Teacher’s Guide: Area of an N-gon Curricular Objectives: 1. Students will learn how incongruent polygons can have similar areas. 2. Students will explore different ways to calculate the area of different polygons. 3. Students will explore ways to change the number of sides of a polygon, but keep the area of the polygon consistent. Construct and Investigate: 1. The purpose of this first question is to assess the students’ prior knowledge on how to calculate the area of a triangle and different quadrilaterals. This first question also asks students think about how the side lengths of a triangle can be changed, while the area remains the same. Triangle The students’ should remember from previous classes that the area of a triangle can be calculated be the formula A=(1/2)*b*h, where b is the base of the triangle and h is the height of the triangle. (as shown) As long as the length of b and h remain constant, the area of the triangle remains constant. One way to construct two incongruent triangles with equal area is shown in the following table. Construct triangle ABC Construct a parallel to side AB through point C Construct a new triangle ABC’ by plotting point C’ on the constructed parallel. Triangle ABC’ maintains a constant area when point C’ is moved along the constructed parallel. Triangles ABC and ABC’ are visually not congruent, but the area remains the same because the base and height of the triangle stays the same. 2. Kite 3. Students may have trouble with the construction of a kite, if help is needed, a simple construction is shown below. Plot two points, A and C Construct a perpendicular Plot points B and D on the bisector to points A and C perpendicular bisector of A and C. Connect consecutive points ABCD using the polygon tools (or line segment tool) to complete the construction. This construction also passes the mess up test. The area of a kite can be measured by dividing the kite into two congruent triangles with a diagonal line. The area of the kite is twice the area of one of those triangles (below) The formula for twice the area of a triangle is A=2*(1/2)b*h=b*h. This means that the area of a kite can be found by calculating the product of the two diagonals. Using the given construction, we can construct a kite that is not congruent but has the same area by a method similar to our method of construct two triangles that are not congruent bit have equal area. This construction is shown in the table below. Starting with the kite construction above, we already have a kite that is divided into two congruent triangles. Construct two parallel lines to segment BD, one through point A and one through point C. Using the polygon tool, or segment tool, connect consecutive points A’BC’D The two kites have equal area because the triangles ABD and A’BD have the same base and height. Plot point A’ on one constructed parallel and reflect A over the perpendicular bisector of BD to point C’. 1. Constructing trapezoids that are not congruent, but have the same area can be done by finding triangles within a trapezoid. A general example is shown in the following table. Start by constructing a trapezoid using the parallel line tool. Connect the two endpoints of segment AB to the parallel line to obtain a trapezoid. Now, there are infinitely many triangles that can be manipulated to change the congruence of the original triangle, but not the area. Here are a few examples. And a few more For one specific example we will use the triangle ADD’, with point D’ on one of the two parallel lines. Consider AD’ to be the base of the triangle, Find the point where the extension of line AB and the parallel of segment AD’ through point D intersect. Label to the intersection point A’. Now connect consecutive points A’D’CB to construct a new trapezoid that is not congruent to ABCD, but has the same area. This is true because triangle ADD’ and A’AD’ have the same area and trapezoid AD’CB remains constant. Parallelogram One of the properties of a parallelogram is that each diagonal divides the quadrilateral into two congruent triangles. Each triangle is a symmetry of the other through the midpoint of the diagonal. The triangles can be changed, but they have to be symmetrical. Example below. Starting with parallelogram ABCD Construct a segment between points A and C Plot point B’ on the constructed parallel, to create triangle AB’C with area equal to ABC. To construct our new parallelogram, perform a symmetry of triangle AB’C through the midpoint of any side of triangle ABC. Construct parallel to segment AC through point B. Rectangle or Square We know that our formula for area of a rectangle is A=b*h so as for whatever value of A we chose, there are infinitely many values for b and h as long as the product is A. The area for a square is A=b^2, the length of segments can’t be changed. Also, given the fact that every angle must be right, there are no incongruent squares with equal area. Exploration 2 1. Like the previous exploration, we can solve this problem by breaking the decagon into triangles. Students may immediately see why there is a way to change a convex decagon into a triangle of equal area, but will have trouble figuring out if this will work when the decagon is concave. First look at a quadrilateral. Turning a concave or convex quadrilateral into a triangle is shown below. The process is the same for a decagon, but it is repeated many times. Convex Quadrilateral Divide into triangle ACD then find where the parallel line from the base AC through point D intersects the extension of any side. Use one of the points as for triangle ACD’ and now triangle ABD’ has the same area as ABCD. Concave Quadrilateral Divide into triangle ACD then find where the parallel line from the base AC through point D intersects the extension of any side. Use one of the points as for triangle ACD’ and now triangle ABD’ has the same area as ABCD. 2. Changing a triangle into a decagon is simply the reversed process of solving for changing a decagon into a triangle. If students have trouble following the process, looking back at how to construct incongruent trapezoids with the same area could help. Because the third point of your triangle will be on one of the triangles sides. 3. The idea here is if a triangle can be changed to have n-many sides and keep the same area, we can let n=infinity, and the resulting n-gon would be a circle. But, only if it is a regular n-gon. This process is similar to Rieman Sums and integral calculus. Exploration 3 The formula for area of an n-gon is given. where S is the length of any side N is the number of sides π is PI, approximately 3.142 TAN http://www.mathopenref.com/polygonregulararea.html Students are not expected to know or come to this equation. The idea is to have students understand that the area of a decagon can be broken into the sum of smaller polygons that are easier to calculate. 2. These measurements are rounded to whole numbers for simplicity reasons. The idea is to break this into 10 triangles with a base length of 4 and a height of 12.