You will need
- - cylinder with the specified parameters;
- - location section.
The cross section of the cylinder by a plane passing through its base, is always a rectangle. But depending on location these rectangles will be different. Find the area of the axial section perpendicular to the bases of the cylinder. One side of this rectangle is equal to the height of the cylinder, the second diameter of the rim base. Accordingly, the cross-sectional area in this case will be equal to the product of the sides of the rectangle. S=2R*h, where S is the cross-sectional area, R is the radius of the circular base, given the conditions of the problem, and h is the height of the cylinder, also specified the conditions of the problem.
If the cross-section perpendicular to the bases, but does not pass through the axis of rotation side of the rectangle equals the diameter of the circle. It has to be calculated. To this end, the conditions of the problem must be stated at what distance from the axis of rotation is the plane of the section. For computational convenience draw a circle of the cylinder base, slide the radius and mark on it the distance from the center of the circle is the cross section. From that point, swipe to the radius of the perpendiculars until they intersect with the circle. Connect the point of intersection with the center. You need to find the size of the chord. Find the size of half the chord by Pythagorean theorem. It will be equal to the square root of the difference of the squares of the radius of the circle and distance from center-line section. a2=R2-b2. The whole chord will be respectively equal to 2A. Calculate the cross-sectional area which is equal to the product of the sides of the rectangle, i.e. S=2a*h.
The cylinder can be cut and the plane not passing through the ground plane. If the cross section is perpendicular to the axis of rotation, it will be a circle. The area it is in this case equal to the area of grounds that is calculated by the formula S=πR2.
In order to better imagine the cross section, make a drawing and additional constructions to it.
Advice 2: How to find the area of an axial section of a truncated cone
To solve this problem, it is necessary to remember that such a truncated cone and which properties it possesses. Be sure to make a drawing. This will allow you to determine what geometric figure represents a cross section of a cone. It is possible that after the solution of the problem will not be present for you difficulty.
A round cone is the body obtained by rotating a triangle around one of its legs. Direct, outbound from the vertex of the cone and intersecting the base, called forming. If all the generators are equal, then the cone is straight. At the base of a circular cone is the circle. The perpendicular on the base from the top, is the height of the cone. Have a round straight cone height coincides with its axis. The axis is the line connecting the vertex with the centre of the base. If horizontal clipping plane of a circular cone parallel to the base, its upper base is a circle.
Because the clause is not specified what kind of cone is given in this case, we can conclude that this is a straight circular truncated cone, horizontal section which is parallel to the base. Its axial cross-section, i.e. a vertical plane which passes through the axis of the circular truncated cone, is ravnovesnoi trapeze. All of the axial cross section of a round straight cone are equal. Therefore, in order to find the area of the axial cross-section, it is required to find the area of a trapezoid, bases of which are the diameters of the bases of the truncated coneand the side of his form. The height of the truncated cone is both the height of the trapezoid.
Area of a trapezoid is given by:S = ½(a+b) h, where S is the area of the trapezoid;a is the value of the lower bases of the trapezoid;b – the value of its upper base;h = height of the trapezoid.
Because the condition is not specified, what values this can be considered that the diameters of the two bases and the height of the truncated cone are known: AD = d1 – the diameter of the lower base of the truncated cone;BC = d2 is the diameter of its upper base; EH = h1 – the height of the cone.Thus, the area of the axial cross-section of the truncated cone is determined by: S1 = ½ (d1+d2) h1
Advice 3: How to calculate area of cylinder
A cylinder is a three-dimensional figure and consists of two equal bases which are circles and the lateral surface of the connecting line, bounding the base. To calculate the area of a cylinder, find the area of all its surfaces and fold them.
You will need
- the notion of circle area and circumference.
Determine the area of the bases of the cylinder. To do this, measure with a ruler the diameter of the base, then divide it by 2. This will be the base radius of the cylinder. Calculate the area of a single base. To do this, lift the value of its radius squared and multiply by a constant π, Kr= π∙R2 where R is the radius of the cylinder, and π≈3,14.
Find the total area of the two bases, based on the definition of the cylinder, which suggests that its base are equal. The area of one circle of the base, multiply by 2, On=2∙Kr=2∙π∙R2.
Calculate the area of the lateral surface of the cylinder. To do this, find the circumference, which restricts one of the bases of the cylinder. If the radius is already known, we calculate it by multiplying the number of 2 π and the base radius R, l= 2∙π∙R, where l is the circumference of the base.
Measure the length of the generatrix of the cylinder, which is equal to the length of a segment connecting corresponding points of the base or their centers. In a typical straight cylinder forming L is numerically equal to its height H. Calculate the area of the lateral surface of a cylinderby multiplying the length of its base for forming BOC= 2∙π∙R∙L.
Calculate the area of the surface of the cylinder, by adding up the area of bases and lateral surface. S=On+ Bok. Substituting the formulae the values of the surfaces, obtain S=2∙π∙R2+2∙π∙R∙L, bring the total multipliers S=2∙π∙R∙(R+L). This will calculate the surface of the cylinder by means of a single formula.
For example, the diameter of the base of a direct cylinder is 8 cm and its height is 10 cm, Determine the area of its lateral surface. Calculate the radius of the cylinder. It is equal to R=8/2=4 cm Forming a direct cylinder is equal to its height, i.e. L=10 cm For calculations using a single formula, it's more convenient. Then S=2∙π∙R∙(R+L), substitute the corresponding numerical values S=2∙3,14∙4∙(4+10)=351,68 cm2.