Topicm22921afdd302ddb0_1528449000663_0Topic

The area of a circumscribed polygoncircumscribed polygoncircumscribed polygon

Levelm22921afdd302ddb0_1528449084556_0Level

Third

Core curriculumm22921afdd302ddb0_1528449076687_0Core curriculum

VIII. Plane geometry. Basic level. The student:

10) identifies basic specific points in triangles: the incenterincenterincenter, the center of the circumscribed circle, the orthocenter, the center of gravity and applies their properties in practice.

Timingm22921afdd302ddb0_1528449068082_0Timing

45 minutes

General objectivem22921afdd302ddb0_1528449523725_0General objective

Noticing regularities, similarities and analogies and formulating relevant conclusions.

Specific objectivesm22921afdd302ddb0_1528449552113_0Specific objectives

1. Derivation of the formula for the area of a circumscribed polygoncircumscribed polygoncircumscribed polygon.

2. Calculation of the area of a circumscribed trianglecircumscribed trianglecircumscribed triangle.

3. Communicating in English, developing basic mathematical, computer and scientific competences, developing learning skills.

Learning outcomesm22921afdd302ddb0_1528450430307_0Learning outcomes

The student:

- derives the formula for the area of a circumscribed polygoncircumscribed polygoncircumscribed polygon,

- calculates the area of a circumscribed polygon.

Methodsm22921afdd302ddb0_1528449534267_0Methods

1. Situational analysis.

2. Discussion.

Forms of workm22921afdd302ddb0_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm22921afdd302ddb0_1528450127855_0Introduction

The teacher informs the students about main goal of the lesson: to derive the formula for the area of a circumscribed polygoncircumscribed polygoncircumscribed polygon.

Students recall the definition of a circumscribed polygoncircumscribed polygoncircumscribed polygon and present the way of determining the incenterincenterincenter of the triangle.

Task
Draw a triangle. Inscribe the circle into this triangle. Describe this construction.

The triangle incenterincenterincenter is an intersection point of the triangle angle bisectors. The incenter is an equidistant point from the sides of the triangle.

Definition
A circumscribed polygon is a polygon in which each of its sides is tangent to the circle.m22921afdd302ddb0_1527752263647_0A circumscribed polygon is a polygon in which each of its sides is tangent to the circle.

Procedurem22921afdd302ddb0_1528446435040_0Procedure

Students works individually or in pairs, using computers. They discover the way of calculating the area of a circumscribed polygoncircumscribed polygoncircumscribed polygon when its sides are given.

Task
Open the applet. Observe the consecutive circumscribed polygons. Polygons were divided into triangles with a common vertex in the center of the circle. Note that the radii of the circle are the altitudes of these triangles, from the incenterincenterincenter to the sides of the polygon. Answer the following questions:

- How many triangles has the polygon been divided into?

- How to calculate the area of a polygon described on a circle with the radius r, if we know the length of its sides?

- How to calculate the area of a polygon described on a circle with the radius r, if we know the perimeter of this polygon?

Conclusion:

The area of the circumscribed polygon is equal to its semiperimeter (a half of its perimeter) multiplied by the radius of its incircle.m22921afdd302ddb0_1527752256679_0The area of the circumscribed polygon is equal to its semiperimeter (a half of its perimeter) multiplied by the radius of its incircle.

Task
Verify the correctness of the derived formula for the area of the square and the area of the hexagon.

The following theorem is true:

Theorem – The triangle area.
The area P of the triangle with the sides of lengths a, b, c and with the radius r of its incircle can be calculated with the following formula:m22921afdd302ddb0_1527712094602_0The area P of the triangle with the sides of lengths a, b, c and with the radius r of its incircle can be calculated with the following formula:

If we mark half of triangle perimeter as $P=\frac{a+b+c}{2}$, then the formula can be rewritten as $P=p·r$.m22921afdd302ddb0_1527712094602_0If we mark half of triangle perimeter as $P=\frac{a+b+c}{2}$, then the formula can be rewritten as $P=p·r$.

Students work individually, solving the following problems. Having completed the exercises, they present the results and discuss them.

Task
The area of a triangle equals 12 cmIndeks górny 2² and its perimeter equals 16 cm. Calculate the radius of its incircle.

Task
A trapezoid with perimeter of 54 cm is circumscribed on the circle with the radius of 4 cm. One of its bases is eight times longer than other base. Find the lengths of both bases of this trapezoid.

A hint:

Compare the area of the trapezoid calculated with the following formulas $P=\frac{a+b}{2}·h$ and $P=p·r$.

An extra task
The circle with the radius r is inscribed into an isosceles triangle with the base a and the legs b. Provide the area of this triangle.

Lesson summarym22921afdd302ddb0_1528450119332_0Lesson summary

Students do the revision exercises. Then together they summarize the class, by formulating the conclusions to memorize.

The area of a circumscribed polygoncircumscribed polygoncircumscribed polygon is equal to its semiperimetersemiperimetersemiperimeter (half of its perimeter) multiplied by the radius of its incircle.

The area of a triangle with the sides a, b, c circumscribed on the circle with the radius r can be calculated with the following formula:

Selected words and expressions used in the lesson plan

altitude of trianglealtitude of trianglealtitude of triangle

circumscribed polygoncircumscribed polygoncircumscribed polygon

circumscribed trianglecircumscribed trianglecircumscribed triangle

derivation of formuladerivation of formuladerivation of formula

equidistant pointsequidistant pointsequidistant points

formula for area of a triangleformula for area of a triangleformula for area of a triangle

incenterincenterincenter

incircle of a triangleincircle of a triangleincircle of a triangle

perimeter of a figureperimeter of a figureperimeter of a figure

semiperimetersemiperimetersemiperimeter