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A cone (more precisely, a circular cone) is a body that consists of a circle - the base of the cone, a point that does not lie in the plane of this circle - the top of the cone and all segments connecting the top of the cone with the points of the base. The segments connecting the top of the cone with the points of the circle of the base are called generators of the cone. The surface of the cone consists of a base and a side surface.
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R
vertex
generators
base
ABOUT
base center
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A cone is called straight if the line connecting the vertex of the cone with the center of the base is perpendicular to the plane of the base. In what follows, we will consider only the right cone, calling it simply the cone for brevity. Visually, a straight circular cone can be thought of as a body obtained by rotating a right triangle around its leg as an axis.
The height of a cone is the perpendicular drawn from its top to the plane of its base. For a right cone, the base of the height coincides with the center of the base. The axis of a right circular cone is called a straight line containing its height.
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The section of a cone by a plane passing through its vertex is an isosceles triangle, in which sides are generators of the cone (Fig. 3).
In particular, an isosceles triangle is an axial section of a cone. This is a section that passes through the axis of the cone (Fig. 4).
(Fig. 3).
(Fig. 4)
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Frustum
A plane perpendicular to the axis of the cone cuts off a smaller cone from it. The rest is called a truncated cone. A truncated cone can also be obtained as a body of revolution. A truncated cone is a body of revolution formed by the rotation of a rectangular trapezoid near the side perpendicular to the bases. The circles O and O1 are its bases, its generators AA1 are equal to each other, the straight line OO1 is the axis, the segment OO1 is the height. Its axial section is an isosceles trapezoid.
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Theorem. A plane parallel to the plane of the base of the cone intersects the cone in a circle, and the side surface - in a circle centered on the axis of the cone.
Proof. Let - a plane parallel to the plane of the base of the cone and intersecting the cone (Fig. 5). A homothety transformation with respect to the vertex of the cone, which combines the plane with the plane of the base, combines the section of the cone by the plane with the base of the cone. Therefore, the section of the cone by the plane is a circle, and the section of the lateral surface is a circle centered on the axis of the cone. The theorem has been proven.
(fig.5)
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Lateral area of the truncated cone: $$S = pi(R_(1) + R_(2)) cdot l $$ Volume of the truncated cone: $$V = frac(1)(3)pi H(R^(2)_ (1) + R_(1) cdot R_(2) + R^(2)_(2))$$, where h is the height of the truncated cone; R1,R2 - radii of the upper and lower bases; l - generatrix.
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In geology, there is the concept of "removal cone". This is a relief form formed by an accumulation of clastic rocks (pebbles, gravel, sand) carried by mountain rivers to a foothill plain or to a flatter wide valley. In biology, there is the concept of "cone of growth". This is the top of the shoot and root of plants, consisting of cells of the educational tissue. "Cones" is a family of marine molluscs of the subclass of the anterior gills. The shell is conical (2–16 cm), brightly colored. There are over 500 types of cones. They live in the tropics and subtropics, are predators, have a poisonous gland. The bite of the cones is very painful. Known deaths. The shells are used as decorations and souvenirs.
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Additional information about the cone
According to statistics, 6 people per 1 million inhabitants per year die from lightning strikes on Earth (more often in southern countries). This would not happen if there were lightning rods everywhere, as a safety cone is formed. The higher the lightning rod, the larger the volume of such a cone. Some people try to hide from discharges under a tree, but the tree is not a conductor, charges accumulate on it and the tree can be a source of voltage. In physics, there is the concept of "solid angle". This is a tapered corner carved into the ball. The unit of solid angle is 1 steradian. 1 steradian is a solid angle whose square of radius is equal to the area of the part of the sphere it cuts out. If a light source of 1 candela (1 candle) is placed in this corner, then we get a luminous flux of 1 lumen. The light from a movie camera, a searchlight, spreads in the form of a cone.
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Slides captions:
Geometry lesson in grade 11 The work was done by a mathematics teacher at the MBOU "Ostroh Secondary School" Nokhrina T.A.
