A regular heptagon has a side length of 36 centimeters. Find the area giving the answer to two decimal places.
In this question, we need to find the area of a regular heptagon which has a side length of 36 centimeters. And we need to give our answer to two decimal places. Since this question is asking us to find the area of a regular polygon, we can do this by using our formula for finding the area of regular polygons. We recall the area of an 𝑛-sided regular polygon with a side length of 𝑥 is given by the following formula. Its area is equal to 𝑛𝑥 squared over four multiplied by the cot of 180 divided by 𝑛.
And there’s a few things worth pointing out about this formula. First, we’re taking our angle in degrees. And secondly, units of our area is going to be the square of whichever units we use for our length. So in our question, since we’re taking the length in centimeters, our area is going to be in centimeters squared. So to find the area of any regular polygon, we only need to know two things. We need to know how many sides our polygon has, and we need to know one of the lengths of the sides.
First, in the question, we’re told we’re working with a regular heptagon. And a heptagon is any polygon with seven sides, so our value of 𝑛 is going to be seven. We’re also told in the question the side length of our regular heptagon is 36. So we set our value of 𝑥 equal to 36 centimeters. Now, all we need to do is substitute these values into our formula. Substituting 𝑛 is equal to seven and 𝑥 is 36 centimeters into our formula, we get the area of our regular heptagon is equal to seven times 36 centimeters all squared all over four multiplied by the cot of 180 degrees divided by seven.
Now we can start evaluating this expression. First, of course, because this is an area and we’re squaring the centimeters value, our units is going to be centimeters squared. Next, we’ll evaluate our coefficient. Seven multiplied by 36 squared over four is equal to 2268.
Finally, we recall the cotangent of an angle is the same as one divided by the tangent of that angle. So we can simplify the cot of 180 divided by seven degrees to be one divided by the tan of 180 divided by seven degrees. Therefore, the area of our regular heptagon is 2268 multiplied by one divided by the tan of 180 divided by seven degrees centimeters squared. And now we can calculate this expression. Well, we remember we need to have our calculator set to degrees mode. We get that this is equal to 4709.550 and this expansion continues centimeters squared.
But remember, the question wants us to give our answer to two decimal places. So we look at the third decimal place in our expansion, and we see that this is equal to zero. Since this is less than five, this means we need to round down. And this gives us our final answer. In this question, we were able to find the area of a regular heptagon with side length 36 centimeters. We were able to show its area to two decimal places is 4709.55 centimeters squared.