calculators have a “tangent” or “tan” function key, and others don’t. If your calculator doesn’t have a tangent key, the tangent of an angle can be found by dividing its sine by its cosine. Many calculators lack a “cotangent” or “cot” key, but the cotangent of an angle is equal to the reciprocal of its tangent, or the cosine divided by the sine.

Interior Area Of Regular Polygon

Let V be a regular, n -sided polygon, each of whose sides have length s as defined above and in Fig. 4-8. The interior area, A , is given by the following formula if angles are specified in degrees:

A = ( ns 2 /4) cot (180/ n )

If angles are specified in radians, then:

A = ( ns 2 /4) cot ( π / n )

Regular Polygon Area Practice Problems

PROBLEM 1

What is the interior area of a regular, 10-sided polygon, each of whose sides is exactly 2 units long? Express your answer to two decimal places.

SOLUTION 1

In this case, n = 10 and s = 2. Let’s use degrees for the angles. Then we can plug our values of n and s into the first formula, above, getting this:

A = (10 × 2 2 ) cot (180/10)

= (10 × 4/4) cot 18

= 10 cot 18

= 10 cos 18/sin 18

= 10 × 0.951057/0.309017

= 10 × 3.07769

= 30.7769

= 30.78 square units (to two decimal places)

In order to obtain an answer to two decimal places, it’s best to use five or six decimal places throughout the calculation, rounding off only at the end. This will ensure that cumulative errors are kept to a minimum.

PROBLEM 2

What is the interior area of a regular, 100-sided polygon, each of whose sides is exactly 0.20 units long? Express your answer to two decimal places.

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