A cone is a geometric figure with two definitions. Analytical geometry definition: The spatial geometric figure composed of a conical surface and a plane that cuts it (satisfying that the intersection is a circle) is called a cone. Solid geometry definition: The geometry surrounded by a curved surface formed by rotating the other two sides 360 degrees with the straight line of the right-angled side of the right triangle as the axis of rotation is called a cone. The axis of rotation is called the axis of the cone. The curved surface formed by rotating the side perpendicular to the axis is called the base of the cone. The curved surface formed by the rotation of the side that is not perpendicular to the axis is called the side surface of the cone. Regardless of the position of rotation, the sides that are not perpendicular to the axis are called the generatrix of the cone. (The sides refer to the two rotating sides of a right triangle) Basic introduction Chinese name: cone Foreign name: cone Volume formula: base area × height × 1/3 ?V cone = 1/3sh Surface area formula: side area + base area? Classification: Mathematics belongs to: classification of geometric figures, definition, composition, measurement, height, base perimeter, surface area, volume, drawing method, expansion diagram, three views, application, classification. A cone is a geometric figure with two definitions. Analytical geometry definition: The spatial geometric figure composed of a conical surface and a plane that cuts it (satisfying that the intersection is a circle) is called a cone. Solid geometry definition: The geometry surrounded by a curved surface formed by rotating the other two sides 360 degrees with the straight line of the right-angled side of the right triangle as the axis of rotation is called a cone. The axis of rotation is called the axis of the cone. The curved surface formed by rotating the side perpendicular to the axis is called the base of the cone. The curved surface formed by the rotation of the side that is not perpendicular to the axis is called the side surface of the cone. Regardless of the position of rotation, the sides that are not perpendicular to the axis are called the generatrix of the cone. (The sides refer to the two rotating sides of a right triangle) Definition: The spatial geometric figure composed of a conical surface and a plane that cuts it (satisfying that the intersection is a circle) is called a cone. The geometry surrounded by the curved surfaces formed by rotating the other two sides is called a cone. Note: A cone is not a special type of cylinder. Composition Height of the cone: The shortest distance between the apex of the cone and the center of the base of the cone is called the height of the cone; Generatus of the cone: the radius of the sector formed by the side expansion of the cone, and the distance from any point on the base circumference to the apex. Side area of ??a cone: Expand the side of the cone along the generatrix to form a sector. The arc length of this sector is equal to the circumference of the cone base, and the radius of the sector is equal to the length of the cone's generatrix. The side area of ??a cone is the arc length of the cone base. Perimeter × bus/2; it is a curved surface when not unfolded. A cone has a base, a side, a vertex, a height, and countless bus lines. The expanded view of the base is a circle, and the expanded view of the side is a fan shape. Measurement Height (l: bus length, r: base radius) Base perimeter (r: base radius, : center angle radian of side expansion diagram, l: bus length) Surface area The area of ??a cone surface is called the surface area of ??the cone. The surface area of ??a cone consists of two parts: side area and base area. Total area (S) = S side + S base. Among them, S side = (r: base radius, l: cone generatrix, : central angle radian of side expansion diagram) Volume Cone The size of the space occupied by a cone is called the volume of the cone. . The volume of a cone is equal to 1/3 of the volume of a cylinder with the same base and height. According to the cylinder volume formula V=Sh (V=πr^2h), the cone volume formula is obtained: , where S is the base area of ??the cylinder, h is the height of the cylinder, and r is the base radius of the cylinder. Prove that the cone is divided into k parts along the height. The height of each part is the radius of the nth part: the area of ??the base of the nth part: the volume of the nth part: the total volume: ∵ ∴total volume: cone ∵ As k becomes larger and larger, the total volume is closer For the volume of a cone, the closer it is to 0 ∴ ∵ V cylinder ∴ V cone is a V cylinder with the same base and height. Expanded diagram of the volume of the cone. The expanded diagram of the cone consists of a sector (the side of the cone) and a circle (the base of the cone). )composition. (As shown below) When drawing the expansion diagram of a specified cone, we generally know a (generator length) and d (base diameter) ∵ Arc AB = Perimeter of ⊙O ∴ Arc AB = πd ∵ Arc AB = 2πa (∠1/360 °) ∴2πa(∠1/360°)=πd ∴2a(∠1/360°)=d Put a and d into 2a(∠1/360°)=d to get the value of ∠1. In this way, all the data required to draw the expanded graph are obtained. Based on the data, the expansion diagram of the cone can be drawn.
For a cone whose generatrix length is equal to the diameter of the base circle, the expanded fan shape is a semicircle. The fan angle of all cones is equal to (base diameter ÷ busbar) × 180 degrees. Three views of a cone is a figure drawn by an observer looking at it from three different positions. Its front view and side view are both isosceles triangles, and its top view is a circle and its center. Application In life, sand piles, funnels, hats, tops, bamboo hats, pencil heads, drill bits, plumb bobs, etc. can all be approximately regarded as cones. Cones are also indispensable in daily life.