Surface area of an irregular pyramid. Correct pyramid. Definition. At the base is a regular polygon
Pyramid- This is a polyhedral figure, at the base of which lies a polygon, and the remaining faces are represented by triangles with a common vertex.
If the base is a square, then a pyramid is called quadrangular if the triangle is triangular. The height of the pyramid is drawn from its top perpendicular to the base. Also used to calculate the area apothem is the height of the side face lowered from its vertex.
The formula for the area of the lateral surface of a pyramid is the sum of the areas of its lateral faces, which are equal to each other. However, this method of calculation is used very rarely. Basically, the area of \u200b\u200bthe pyramid is calculated through the perimeter of the base and the apothem:
Consider an example of calculating the area of the lateral surface of a pyramid.
Let a pyramid be given with base ABCDE and vertex F. AB=BC=CD=DE=EA=3 cm. Apothem a = 5 cm. Find the area of the lateral surface of the pyramid.
Let's find the perimeter. Since all the faces of the base are equal, then the perimeter of the pentagon will be equal to:
Now you can find the side area of the pyramid:
Area of a regular triangular pyramid
A regular triangular pyramid consists of a base in which a regular triangle lies and three side faces that are equal in area.
The formula for the lateral surface area of a regular triangular pyramid can be calculated in many ways. You can apply the usual formula for calculating through the perimeter and apothem, or you can find the area of \u200b\u200bone face and multiply it by three. Since the face of the pyramid is a triangle, we apply the formula for the area of a triangle. It will require an apothem and the length of the base. Consider an example of calculating the lateral surface area of a regular triangular pyramid.
Given a pyramid with an apothem a = 4 cm and a base face b = 2 cm. Find the area of the lateral surface of the pyramid.
First, find the area of one of the side faces. In this case it will be:
Substitute the values in the formula:
Since in a regular pyramid all sides are the same, the area of the side surface of the pyramid will be equal to the sum of the areas of the three faces. Respectively:
The area of the truncated pyramid
truncated A pyramid is a polyhedron formed by a pyramid and its section parallel to the base.
The formula for the lateral surface area of a truncated pyramid is very simple. The area is equal to the product of half the sum of the perimeters of the bases and the apothem:
Consider an example of calculating the area of the lateral surface of a truncated pyramid.
Given a regular quadrangular pyramid. The lengths of the base are b = 5 cm, c = 3 cm. Apothem a = 4 cm. Find the area of the lateral surface of the figure.
First, find the perimeter of the bases. In a larger base, it will be equal to:
In a smaller base:
Let's calculate the area:
Before studying questions about this geometric figure and its properties, it is necessary to understand some terms. When a person hears about the pyramid, he imagines huge buildings in Egypt. This is what the simplest ones look like. But they come in different types and shapes, which means that the calculation formula for geometric shapes will be different.
Figure types
Pyramid - geometric figure, denoting and representing multiple faces. In fact, this is the same polyhedron, at the base of which lies a polygon, and on the sides there are triangles that connect at one point - the vertex. The figure is of two main types:
- correct;
- truncated.
In the first case, the base is a regular polygon. Here all side surfaces are equal between themselves and the figure itself will please the eye of a perfectionist.
In the second case, there are two bases - a large one at the very bottom and a small one between the top, repeating the shape of the main one. In other words, a truncated pyramid is a polyhedron with a section formed parallel to the base.
Terms and notation
Basic terms:
- Regular (equilateral) triangle A figure with three identical angles and equal sides. In this case, all angles are 60 degrees. The figure is the simplest of the regular polyhedra. If this figure lies at the base, then such a polyhedron will be called a regular triangular one. If the base is a square, the pyramid will be called a regular quadrangular pyramid.
- Vertex- the highest point where the edges meet. The height of the top is formed by a straight line emanating from the top to the base of the pyramid.
- edge is one of the planes of the polygon. It can be in the form of a triangle in the case of a triangular pyramid, or in the form of a trapezoid for a truncated pyramid.