Test on the topic: “Cylinder. Its surface area
Question number 1: What figure is the base of the cylinder? a) Oval b) Circle c) Square
Question number 2: What is the area of the base of a cylinder with a radius of 2cm? a) 4 π b) 8 π c) 4
Question number 3: What is the name of the segment marked in red? a) diagonal of a cylinder b) apothem of a cylinder c) generatrix of a cylinder
Question number 4: What formula can be used to calculate the lateral surface of a cylinder? a) 2 π Rh b) 2 π R(h+R) c) π R 2 h
Question #5: What is the formula for calculating the total area of a cylinder? a) π R 2 h b) 2 π Rh c) 2 π R(h+R)
Question #6: Calculate the lateral surface of the given cylinder. a) 15 π cm 2 b) 30 π cm 2 c) 48 π cm 2 3cm 5cm 3cm
Question #7: Calculate the total surface of the given cylinder. a) 32 π cm 2 b) 24 π cm 2 c) 16 π cm 2 2cm 6cm
Question No. 8: What is the area of the axial section of a cylinder with a radius of 1cm and a generatrix of 3cm? a) 6 cm 2 b) 3 cm 2 c) 6 π cm 2
Correct answers: question number answer 1 b 2 a 3 c 4 a 5 c 6 b 7 a 8 a For a mark of "5" - 8 correct answers. On the assessment of "4" - 6 - 7 correct answers. For a mark of "3" - 5 correct answers. For a mark of "2" - 4 or less correct answers.
“... I read somewhere that the king once ordered his soldiers to demolish the land handfuls in a heap. And the proud hill rose, and the king could look from the height with joy and the valley covered with white tents, and the sea where the ships ran. A.S. Pushkin "The Miserly Knight"
Lesson topic:
Cone in Greek "konos" means "pine cone". Historical information about the cone
The concept of a cone Definition: a body bounded by a conical surface and a circle with boundary L is called a cone. L Textbook p. 135
lateral (conical) surface cone height (RO) cone axis cone apex (P) cone base cone radius (r) Cone elements B r generators P
Cones all around us
Bonsai
Cone-shaped houses - trulli
Ice cream
Cones
Tuff houses (carved into the rock)
Bushes in the royal garden
Cones - shells
Roof-cone
inflatable cones
Cone - a body of revolution A cone is obtained by rotating a right triangle around the leg
We work in a notebook: BASE TOP HEIGHT h R RADIUS GENERATOR L L h
Side surface of the cone If we cut the cone along the generatrix, we obtain a development of the cone. L A B C S side = π RL
Total surface of the cone Knowing the formula for the lateral surface of the cone, derive the formula for finding the total surface of the cone R S total =S side +S main S side = π RL S main = π R 2 S total = π RL+ π R 2 S total = π R(L+R )
SECTION OF A CONE The section of a cone by a plane passing through its vertex is an isosceles triangle.
CONE SECTION The axial section of a cone is the section passing through its axis.
SECTION OF A CONE A section of a cone by a plane parallel to its base is a circle centered on the axis of the cone.
Generator L Vertex Height h Radius R Lateral surface S side = π RL Full surface S total = π R(L+R) Reference abstract
Sources: Textbook "Geometry 10-11" ed. L.S. Atanasyan 2012 900igr.net Presentation by Sivak Svetlana Olegovna Gymnasium No. 56 St. Petersburg 2011
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The concept of a cone
A body bounded by a conical surface and a circle with boundary L is called a cone.
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Historical background Cone in Greek "konos" means "pine cone". With a cone, people are familiar with ancient times. In 1906, the book of Archimedes (287–212 BC) “On the Method” was discovered, in which the solution of the problem of the volume of the common part of intersecting cylinders is given.
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Cone elements
lateral (conical) surface
cone height (RO)
cone axis
cone apex (P)
cone base
cone radius (r)
generators
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Cone figure of revolution
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Axial section
If the cutting plane passes through the axis of the cone, then the section is an isosceles triangle, the base of which is the diameter of the base of the cone, and the sides are the generatrix of the cone. This section is called axial.
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If the cutting plane is perpendicular to the axis OP of the cone, then the section of the cone is a circle with center O and located on the axis of the cone. The radius r1 of this circle is equal to (OP/PO1)*r, where r is the radius of the base of the cone.
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Cone surface area
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The area of the lateral surface of the cone is equal to the product of half the circumference of the base and the generatrix.
The total surface area of a cone is the sum of the areas of the lateral surface and the base. To calculate the area SCON of the total surface of the cone, the formula is obtained
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Sside =πr(l+r)
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Frustum
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A plane perpendicular to the axis of the cone cuts off a smaller cone from it. The rest is called a truncated cone. A truncated cone can also be obtained as a body of revolution. A truncated cone is a body of revolution formed by the rotation of a rectangular trapezoid near the side perpendicular to the bases.
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Truncated Cone Elements
Base
Formative
Base
Side surface
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Cones all around us
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Cones all around us
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