Goals: Each student will use trigonometric ratios to find area of regular polygons.
Objectives:
Engage: Finding the area of a regular hexagon, triangle, and quadrilateral are more simplistic because of the special triangles that can be formed with the apothem and radius. However, pentagons, decagons, and octagons are more difficult. Student always see regular octagons as they drive down the road (i.e. stop signs). We have a stop sign on campus so I will ask students to find the area of that stop sign. Most of my students will ask why they have to learn all of these formulas and know how to find area when “in real life, there’s a computer I can just enter all of this into.” So, I will tell them that, for this lesson, we are the computer programmers. Each of them must come up with a spreadsheet that someone else could just “enter everything into” to get the answers.
Lesson Procedure:
1) Students must review the trigonometric ratios using the mnemonic SOHCAHTOA
Sin = Opp Cos = Adj Tan = Opp
Hyp Hyp Adj
2) Students must recall formula for area of regular polygons: A = ½ aP where a is the apothem and P is the perimeter.
3) I will ask students to draw a pentagon, octagon, and decagon; and draw the radii of each.
We will find the central angles of the pentagon. We will be reminded that a circle is
360◦ and there are 5 central angles. 360◦/5 = 72◦. The apothem will divide 72◦ in
half to make an angle of 36◦ at the top. Have students find the angles in an octagon
and decagon formed by the apothem and radius just like we did for the pentagon.
4) Work through an example for finding area of a regular pentagon when various lengths are given:
Ex. 1 Find the area of a regular pentagon when the apothem = 6 cm.
Ex. 2 Find the area of a regular pentagon when the side length = 10 in.
5) Teach students how to find the area of a triangle when 2 sides and the included angle are given.
Recall that the area of a triangle = ½ bh.
area = ½ ab sin C where a and b are the 2 sides and C is the included angle.
7) Students need to create an Excel spreadsheet that can be used to find the area of regular polygons.
Explore: For each polygon, compare the ratio of perimeters and areas for 2 different side lengths. Include findings on homework sheet.
Closure: This lesson should take one class period to teach. Spreadsheets must be completed outside of class as there is no computer access during class time. Arrangements will be made for students who need to use school computer lab after school to complete assignment.
Extension/Elaboration: Spreadsheet could be continued for any regular polygon. Students could possibly find polygons in the real world to either bring in or take measurements so that they can find the area of that figure.
Evaluation/Assessment: Click for scoring rubric
Objectives:
- Using an Excel spreadsheet, the student will be able to find the area of a regular pentagon, octagon, and decagon using trigonometric ratios.
- The student will be able to find the area of a triangle when given two side lengths and one angle measure.
Engage: Finding the area of a regular hexagon, triangle, and quadrilateral are more simplistic because of the special triangles that can be formed with the apothem and radius. However, pentagons, decagons, and octagons are more difficult. Student always see regular octagons as they drive down the road (i.e. stop signs). We have a stop sign on campus so I will ask students to find the area of that stop sign. Most of my students will ask why they have to learn all of these formulas and know how to find area when “in real life, there’s a computer I can just enter all of this into.” So, I will tell them that, for this lesson, we are the computer programmers. Each of them must come up with a spreadsheet that someone else could just “enter everything into” to get the answers.
Lesson Procedure:
1) Students must review the trigonometric ratios using the mnemonic SOHCAHTOA
Sin = Opp Cos = Adj Tan = Opp
Hyp Hyp Adj
2) Students must recall formula for area of regular polygons: A = ½ aP where a is the apothem and P is the perimeter.
3) I will ask students to draw a pentagon, octagon, and decagon; and draw the radii of each.
We will find the central angles of the pentagon. We will be reminded that a circle is
360◦ and there are 5 central angles. 360◦/5 = 72◦. The apothem will divide 72◦ in
half to make an angle of 36◦ at the top. Have students find the angles in an octagon
and decagon formed by the apothem and radius just like we did for the pentagon.
4) Work through an example for finding area of a regular pentagon when various lengths are given:
Ex. 1 Find the area of a regular pentagon when the apothem = 6 cm.
Ex. 2 Find the area of a regular pentagon when the side length = 10 in.
5) Teach students how to find the area of a triangle when 2 sides and the included angle are given.
Recall that the area of a triangle = ½ bh.
area = ½ ab sin C where a and b are the 2 sides and C is the included angle.
7) Students need to create an Excel spreadsheet that can be used to find the area of regular polygons.
Explore: For each polygon, compare the ratio of perimeters and areas for 2 different side lengths. Include findings on homework sheet.
Closure: This lesson should take one class period to teach. Spreadsheets must be completed outside of class as there is no computer access during class time. Arrangements will be made for students who need to use school computer lab after school to complete assignment.
Extension/Elaboration: Spreadsheet could be continued for any regular polygon. Students could possibly find polygons in the real world to either bring in or take measurements so that they can find the area of that figure.
Evaluation/Assessment: Click for scoring rubric
sample_spreadsheet_for_area_of_regular_polygons.xls | |
File Size: | 39 kb |
File Type: | xls |