How To Find The Area Of ​​The Axial Section Of The Cone | The Science



The Science

How to find the area of ​​the axial section of the cone

The cone is a geometric body whose base is a circle and the lateral surface - all segments drawn from a point located outside the base plane to the ground. Direct cone, which is usually seen in the school geometry course, can be represented as a body formed by rotating a right triangle around one of the legs. Perpendicular section of the cone is a plane passing through its vertex perpendicular to the base.

How to find the area of ​​the axial section of the cone

You will need:

Drawing a cone with the specified parameters Ruler Pencil Mathematical formulas and definitions The height of the cone radius cone base circle area of ​​a triangle formula

Instruction how to find the area of ​​the axial section of the cone

Step 1:

Draw a cone with the specified parameters. Mark the center of the circle as O, and the top of the cone - as P. You need to know the radius of the base and the height of the cone. Remember the properties of the height of the cone. It is a perpendicular drawn from the apex to the base. The intersection point of the plane with the height of the cone at the base of the cone coincides with the forward center of the base circle. Build an axial section of the cone. It is formed by forming a base diameter of the cone, the diameter of which pass through the points of intersection with the circle. Mark received points A and B.

Build an axial section of the cone

Step 2:

Axial section is formed by two rectangular triangles lying in one plane and have a common leg. Calculate the axial sectional area is possible in two ways. The first way - to find the area of ​​a triangle and put them together. This is the most obvious method, but in fact it is no different from the classical calculation of an isosceles triangle area. So, you got two right-angled triangles, the overall height of the leg which is cone h, the second legs of - the base circle radius R, and the hypotenuses - generators of the cone. Since all three sides of the triangle are equal, then the triangles themselves also get equal, according to the third property of equal triangles. The area of ​​a right triangle is equal to half the product of the other two sides, that is, S = 1 / 2Rh. The area of ​​the two triangles will correspondingly equal to the product of the radius of circumference of the base to a height, S = Rh.

Step 3:

Axial section often considered as an isosceles triangle, whose height is the height of the cone. In this case, APV triangle whose base is equal to the circumference of the cone base diameter D, and a height equal to the height of the cone h. Its area is calculated according to the classical formula area of ​​a triangle, that is, in the end we get the same formula S = 1 / 2Dh = Rh, where S - area of ​​an isosceles triangle, R - base circle radius, and h - height of the triangle, which is both a cone height .