A1 10.7 Concept Byte-Exploring Area & Circumference
Geogebra Activity
Open Geogebra. Verify that View…Algebra is checked and View…Input Bar is checked.
1. Use the point tool and make a point A anywhere on the grid. Select Circle with Center
Through Point and click on the screen a good distance away from point A (creates point
B) and then click on point A. You should now have a circle on the screen.
2. Select Point On Object and make three additional points on the
circle that are evenly spaced apart (points C, D, E). Select the Polygon
tool and click points C, D, E and C again to create an equilateral triangle.
Move the points again, if you need to, in order to make sure the sides
are equal.
3. We want to have Geogebra calculate the circumference and area of
the circle and the perimeter and area of the triangle. Using the
command line…enter the following commands:
Note: poly1 should be the name of your triangle in your algebra pane and c
should be the name of your circle in the conic section.
trip=perimeter[poly1] and hit enter…this is the perimeter for the Δ
circp=circumference[c] and hit enter…this is the circumference for the ⨀
tria=area[poly1] and hit enter…this is the area for the Δ
circa=area[c] and hit enter…this is the area for the ⨀
4. Fill out row-1 in the table below with the measures for the Δ. Click on the selector tool
and click on point A and drag the circle to make it larger or smaller. Record the new
measures in row-2.
5. Click on the triangle and hit the delete key to remove it from the
grid. Select Point On Object and create a new point (point F) on
your circle. Click on the Polygon tool and click on points to create
a square (do not use A). Use the selection tool to move the points
around until all sides are approximately the same length.
6. Enter the following commands in the command line:
sqp=perimeter[poly1] and hit enter…this is the perimeter for the ⊡
sqa=area[poly1] and hit enter…this is the area for the ⊡
7. Fill out the row-3 in the table with the measures for the ⊡. Click on the selector tool
and click on point A and drag the circle to make it larger or smaller. Record the new
measures for the ⊡ in row-4.
8. Click on the square and hit the delete key to remove it from the grid. Select Point On
Object and create a new point (point G) on your circle. Click on the Polygon tool and
click on points to create a pentagon (do not use A). Use the selection tool to move the
points around until all sides are approximately the same length.
9. Enter the following commands in the command line:
pentp=perimeter[poly1] and hit enter…this is the perimeter for the pentagon.
penta=area[poly1] and hit enter…this is the area for the pentagon.
10. Fill out row-5 in the table below with the measures for the pentagon. Click on the
selector tool and click on point A and drag the circle to make it larger or smaller. Record
the new measures for the pentagon in row-6.
Row
Sides
1
3
2
3
3
4
4
4
5
5
6
5
Polygon
Perimeter
Area
Circle
Circumference
Area
Ratios
Perimeter
Polygon Area
Circumference
Circle Area
When the size of the circle was changed, did the ratios you calculated stay the same or
change?
What will happen to the ratios as you increase the number of sides of the polygon, based
on your data in the table above?
How are the ratios of perimeters different from the ratios of the areas?
Estimate the perimeter and area of a polygon that has 100 sides that is inscribed in a
circle with a radius of 10 cm. Explain how you made your estimate.
G 10.7 Extension Activity
Calculating Area in Ancient Egypt
In ancient Egypt, when the yearly floods of the Nile River receded, the river often
followed a different course, so the shape of farmers’ fields along the banks could change
from year to year. Officials then needed to measure property areas, in order to keep
records and calculate taxes. Partly to keep track of land and finances, ancient Egyptians
developed some of the earliest mathematics.
Historians believe that ancient Egyptian tax assessors used this formula to find the area
of any quadrilateral:
A = ½ (a + c) * ½ (b + d)
Where a, b, c and d are lengths, in consecutive order, of the figure’s four sides.
Let’s see if this formula works for all quadrilaterals.
1. Construct and irregular quadrilateral in Geogebra.
2. Make sure the sides are labeled a, b, c and d consecutively (control-click and select
Show Label). Use control-click…Object Properties to rename.
3. Enter the Egyptian area formula in the command line.
Egypt=(1/2(a+c))*(1/2(b+d))
4. Find the actual area of the polygon by entering the area formula in the command
line:
Area=area[poly1]
5. Compare the two areas. Using the selector tool, move the points of the
quadrilateral and compare again. How does the Egyptian formula compare to the
actual area? Is the ancient Egyptian formula correct?
6. Try with different quadrilaterals (move the points of the quadrilateral into the
shape you want)…parallelograms, squares, rectangles, rhombus, kites, and
trapezoids. Is the formula more accurate for certain kinds of quadrilaterals?
Which ones?
7. Does the ancient Egyptian formula favor the tax collector or the landowner? Some
of the time or always? Explain.
8. What is the ancient Egyptian formula really computing?