- cross section- a flat figure formed as a result of dissection. Not to be confused with a section, as a section also shows what is behind the section.
- Apothem- a segment drawn from the top of the pyramid to its base. It is also the height of the face where the second height point is. This definition is valid only in relation to a regular polyhedron. For example - if it is not a truncated pyramid, then the face will be a triangle. In this case, the height of this triangle will become an apothem.
Area formulas
Find the area of the lateral surface of the pyramid any type can be done in several ways. If the figure is not symmetrical and is a polygon with different sides, then in this case it is easier to calculate the total surface area through the totality of all surfaces. In other words, you need to calculate the area of \u200b\u200beach face and add them together.
Depending on what parameters are known, formulas for calculating a square, a trapezoid, an arbitrary quadrilateral, etc. may be required. The formulas themselves in different cases will also be different.
In the case of a regular figure, finding the area is much easier. It is enough to know just a few key parameters. In most cases, calculations are required precisely for such figures. Therefore, the corresponding formulas will be given below. Otherwise, you would have to paint everything on several pages, which will only confuse and confuse.
Basic formula for calculation the lateral surface area of a regular pyramid will look like this:
S \u003d ½ Pa (P is the perimeter of the base, and is the apothem)
Let's consider one of the examples. The polyhedron has a base with segments A1, A2, A3, A4, A5, and they are all equal to 10 cm. Let the apothem be equal to 5 cm. First you need to find the perimeter. Since all five faces of the base are the same, it can be found as follows: P \u003d 5 * 10 \u003d 50 cm. Next, we apply the basic formula: S \u003d ½ * 50 * 5 \u003d 125 cm squared.
Lateral surface area of a regular triangular pyramid the easiest to calculate. The formula looks like this:
S =½* ab *3, where a is the apothem, b is the facet of the base. The factor of three here means the number of faces of the base, and the first part is the area of the side surface. Consider an example. Given a figure with an apothem of 5 cm and a base face of 8 cm. We calculate: S = 1/2 * 5 * 8 * 3 = 60 cm squared.
Lateral surface area of a truncated pyramid it's a little more difficult to calculate. The formula looks like this: S \u003d 1/2 * (p _01 + p _02) * a, where p_01 and p_02 are the perimeters of the bases, and is the apothem. Consider an example. Suppose, for a quadrangular figure, the dimensions of the sides of the bases are 3 and 6 cm, the apothem is 4 cm.
Here, for starters, you should find the perimeters of the bases: p_01 \u003d 3 * 4 \u003d 12 cm; p_02=6*4=24 cm. It remains to substitute the values into the main formula and get: S =1/2*(12+24)*4=0.5*36*4=72 cm squared.
Thus, it is possible to find the lateral surface area of a regular pyramid of any complexity. Be careful not to confuse these calculations with the total area of the entire polyhedron. And if you still need to do this, it’s enough to calculate the area of \u200b\u200bthe largest base of the polyhedron and add it to the area of \u200b\u200bthe lateral surface of the polyhedron.
Video
To consolidate information on how to find the lateral surface area of different pyramids, this video will help you.
Definition 1. A pyramid is called regular if its base is a regular polygon, and the top of such a pyramid is projected into the center of its base.
Definition 2. A pyramid is called regular if its base is a regular polygon and its height passes through the center of the base.
Elements of a regular pyramid
- The height of a side face drawn from its vertex is called apothem. In the figure it is designated as segment ON
- The point connecting the side edges and not lying in the plane of the base is called top of the pyramid(ABOUT)
- Triangles that have a common side with the base and one of the vertices coinciding with the vertex are called side faces(AOD, DOC, COB, AOB)
- The segment of the perpendicular drawn through the top of the pyramid to the plane of its base is called pyramid height(OK)
- Diagonal section of a pyramid- this is the section passing through the top and the diagonal of the base (AOC, BOD)
- A polygon that does not have a pyramid vertex is called the base of the pyramid(ABCD)
If at the base correct pyramid lies a triangle, quadrilateral, etc. then it's called regular triangular , quadrangular etc.
