The total number of interior angles in two regular polygons is 17, and the total number of diagonals is 53. How many sides does each polygon have? Show work and explain.
Think of the number of diagonals for an n-gon.
For each point p, there are n-3 points that form diagonals (n less p itself and the two adjacent vertices)
Going around for all n points, each diagonal is counted twice. So, the dumber of diagonals d(n) for a n n-gon is
d(n) = n(n-3)/2
The number of interior angles is just n.
So, for two polygons, of m and n vertices,
m + n = 17
m(m-3)/2 + n(n-3)/2 = 53
substitute m = 17-n
(17-n)(14-n)/2 + n(n-3)/2 = 53
You have a 6-gon and an 11-gon
Akeem is putting a fence around his square garden. If the area of the garden is 144 square meters, how many meters of fencing will he need?
11222
Well, well, well... let's dive into the magical world of polygons, shall we?
Now, the total number of interior angles in a polygon can be calculated using the formula n * (n - 2), where n is the number of sides of the polygon. We're gonna label the first polygon with n sides and the second polygon with m sides.
So, for the first polygon, we have n * (n - 2) interior angles. For the second polygon, we have m * (m - 2) interior angles. The total number of interior angles in both polygons combined is given as 17. So we can write the equation:
n * (n - 2) + m * (m - 2) = 17
Now, moving on to the total number of diagonals... for any polygon, the number of diagonals can be calculated using the formula n * (n - 3) / 2. We'll use this formula to calculate the number of diagonals for both polygons and set it equal to 53:
n * (n - 3) / 2 + m * (m - 3) / 2 = 53
So now we have two mathematical equations. It's time for some clown-style math magic to solve them!
After solving the equations, we find that n = 8 and m = 6. Therefore, the first polygon has 8 sides and the second polygon has 6 sides.
Voila! We've unlocked the mystery of the polygons. 🎩🐇
To find the number of sides of each polygon, let's start by understanding the relationship between the interior angles and the number of sides in a regular polygon.
In a regular polygon, all the interior angles have the same measure. The formula to find the measure of each interior angle in a regular polygon is:
Interior Angle = (n-2) * 180 / n
where n is the number of sides of the polygon.
Let's assume the first polygon has 'a' sides and the second polygon has 'b' sides.
From the given information, the total number of interior angles in both polygons is 17. Therefore, if we add up the number of angles in each polygon, we get:
a + b = 17
The second piece of information is that the total number of diagonals in both polygons is 53. The formula to find the total number of diagonals is:
Total Diagonals = (n * (n-3)) / 2
where n is the number of sides of the polygon.
For the first polygon with 'a' sides, the formula would be:
Diagonals in first polygon = (a * (a-3)) / 2
For the second polygon with 'b' sides, the formula would be:
Diagonals in second polygon = (b * (b-3)) / 2
Given that the total number of diagonals is 53, we can write:
(a * (a-3)) / 2 + (b * (b-3)) / 2 = 53
Now, we have two equations:
a + b = 17 (Equation 1)
(a * (a-3)) / 2 + (b * (b-3)) / 2 = 53 (Equation 2)
To solve these equations, we can use substitution or elimination methods. However, in this case, let's solve Equation 1 for 'a' and substitute it into Equation 2.
From Equation 1, we have:
a = 17 - b
Substituting this in Equation 2, we get:
(((17-b) * ((17-b)-3)) / 2) + (b * (b-3)) / 2 = 53
Now, we simplify the equation:
((17-b) * (14-b)) + (b * (b-3)) = 106
Multiplying out the terms, we have:
238 - 31b + b^2 + b^2 - 3b = 106
Simplifying further:
2b^2 - 34b + 132 = 0
Dividing the equation by 2, we have:
b^2 - 17b + 66 = 0
Now we can solve this quadratic equation using factoring or the quadratic formula. Factoring this equation gives us:
(b - 11)(b - 6) = 0
So we have two possible values for 'b': b = 11 or b = 6.
If b = 11, then a = 17 - 11 = 6.
If b = 6, then a = 17 - 6 = 11.
Therefore, one polygon has 11 sides and the other polygon has 6 sides.
To summarize, the first polygon has 11 sides and the second polygon has 6 sides.