Comparative Examination of Polygons
(Triangle). What is the distinctive characteristic of the triangle? What did it have? Everything. This is because the triangle is the constructor. The only thing the triangle did not have was a diagonal. It does not need a diagonal because it is stable without it. All of the angles are successive.
(Quadrilaterals). The common most fundamental characteristic of quadrilaterals is their diagonal. All, except the common quadrilateral, had bases, and therefore they had heights. the common quadrilateral has no base, thus no height. It had to be considered in terms of the triangles formed by the diagonal.
(Polygons). The polygons have no base and therefore, no height. The regular polygon had improved nomenclature – center, radius, apothem. this is true also for the equilateral triangle and the square, because they are also regular polygons.
A. Study of the apothem
Intuition of the apothem
This was given when the child studied the Parts of the Polygons. The regular polygons had three elements of nomenclature that was characteristic: the center, the radius, and a special radius called the apothem. The presence of two different radii indicates that two circles are also involved.
Identification of the apothem
Geometry cabinet: presentation tray, drawer of polygons Inset of a triangle inscribed in a circle
Fraction inset of the square divided into four triangles
A special cardboard square (10cm diagonal and black dot in center
Box, entitled “Apothem”, containing white cardboard circles, each radius is drawn in red and each circle corresponds to a regular polygon:
Polygon Sides Radius
triangle 3 approx. 2.5cm
square 4 approx. 3.5
pentagon 5 approx. 4.0
hexagon 6 approx. 4.3
heptagon 7 approx. 4.5
octagon 8 approx. 4.6
nonagon 9 approx. 4.7
decagon 10 approx. 4.8
Note: All of the measurements of the radii are irrational numbers with the exception of that which is relative to the equilateral triangle. In that case, the radius of the inscribed circle is half of the radius of the circumscribing circle. Theorem: The area of the circumscribing circle is four times the area of the inscribed circle.
Note: The two new figures of the equilateral triangle and the square are introduced, because those same figures found in the presentation tray have sides of 10cm. Therefore it is impossible for the two original figures to be circumscribed by a circle which has a diameter of 10 cm.
Presentation: Recall the special nomenclature pertinent to regular polygons: center, radius, and a special radius called the apothem. Now we’ll find out what’s special about the apothem. The child may remember these three elements of nomenclature, but at this point he hasn’t understood the reasoning.
Present the inset of the equilateral triangle inscribed in a circle. remove the circle insets and invite the child to classify the triangle. Verify that the sides are equal by tracing one side on a piece of paper and matching the other sides. Isolate the triangle at center stage left and get rid of the frame and circle segments.
Bring out the square fraction inset. Lift our two opposite triangle pieces and juxtapose to form a square. Take the red cardboard square. Superimpose the fraction pieces to demonstrate congruency. Place the cardboard square next to the triangle and get rid of the inset, since these constructive triangles have served their purpose.
Bring out the drawer of polygons and ask the child to line up the polygons (in a row with the triangle and square) naming them as he goes along.
We have eight regular polygons, all of which have a center, a radius, and a special radius – the apothem. Since there are two different radii, there must be two different circles. Bring out the circle inset and frame from the presentation tray, and place it above the series of polygons. This is the first of the two circles. this series contains the second circle. Invite the child to put the circles in order from smallest to largest. Place each circle under a polygon. How interesting! The first circle is common to all polygons, but each polygon has its own circle. The triangle has this circle (indicate the circle inset) and this smaller circle (indicate the corresponding small cardboard circle) the square has … and so on, for all polygons.
B. The polygon in relation to the first circle
Remove the inset of the circle from its frame. Place the inset of the triangle in this frame. The circle holds it inside (circumscribe: Latin circum – around, scribere – write or draw). Place the two insets back to back to see this in another way. Hold the two insets back to back using two fingers of one hand on the little knobs. We can see that the center of the triangle coincides with the center of the circumscribing circle.
Place the triangle inset in the circle frame. Now we can try to find the radius. It is the segment which joins the center to one of the vertices. This radius is also the radius of the circumscribing circle. The teacher identifies the radius of the figure, then removes the triangle from the frame, inviting the child to identify the radius without the help of the circle.[top]
C. The polygon in relation to the 2nd circle
Take the white cardboard circle and superimpose it on the back of the triangle inset so that it is inscribed. Then rotate the circle so that the radius is perpendicular to the side of the triangle.
Because this line segment is perpendicular, it meets the midpoint of the side. This circle is inscribed by the triangle; it is contained within the triangle. Hold the triangle and circle as before to show that the centers coincide.
This line segment which joins the center of the polygon with the midpoint of one of the sides is that special radius called the apothem. It is also the radius os the inscribed circle.
(apothem: from a Greek verb meaning to bring down, thus the line segment is brought down (dropped) from the center of a regular polygon to one of its sides)
Ask the child to identify the apothem without the aid of the circle.
D. The polygon in relation to the 1st and 2nd circles simultaneously
We’ve seen that the triangle is embraced by the first circle and it embraces the second circle. Place the inset in the frame wrong side up (extend off the edge of a table so the triangle is flush with the frame) and place the small circle on the triangle inset. We know that the three centers coincide; they have become one center.
Repeat the same procedure with all of the other seven polygons. After we can bring these facts to the child’s attention: 1) Each time the number of sides of the regular polygon increases, so too, the size of the inscribed circle increases. 2) If the size of the circle increases, the length of the radius increases. The length of the radius varies from a minimum of 25 cm for the equilateral triangle to a maximum of just under 5 cm for the decagon. 3) As the number of sides of the regular polygon approaches infinity, as the inscribed circle becomes larger, the radius approached 5 cm the radius of the circumscribing circle. 4) When the inscribed circle of the polygon coincides with the circumscribing circle of the same polygon, the polygon no longer exists; it is identified in the circles. 5) The two radii coincide, because since there is no polygon; the circle are one. The radius of a circle can be regarded as the apothem of a polygon having an infinite number of sides,
Age: after 9 years