A triangular pyramid is a tetrahedron - a tetrahedron.
Properties of a regular pyramid
To solve problems, it is necessary to know the properties of individual elements, which are usually omitted in the condition, since it is believed that the student should know this from the very beginning.
- side ribs are equal between themselves
- apothems are equal
- side faces are equal among themselves (at the same time, their areas, sides and bases are equal, respectively), that is, they are equal triangles
- all side faces are congruent isosceles triangles
- in any regular pyramid, you can both inscribe and describe a sphere around it
- if the centers of the inscribed and circumscribed spheres coincide, then the sum of the plane angles at the top of the pyramid is π, and each of them is π/n, respectively, where n is the number of sides of the base polygon
- the area of the lateral surface of a regular pyramid is equal to half the product of the perimeter of the base and the apothem
- a circle can be circumscribed near the base of a regular pyramid (see also the radius of the circumscribed circle of a triangle)
- all side faces form equal angles with the base plane of a regular pyramid
- all heights of the side faces are equal to each other
Instructions for solving problems. The properties listed above should help in a practical solution. If you need to find the angles of inclination of the faces, their surface, etc., then the general technique is to split the entire three-dimensional figure into separate flat figures and use their properties to find individual elements of the pyramid, since many elements are common to several figures.
It is necessary to break the entire three-dimensional figure into separate elements - triangles, squares, segments. Further, to apply knowledge from the planimetry course to individual elements, which greatly simplifies finding the answer.
Formulas for the correct pyramid
Formulas for finding volume and lateral surface area:
Notation:
V - volume of the pyramid
S - base area
h - the height of the pyramid
Sb - side surface area
a - apothem (not to be confused with α)
P - base perimeter
n - number of base sides
b - side rib length
α - flat angle at the top of the pyramid
This formula for finding volume can be used only For correct pyramid:
, Where
V - volume of a regular pyramid
h - the height of the regular pyramid
n is the number of sides of the regular polygon that is the base for the regular pyramid
a - side length of a regular polygon
Correct truncated pyramid
If we draw a section parallel to the base of the pyramid, then the body enclosed between these planes and the side surface is called truncated pyramid. This section for a truncated pyramid is one of its bases.
The height of the side face (which is an isosceles trapezoid) is called - apothem of a regular truncated pyramid.
A truncated pyramid is called correct if the pyramid from which it was obtained is correct.
- The distance between the bases of a truncated pyramid is called truncated pyramid height
- All faces of a regular truncated pyramid are isosceles (isosceles) trapezoids
Notes
See also: special cases (formulas) for a regular pyramid:
How to use the theoretical materials given here to solve your problem:
triangular pyramid A polyhedron is called a polyhedron whose base is a regular triangle.
In such a pyramid, the faces of the base and the edges of the sides are equal to each other. Accordingly, the area of the side faces is found from the sum of the areas of three identical triangles. You can find the lateral surface area of a regular pyramid using the formula. And you can make the calculation several times faster. To do this, apply the formula for the area of the lateral surface of a triangular pyramid:
where p is the perimeter of the base, all sides of which are equal to b, a is the apothem lowered from the top to this base. Consider an example of calculating the area of a triangular pyramid.
Task: Let the correct pyramid be given. The side of the triangle lying at the base is b = 4 cm. The apothem of the pyramid is a = 7 cm. Find the area of the lateral surface of the pyramid.
Since, according to the conditions of the problem, we know the lengths of all the necessary elements, we will find the perimeter. Remember that in a regular triangle, all sides are equal, and, therefore, the perimeter is calculated by the formula:
Substitute the data and find the value:
Now, knowing the perimeter, we can calculate the lateral surface area:
To apply the formula for the area of a triangular pyramid to calculate the full value, you need to find the area of the base of the polyhedron. For this, the formula is used:
The formula for the area of \u200b\u200bthe base of a triangular pyramid may be different. It is allowed to use any calculation of parameters for a given figure, but most often this is not required. Consider an example of calculating the area of the base of a triangular pyramid.
