Advice 1: How to find the lateral surface area of a pyramid

Under the pyramid is understood one of the varieties of polyhedra, which is formed from the base of the polygon and triangles, which are its faces, and unite in one point - the top of the pyramid. Find the area of the lateral surface of the pyramid will not make special difficulties.
Instruction
1
First of all, it should be understood that the lateral surface of the pyramid is represented by several triangles, the square of which can be found through a variety of formulas, depending on known data:

S = (a*h)/2, where h is the height lowered on the side of a;

S = a*b*sinβ, where a, b are the sides of a triangle, and β is the angle between these sides;

S = (r*(a + b + c))/2, where a, b, c be the sidelengths of triangle and r is the radius of the inscribed in the triangle circumference;

S = (a*b*c)/4*R, where R is the radius of the circle circumscribed around the triangle;

S = (a*b)/2 = r2 + 2*r*R (if the triangle is rectangular);

S = S = (a2*√3)/4 (if the triangle is equilateral).

In fact, it is only the most basic of well-known formulas for finding the area of a triangle.
2
Calculating using the above formulas the area of all the triangle faces of the pyramid, you can begin calculating the square of the side surface of the pyramid. This is done very simply: you add together the area of all triangles forming the lateral surface of the pyramid. The formula can be expressed as:

SN = ΣSi, where SN is the lateral surface area of a pyramid, Si - the area of the ith triangle as part of its side surface.
3
For clarity, we can consider a small example: given a right pyramid, the side faces of which is formed by the equilateral triangles and base is square. The edge length of the pyramid is 17 cm you want to find the lateral surface area of the pyramid.

Solution: a known edge length of the pyramid, we know that her face is an equilateral triangle. Thus, we can say that all sides of all triangles of the side surface is equal to 17 cm is Therefore in order to calculate the area of any of these triangles, you will need to use the formula:

S = (172*√3)/4 = (289*1.732)/4 = 125.137 cm2

It is known that at the base of the pyramid is a square. Thus, it is clear that these equilateral triangles four. Then the lateral surface area of a pyramid is calculated as follows:

125.137 cm2 * 4 = 500.548 cm2

Answer: the lateral surface area of a pyramid is 500.548 cm2

Advice 2: How to calculate the area of a polygon

A polygon is a geometric figure constructed by closing the polyline. There are several types of polygon, which differ depending on the number of vertices. The calculation is made for each polygon in certain ways.
Instruction
1
Multiply the lengths of the sides if you need to calculate the area of a square or rectangle. If you want to know the area of a right triangle, extend it to a rectangle to calculate its area and divide it into two.
2
Use to compute the area of polygons the following method, if the shape has interior angles greater than 180 degrees (a convex polygon), and all its vertices are in the mesh nodes coordinates and polyline itself does not cross.
Describe around such a polygon rectangle so that its sides were parallel to the grid lines (coordinate axes). At least one of the vertices of the polygon must be a vertex of a rectangle.
3
Divide the space inside the rectangle in the basic shapes (triangles and squares). Find the area of each and add up all the resulting squares. Subtract the area of the rectangle calculated square inner shapes. So you will find the area of a polygon fast enough.
4
Use one of the following formulas to calculate the area of a regular polygon (angles and sides are equal):

– multiply the number of corners (sides) of the polygon n at two radius r of the inscribed circle and the tangent (π/n);

– multiply (n/2) for two radius R of the circumscribed circle and the sine (2π/n);

– multiply (n/4) for two radius r of the incircle and the cotangent (π/n).
5
Use the following formula, when known side of the triangle:

– multiply the side of the triangle that are adjacent to the corner C, the sine of this angle;

– subtract properiety (p) of the triangle first, one side (p-a), then the other (p-b) and third (p-c). Multiply the values obtained with pauperisation and divide the result by 2.
6
Use to find the area of a trapezoid the formula S = h * ( a + b ) / 2, if we know the height and both bases of the trapezoid.
7
Use a method of calculating square of polygon with the help of palettes. Draw a polygon around the square grid, in which the side of one cell will be equal to one for all vertices of the polygon were located in the nodes.
8
Calculate the area of this figure by the formula of Peak: S = B + G/2 – 1. Here is the number of grid points located inside the polygon, and G is the number of nodes of a square grid located at the border of the polygon.

Advice 3: How to find the area of the bases of the pyramid

Two grounds can only be a truncated pyramid. In this case, the second base is formed with a cross-section parallel to the larger base of the pyramid. Find one reason in that case, if you know the area or the linear elements of the second.
You will need
  • properties of the pyramid;
  • - trigonometric functions;
  • the similarity of figures;
  • - finding the areas of polygons.
Instruction
1
The area of the larger base of the pyramid is the area of the polygon that represents it. If this is the right pyramid, its base lies a regular polygon. To know his areaenough to know the only one of its sides.
2
If the large base is a right triangle, find its areaby multiplying the square of side square root of 3 divided by 4. If the base is a square, erect his side in the second degree. In General, for any regular polygon, use the formula S=(n/4)•a2•ctg(180º/n), where n is the number of sides of a regular polygon, a is the length of its side.
3
The side of the smaller base of the find, according to the formula b=2•(a/(2•tg(180º/n))-h/tg(α))•tg(180 ° /n). Here the a – side of the larger base, h – height of the truncated pyramid, α is the dihedral angle at its base, n is the number of sides of the bases (it is the same). The area of the second base look similar to the first, using the formula the length of its sides S=(n/4)• b2•ctg(180º/n).
4
If the reasons are other types of polygons, all known one reason, and one of the sides of the other, the other hand, calculate as such. For example, the side of larger base 4, 6, 8 see the Big side of the smaller base of the wound is 4 cm. Calculate the coefficient of proportionality, 4/8=2 (take the large side in each of the bases), and calculate the other sides 6/2=3 cm, 4/2=2, see Receive side 2, 3, 4 cm smaller base side. Then, calculate their area, as the areas of triangles.
5
If you know the ratio of the corresponding elements of the truncated pyramid, the ratio of the areas of the bases is equal to the ratio of the squares of these elements. For example, if you know the corresponding sides of the bases a and A1, A2/A12=S/S1.

Advice 4: How to calculate the area of a pyramid

Under area of a pyramid is usually defined as the area of its side or full surface. The basis of this geometry is the polygon. The side faces have a triangular shape. They have a common vertex, that is simultaneously the pinnacle of the pyramid.
You will need
  • - a sheet of paper;
  • - handle;
  • calculator;
  • the pyramid with the specified parameters.
Instruction
1
Consider this in the task pyramid. Determine correct or incorrect polygon lying at its base. On the right, all sides are equal. The area in this case is equal to half the product of the perimeter to the radius of the inscribed circle. Find the perimeter by multiplying the side length l on the number of parties n, that is, P=l*n. To Express the area of the Foundation can be a formula of Ln=1/2P*r, where P is perimeter and r is the radius of the inscribed circle.
2
Perimeter and area of an irregular polygon are calculated differently. The sides are of different length. To calculate the perimeter is to add all the segments, limiting the basis. To compute the area run an additional build. Divide the irregular polygon in shape, the parameters of which you are aware of, and the area you can easily find using the most common formulas and trigonometric functions.
3
The lateral surface of a pyramid is the sum of all the lateral faces. The right pyramid of height falls in the center of the base regular polygon. For clarity, it is very helpful to build most of the height of the pyramid and one of its sides. The intersection of the second height to the bottom face connect with the center of the base. In any case, get a rectangular triangle, in which you need to calculate the hypotenuse, and the height of the side faces. Do so using your known parameters (for example, the height of the pyramid and the radius of the inscribed in the polygon of the base circle).
Build the height of the side faces
4
Knowing the height of the right side face of the pyramid, compute the lateral surface area. It is equal to half the product of the perimeter of the base to the height of the side faces, that is to calculate it by the formula SB=1/2P*h, where P is the already known perimeter, and h is the height of the side faces.
5
The calculation of the lateral surface of the irregular pyramid will require several time-consuming. It is equal to the sum of the areas of all lateral faces. Remember, what is the area of a triangle. It can be found by the formula S=1/2l*h, that is, polypropilenium the base of the triangle to its height.
6
Find the surface area of the pyramid. To do this, fold the already known base area and lateral surface.

Advice 5: How to find the area of a regular quadrangular pyramid

Pyramid - a polyhedron made up of a certain number having one common vertex flat lateral surfaces and one base. The Foundation, in turn, has on each side a common edge, and therefore its form determines the total number of faces of the shape. In a regular quadrangular pyramid such faces five, but to determine the total surface area is sufficient to calculate areas of only two of them.
Instruction
1
The total surface area of any polyhedron is the sum of the areas of its faces. In a regular quadrangular pyramid there are two forms of the polygons in the base has a square, the sides have a triangular configuration. Start calculations, e.g. calculation of the area of the quadrangular base of the pyramid (Sₒ). By definition, the right of the pyramid at its base must lie a regular polygon, in this case a square. If in terms of the length of an edge of (a), just erect it in the second degree: Sₒ = a2. If you know only the length of the diagonal of the base (l), to compute the area find the half of its square: Sₒ = l2/2.
2
Determine the area of the triangular lateral faces of the pyramid Sₐ. If you know the length of its shared base edge (a) and apofema (h), calculate half of the product of these two quantities: Sₐ = a*h/2. Under these conditions, the lengths of a side edge (b) and fin base (a) find half of a work the length of its base at the root of the difference between the squared length of a side edge and a quarter of square base length: Sₐ = ½*a*√(b2-a2/4). If in addition the length of the common base edge (a) given a plane angle at the vertex of the pyramid (α), calculate the ratio of the squared length of the edge to twice the cosine of half the plane angle: Sₐ = a2/(2*cos(α/2)).
3
Calculating the area of one side face (Sₐ), increase the value four times to calculate the lateral surface area of the regular quadrangular pyramid. At a certain apofema (h) and the perimeter of the base (P) is the action along with all the previous step can be replaced by calculating half the product of these two parameters: 4*Sₐ = ½*h*P. In any case, the lateral surface area will add up to calculated in the first step, with an area of square base figures - this will be the total surface area of a pyramid: S = Sₒ+4*Sₐ.
Search
Is the advice useful?