Lateral area of the cone formula. The area of the lateral and full surface of the cone
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Attention! The slide preview is for informational purposes only and may not represent the full extent of the presentation. If you are interested this work please download the full version.
Lesson type: a lesson in studying new material using elements of a problem-developing teaching method.
Lesson Objectives:
- cognitive:
- familiarization with a new mathematical concept;
- formation of new ZUN;
- the formation of practical skills for solving problems.
- developing:
- development of independent thinking of students;
- skills development correct speech schoolchildren.
- educational:
- development of teamwork skills.
Lesson equipment: magnetic board, computer, screen, multimedia projector, cone model, lesson presentation, handout.
Lesson objectives (for students):
- get acquainted with a new geometric concept - a cone;
- derive a formula for calculating the surface area of a cone;
- learn to apply the acquired knowledge in solving practical problems.
During the classes
I stage. Organizational.
Handing over notebooks from home verification work on the topic covered.
Students are invited to find out the topic of the upcoming lesson by solving the rebus (slide 1):
Picture 1.
Announcement to students of the topic and objectives of the lesson (slide 2).
II stage. Explanation of new material.
1) Teacher's lecture.
On the board is a table with the image of a cone. new material explained with program material"Stereometry". A three-dimensional image of a cone appears on the screen. The teacher gives a definition of a cone, talks about its elements. (slide 3). It is said that a cone is a body formed during rotation right triangle regarding the catheter. (slides 4, 5). An image of the development of the lateral surface of the cone appears. (slide 6)
2) Practical work.
Actualization of basic knowledge: repeat the formulas for calculating the area of a circle, the area of a sector, the length of a circle, the length of an arc of a circle. (slides 7-10)
The class is divided into groups. Each group receives a scan of the lateral surface of the cone cut out of paper (a circle sector with an assigned number). Students take the necessary measurements and calculate the area of the resulting sector. Instructions for doing work, questions - problem statements - appear on the screen (slides 11-14). The representative of each group writes the results of the calculations in a table prepared on the board. The participants of each group glue the model of the cone from the development they have. (slide 15)
3) Statement and solution of the problem.
How to calculate the lateral surface area of a cone if only the radius of the base and the length of the generatrix of the cone are known? (slide 16)
Each group makes the necessary measurements and tries to derive a formula for calculating the required area using the available data. When doing this work, students should notice that the circumference of the base of the cone is equal to the length of the arc of the sector - the development of the lateral surface of this cone. (slides 17-21) Using the necessary formulas, the desired formula is derived. Students' reasoning should look something like this:
The radius of the sector - sweep is equal to l, the degree measure of the arc is φ. The area of the sector is calculated by the formula: the length of the arc bounding this sector is equal to the Radius of the base of the cone R. The length of the circle lying at the base of the cone is C = 2πR. Note that Since the area of the lateral surface of the cone is equal to the area of the development of its lateral surface, then
So, the area of the lateral surface of the cone is calculated by the formula S BOD = πRl.
After calculating the lateral surface area of the cone model according to the formula derived independently, a representative of each group writes the result of the calculations in a table on the board in accordance with the model numbers. The calculation results in each row must be equal. On this basis, the teacher determines the correctness of the conclusions of each group. The result table should look like this:
|
model no. |
I task |
II task |
(125/3)π ~ 41.67π |
||
(425/9)π ~ 47.22π |
||
(539/9)π ~ 59.89π |
Model parameters:
- l=12 cm, φ=120°
- l=10 cm, φ=150°
- l=15 cm, φ=120°
- l=10 cm, φ=170°
- l=14 cm, φ=110°
The approximation of calculations is associated with measurement errors.
After checking the results, the output of the formulas for the areas of the lateral and full surfaces of the cone appears on the screen (slides 22-26) students keep notes in notebooks.
Stage III. Consolidation of the studied material.
1) Students are offered tasks for oral solution on ready-made drawings.
Find the areas of the total surfaces of the cones shown in the figures (slides 27-32).
2) Question: Are the areas of the surfaces of cones formed by the rotation of one right triangle about different legs equal? Students make a hypothesis and test it. Hypothesis testing is carried out by solving problems and is written by the student on the blackboard.
Given:Δ ABC, ∠C=90°, AB=c, AC=b, BC=a;
BAA", ABV" - bodies of revolution.
Find: S PPC 1 , S PPC 2 .
Figure 5 (slide 33)
Solution:
1) R=BC = a; S PPC 1 = S BOD 1 + S main 1 = π a c + π a 2 \u003d π a (a + c).
2) R=AC = b; S PPC 2 = S BOD 2 + S main 2 = π b c + π b 2 \u003d π b (b + c).
If S PPC 1 = S PPC 2, then a 2 + ac \u003d b 2 + bc, a 2 - b 2 + ac - bc \u003d 0, (a-b) (a + b + c) \u003d 0. Because a, b, c positive numbers (the lengths of the sides of the triangle), the tore-equality is true only if a =b.
Conclusion: The areas of the surfaces of two cones are equal only if the legs of the triangle are equal. (slide 34)
3) Solution of the problem from the textbook: No. 565.
IV stage. Summing up the lesson.
Homework: p.55, 56; No. 548, No. 561. (slide 35)
Announcement of grades.
Conclusions during the lesson, repetition of the main information received in the lesson.
Literature (slide 36)
- Geometry grades 10–11 - Atanasyan, V. F. Butuzov, S. B. Kadomtsev et al., M., Enlightenment, 2008.
- "Mathematical puzzles and charades" - N.V. Udaltsov, library "First of September", series "MATHEMATICS", issue 35, M., Chistye Prudy, 2010.