Task: In a regular pyramid, the side of the triangle lying at the base is a = 6 cm. Calculate the area of the base.
To calculate, we only need the length of the side of a regular triangle located at the base of the pyramid. Substitute the data in the formula:
Quite often it is required to find the total area of a polyhedron. To do this, you need to add the area of \u200b\u200bthe side surface and the base.
Consider an example of calculating the area of a triangular pyramid.
Problem: Let a regular triangular pyramid be given. The side of the base is b = 4 cm, the apothem is a = 6 cm. Find the total area of the pyramid.
First, let's find the lateral surface area using the already known formula. Calculate the perimeter:
We substitute the data in the formula:
Now find the area of the base:
Knowing the area of the base and lateral surface, we find the total area of \u200b\u200bthe pyramid:
When calculating the area of \u200b\u200ba regular pyramid, one should not forget that the base is a regular triangle and many elements of this polyhedron are equal to each other.
The total area of the lateral surface of the pyramid consists of the sum of the areas of its lateral faces.
In a quadrangular pyramid, two types of faces are distinguished - a quadrilateral at the base and triangles with a common vertex, which form the side surface.
First you need to calculate the area of the side faces. To do this, you can use the formulas for the area of a triangle, or you can also use the formula for the surface area of a quadrangular pyramid (only if the polyhedron is regular). If the pyramid is regular and the length of the edge a of the base and the apothem drawn to it h is known in it, then:
If, according to the conditions, the length of the edge c of a regular pyramid and the length of the side of the base a are given, then you can find the value using the following formula:
If the length of the rib at the base and the acute angle opposite it at the apex are given, then the lateral surface area can be calculated by the ratio of the square of the side a to the doubled cosine of half the angle α:
Consider an example of calculating the surface area of a quadrangular pyramid through a side edge and a side of the base.
Problem: Let a regular quadrangular pyramid be given. Edge length b = 7 cm, base side length a = 4 cm. Substitute the given values into the formula:
We have shown calculations of the area of one side face for a regular pyramid. Respectively. To find the area of the entire surface, it is necessary to multiply the result by the number of faces, that is, by 4. If the pyramid is arbitrary and its faces are not equal to each other, then it is necessary to calculate the area for each individual side. If the base is a rectangle or parallelogram, then it is worth remembering their properties. The sides of these figures are parallel in pairs, and, accordingly, the faces of the pyramid will also be identical in pairs.
The formula for the area of \u200b\u200bthe base of a quadrangular pyramid directly depends on which quadrilateral lies at the base. If the pyramid is correct, then the area of \u200b\u200bthe base is calculated by the formula, if the base is a rhombus, then you need to remember how it is located. If the base is a rectangle, then finding its area will be quite simple. It is enough to know the lengths of the sides of the base. Consider an example of calculating the area of the base of a quadrangular pyramid.
Task: Let a pyramid be given, at the base of which lies a rectangle with sides a = 3 cm, b = 5 cm. An apothem is omitted from the top of the pyramid to each side. h-a \u003d 4 cm, h-b \u003d 6 cm. The top of the pyramid lies on the same line with the intersection point of the diagonals. Find the total area of the pyramid.
The formula for the area of a quadrangular pyramid consists of the sum of the areas of all faces and the area of the base. First, let's find the area of the base:
Now consider the faces of the pyramid. They are identical in pairs, because the height of the pyramid intersects the intersection point of the diagonals. That is, in our pyramid there are two triangles with base a and height h-a, as well as two triangles with base b and height h-b. Now we find the area of the triangle using the well-known formula:
Now let's perform an example of calculating the area of a quadrangular pyramid. In our pyramid with a rectangle at the base, the formula will look like this: