You will need

- Depending on the situation, to know the volume of a pyramid, the area of the lateral faces of the pyramid, the edge length the length of the diameter of the polygon at the base.

Instruction

1

One of the ways to find

V = (S*h)/3, where S is the area of all the lateral faces of the pyramid in the sum, h is the height of the pyramid.

Then from this formula we can deduce another, to find the height of the pyramid:

h = (3*V)/S

For example, it is known that the area of the lateral faces of the pyramid 84 cm2, and the volume of a pyramid is equal to 336 cubic cm, Then find

h = (3*336)/84 = 12 cm

Answer: the height of the pyramid is 12 cm

**the height**of the pyramid, and not only right is to Express it through the volume of the pyramid. The formula, with the help of which you can find out its volume, is as follows:V = (S*h)/3, where S is the area of all the lateral faces of the pyramid in the sum, h is the height of the pyramid.

Then from this formula we can deduce another, to find the height of the pyramid:

h = (3*V)/S

For example, it is known that the area of the lateral faces of the pyramid 84 cm2, and the volume of a pyramid is equal to 336 cubic cm, Then find

**the height**:h = (3*336)/84 = 12 cm

Answer: the height of the pyramid is 12 cm

2

Considering the right pyramid, the base of which is a regular polygon, we can conclude that the triangle formed by the height, half the diagonal and one of the faces of the pyramid is a right triangle (e.g., triangle AED in the figure above). According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a2 = b2 + c2). In the case of the right pyramid, the hypotenuse is the face of the pyramid, one of the legs is half the diagonal of the polygon at the base and the other leg is the height of the pyramid. In this case, given the length of edges and diagonals, we can calculate

AE2 = EG2+GA2

Here

GA = √(AE2-EG2).

**the height**. As an example we can consider the triangle AEG:AE2 = EG2+GA2

Here

**the height of**pyramid GA can be expressed as:GA = √(AE2-EG2).

3

To make it more clear how to find

144 = 16 + 128

Thus, the height of the pyramid √128, or about 11.3 cm

**the height**of a regular pyramid, you can see an example: in the right**pyramid of**edge length 12 cm, diagonal length of the polygon at the base is 8 cm Based on these data, it is required to find the length of the height of the pyramid.Solution: 122 = 42 + c2, where C is the unknown side (height) of the pyramid (a right triangle).144 = 16 + 128

Thus, the height of the pyramid √128, or about 11.3 cm

# Advice 2 : How to find the area of a pyramid

Pyramid - complex geometrical body. It is formed of a flat polygon (the base of the pyramid), a point not lying in the plane of the polygon (the top of the pyramid) and all segments that connect the base of the pyramid to the vertex. How to find the area of the pyramid?

You will need

- ruler, pencil and paper

Instruction

1

The lateral surface area of any pyramid is equal to the sum of the areas of the lateral faces.

Because all the lateral faces of a pyramid are triangles, it is necessary to find the sum of the areas of all these triangles. The area of a triangle is calculated by multiplying the length of the base of the triangle to the length of its height.

Because all the lateral faces of a pyramid are triangles, it is necessary to find the sum of the areas of all these triangles. The area of a triangle is calculated by multiplying the length of the base of the triangle to the length of its height.

2

The base of the pyramid is a polygon. If this polygon be divided into triangles, the area of the polygon simple to compute as the sum of the squares poluchivshimsya when dividing triangles by the already known formula.

3

Finding the sum of the lateral surface area of a pyramid and the base of the pyramid, you can find the total surface area of the pyramid.

4

For calculations of area of regular pyramid using a special formula.

Example:

Before us is a pyramid. At the base is a regular n-gon with side a. The height of the side faces h (by the way, is the name apofema pyramid). The area of each lateral face is 1/2ah. The entire side surface of the pyramid has an area of n/2ha, calculated by summing the squares of the side faces. na is the perimeter of the base of the pyramid. The area of this pyramid we find: a work of apogamy of the pyramid and one half of the perimeter of its base equal to the area of lateral surface of regular pyramid.

Example:

Before us is a pyramid. At the base is a regular n-gon with side a. The height of the side faces h (by the way, is the name apofema pyramid). The area of each lateral face is 1/2ah. The entire side surface of the pyramid has an area of n/2ha, calculated by summing the squares of the side faces. na is the perimeter of the base of the pyramid. The area of this pyramid we find: a work of apogamy of the pyramid and one half of the perimeter of its base equal to the area of lateral surface of regular pyramid.

5

With regard to the area of the entire surface, then just add to the side area of the base, according to the principle discussed above.

# Advice 3 : 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).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 4 : How to calculate the height of the pyramid

Problem to the determination of any parameters of the polyhedrons, of course, can cause obstruction. But, if you think a little, it becomes clear that the decision boils down to a consideration of the properties of certain plane figures that make up this geometric solid.

Instruction

1

Pyramid – a polyhedron whose base is the polygon. The side faces represent triangles with a common vertex that is both the pinnacle

**of the pyramid**. If the base**of the pyramid**lies a regular polygon, i.e., one in which all angles and all sides are equal, then the pyramid is called regular. Since the problem statement does not specify what kind of polyhedron should be considered in this case, we can assume that there is a right polygonal pyramid.2

In a right pyramid all edges are equal, all faces are equal isosceles triangles. Height

**of a pyramid**is the perpendicular from the vertex to its base.3

Finding the height

**of a pyramid**depends on what is given in the problem statement. Use of the formula in which to locate any parameters**of the pyramid**is the height. For example, given: V is the volume**of the pyramid**'s square base. Use the formula for finding the volume**of a pyramid**V=SH/3, where H is the height**of the pyramid**. Hence: H=3V/S.4

Moving in the same direction, it should be noted that if the footprint is not given, it in some cases, you can find the formula for finding the area of a regular polygon. Type designation:R - properiter base (properiter easy to find if you know the number of sides and the size of one side);h – apofema polygon (apofema is called the perpendicular from the center of the polygon to any of its sides); a side of the polygon;n = number of sides.Thus, p=an/2, and S=ph= (an/2)h. Whence it follows that: H=3V/ (an/2) h.

5

Of course, there are many other options. For example, given:h - apofema

**of the pyramid**;n is apofema base;H - the height**of the pyramid**.Consider the shape formed by the height**of the pyramid**, its apofema and apofema base. It is a right triangle. Solve the problem using the well-known Pythagorean theorem. In this case we can write h2=n2+H2, where H2=h2-n2. You just have to take the square root of the expression of h2-n2.