Instructions
First of all, it is worth understanding that the side surface of the pyramid is represented by several triangles, the areas of which can be found using a variety of formulas, depending on the known data:
S = (a * h) / 2, where h is the height lowered to side a;
S = a * b * sinβ, where a, b are the sides of the triangle, and β is the angle between these sides;
S = (r * (a + b + c)) / 2, where a, b, c are the sides of the triangle, and r is the radius of the circle inscribed in this triangle;
S = (a * b * c) / 4 * R, where R is the radius of the triangle circumscribed around the circle;
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, these are only the most basic known formulas for finding the area of a triangle.
Having calculated the areas of all triangles that are the faces of the pyramid using the above formulas, we can begin to calculate the area of this pyramid. This is done very simply: it is necessary to add up the areas of all triangles that form the side surface of the pyramid. The formula can express it like this:
Sп = ΣSi, where Sп is the lateral area, Si is the area of the i-th triangle, which is part of its lateral surface.
For greater clarity, you can consider a small example: a regular pyramid is given, the side faces of which are formed by equilateral triangles, and at the base of it lies a square. The length of the edge of this pyramid is 17 cm. It is required to find the area of the lateral surface of this pyramid.
Solution: the length of the edge of this pyramid is known, it is known that its faces are equilateral triangles. Thus, we can say that all sides of all triangles of the lateral surface are 17 cm.Therefore, in order to calculate the area of any of these triangles, you will need to apply the formula:
S = (17² * √3) / 4 = (289 * 1.732) / 4 = 125.137 cm²
It is known that there is a square at the base of the pyramid. Thus, it is clear that there are four given equilateral triangles. Then the area of the side surface of the pyramid is calculated as follows:
125.137 cm² * 4 = 500.548 cm²
Answer: the area of the side surface of the pyramid is 500.548 cm²
First, we calculate the area of the side surface of the pyramid. The lateral surface means the sum of the areas of all lateral faces. If you are dealing with a regular pyramid (that is, one with a regular polygon at the base, and the vertex is projected to the center of this polygon), then to calculate the entire lateral surface, it is enough to multiply the base perimeter (that is, the sum of the lengths of all sides of the polygon lying at the base pyramid) by the height of the lateral face (otherwise called apothem) and divide the resulting value by 2: Sb = 1 / 2P * h, where Sb is the area of the lateral surface, P is the perimeter of the base, h is the height of the lateral face (apothem).
If you have an arbitrary pyramid in front of you, then you will have to separately calculate the areas of all the faces, and then add them up. Since the sides of the pyramid are triangles, use the formula for the area of a triangle: S = 1 / 2b * h, where b is the base of the triangle and h is the height. When the areas of all the faces have been calculated, all that remains is to add them to get the area of the side surface of the pyramid.
Then you need to calculate the area of the base of the pyramid. The choice of the formula for the calculation depends on which polygon lies at the base of the pyramid: correct (that is, one, all sides of which have the same length) or incorrect. The area of a regular polygon can be calculated by multiplying the perimeter by the radius of the circle inscribed in the polygon and dividing the resulting value by 2: Sn = 1 / 2P * r, where Sn is the area of the polygon, P is the perimeter, and r is the radius of the circle inscribed in the polygon ...
A truncated pyramid is a polyhedron that is formed by a pyramid and its section parallel to the base. Finding the area of the lateral surface of a pyramid is not difficult at all. Its very simple: the area is equal to the product of half the sum of the bases over the apothem. Let's consider an example of calculating the lateral surface area of a truncated pyramid. Suppose you are given a regular quadrangular pyramid. The base lengths are b = 5 cm, c = 3 cm. Apothem a = 4 cm. To find the area of the side surface of the pyramid, you must first find the perimeter of the bases. In a large base, it will be equal to p1 = 4b = 4 * 5 = 20 cm.In a smaller base, the formula will be as follows: p2 = 4c = 4 * 3 = 12 cm.Consequently, the area will be equal: s = 1/2 (20 + 12 ) * 4 = 32/2 * 4 = 64 cm.
A pyramid is a polyhedron, one of the faces of which (base) is an arbitrary polygon, and the other faces (side) are triangles with a common vertex. According to the number of angles of the base of the pyramid, there are triangular (tetrahedron), quadrangular, and so on.
The pyramid is a polyhedron with a base in the form of a polygon, and the rest of the faces are triangles with a common vertex. Apothem is the height of the side face of a regular pyramid, which is drawn from its top.
Is a polyhedral figure, at the base of which is a polygon, and the rest of the faces are represented by triangles with a common vertex.
If there is a square at the base, then the pyramid is called quadrangular, if a triangle - then triangular... The height of the pyramid is drawn from its top perpendicular to the base. Also used to calculate the area apothem- the height of the side face, dropped from its top.
The formula for the lateral surface area of a pyramid is the sum of the areas of its lateral faces, which are equal to each other. However, this calculation method is used very rarely. Basically, the area of the pyramid is calculated through the perimeter of the base and the apothem:
Let's consider an example of calculating the area of the lateral surface of a pyramid.
Let a pyramid be given with base ABCDE and top 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, 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 different ways. You can apply the usual formula for calculating through the perimeter and apothem, or you can find the area of one face and multiply it by three. Since the face of the pyramid is a triangle, we will apply the formula for the area of a triangle. It will require apothem and base length. Let's consider an example of calculating the lateral surface area of a regular triangular pyramid.
You are given a pyramid with apothem a = 4 cm and a facet of the base 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 into the formula: 
Since in a regular pyramid all the 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: 
Truncated pyramid area
Truncated A pyramid is a polyhedron that is 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 by the apothem: 
Pyramid- one of the varieties of a polyhedron formed from polygons and triangles that lie at the base and are its faces.
Moreover, at the top of the pyramid (i.e. at one point), all the faces are combined.
In order to calculate the area of a pyramid, it is worth determining that its lateral surface consists of several triangles. And we can easily find their areas by applying
various formulas. Depending on what kind of triangle data we know, we look for their area.
We list some formulas with which you can find the area of triangles:
- S = (a * h) / 2 ... In this case, we know the height of the triangle h which is lowered to the side a .
- S = a * b * sinβ ... Here are the sides of the triangle a , b , and the angle between them is β .
- S = (r * (a + b + c)) / 2 ... Here are the sides of the triangle a, b, c ... The radius of a circle that is inscribed in a triangle is r .
- S = (a * b * c) / 4 * R ... The radius of the circumscribed circle around the triangle is R .
- S = (a * b) / 2 = r² + 2 * r * R ... This formula should only be applied when the triangle is rectangular.
- S = (a² * √3) / 4 ... We apply this formula to an equilateral triangle.
Only after we calculate the areas of all triangles that are the faces of our pyramid, we can calculate the area of its lateral surface. For this we will use the above formulas.
In order to calculate the area of the lateral surface of the pyramid, no difficulties arise: you need to find out the sum of the areas of all triangles. Let's express it with the formula:
Sп = ΣSi
Here Si is the area of the first triangle, and S P - the area of the lateral surface of the pyramid.
Let's look at an example. A regular pyramid is given, its lateral faces are formed by several equilateral triangles,
« Geometry is the most powerful tool for sharpening our mental faculties.».
Galileo Galilei.
and the square is the base of the pyramid. Moreover, the edge of the pyramid has a length of 17 cm. Let us find the area of the lateral surface of this pyramid.
We argue like this: we know that the faces of the pyramid are triangles, they are equilateral. We also know how long a given pyramid's rib is. Hence it follows that all triangles have equal lateral sides, their length is 17 cm.
To calculate the area of each of these triangles, you can use the following formula:
S = (17² * √3) / 4 = (289 * 1.732) / 4 = 125.137 cm²
Since we know that the square lies at the base of the pyramid, it turns out that we have four equilateral triangles. This means that the area of the side surface of the pyramid can be easily calculated using the following formula: 125.137 cm² * 4 = 500.548 cm²
Our answer is as follows: 500.548 cm² - this is the area of the lateral surface of this pyramid.
Definition of the pyramid
Pyramid is a polyhedron, at the base of which is a polygon, and its faces are triangles.
Online calculator
It is worth dwelling on the definition of some of the components of the pyramid.
She, like other polyhedra, has ribs... They converge to one point, which is called apex pyramids. An arbitrary polygon can lie at its base. By the edge called geometric figure formed by one of the sides of the base and two nearest edges. In our case, this is a triangle. Height a pyramid is the distance from the plane in which its base lies to the top of the polyhedron. For the correct pyramid, there is also a concept apothems- this is a perpendicular, lowered from the top of the pyramid to its base.
Types of pyramids
There are 3 types of pyramids:
- Rectangular- one in which any edge forms a right angle with the base.
- Correct- its base is a regular geometric figure, and the vertex of the polygon itself is a projection of the center of the base.
- Tetrahedron- a pyramid made up of triangles. Moreover, each of them can be taken as a basis.
The formula for the surface area of a pyramid
To find the total surface area of the pyramid, add the lateral surface area and the base area.
The simplest is the case of a regular pyramid, so we will deal with it. Let's calculate the total surface area of such a pyramid. The lateral surface area is:
S side = 1 2 ⋅ l ⋅ p S _ (\ text (side)) = \ frac (1) (2) \ cdot l \ cdot pS side = 2 1 ⋅ l ⋅p
L l l- apothem of the pyramid;
p p p- the perimeter of the base of the pyramid.
Total surface area of the pyramid:
S = S side + S main S = S _ (\ text (side)) + S _ (\ text (main))S =S side + S main
S side S _ (\ text (side)) S side
- the area of the lateral surface of the pyramid;
S main S _ (\ text (main)) S main
- the area of the base of the pyramid.
An example of solving the problem.
ExampleFind the total area of a triangular pyramid if its apothem is 8 (see), and at the base is an equilateral triangle with side 3 (see)
Solution
L = 8 l = 8 l =8
a = 3 a = 3 a =3
Find the perimeter of the base. Since at the base lies an equilateral triangle with side a a a, then its perimeter p p p(the sum of all its sides):
P = a + a + a = 3 ⋅ a = 3 ⋅ 3 = 9 p = a + a + a = 3 \ cdot a = 3 \ cdot 3 = 9p =a +a +a =3 ⋅ a =3 ⋅ 3 = 9
Then the side area of the pyramid:
S side = 1 2 ⋅ l ⋅ p = 1 2 ⋅ 8 ⋅ 9 = 36 S _ (\ text (side)) = \ frac (1) (2) \ cdot l \ cdot p = \ frac (1) (2) \ cdot 8 \ cdot 9 = 36S side = 2 1 ⋅ l ⋅p =2 1 ⋅ 8 ⋅ 9 = 3 6 (see sq.)
Now we will find the area of the base of the pyramid, that is, the area of the triangle. In our case, the triangle is equilateral and its area can be calculated by the formula:
S main = 3 ⋅ a 2 4 S _ (\ text (main)) = \ frac (\ sqrt (3) \ cdot a ^ 2) (4)S main = 4 3 ⋅ a 2
A a a- the side of the triangle.
We get:
S main = 3 ⋅ a 2 4 = 3 ⋅ 3 2 4 ≈ 3.9 S _ (\ text (main)) = \ frac (\ sqrt (3) \ cdot a ^ 2) (4) = \ frac (\ sqrt (3 ) \ cdot 3 ^ 2) (4) \ approx3.9S main = 4 3 ⋅ a 2 = 4 3 ⋅ 3 2 ≈ 3 . 9 (see sq.)
Full area:
S = S side + S base ≈ 36 + 3.9 = 39.9 S = S _ (\ text (side)) + S _ (\ text (base)) \ approx36 + 3.9 = 39.9S =S side + S main ≈ 3 6 + 3 . 9 = 3 9 . 9 (see sq.)
Answer: 39.9 cm. Sq.
Another example, a little more complicated.
ExampleThe base of the pyramid is a square with an area of 36 (cm). The apothem of a polyhedron is 3 times the side of the base a a a... Find the total surface area of a given shape.
Solution
S quad = 36 S _ (\ text (quad)) = 36S quad
=
3
6
l = 3 ⋅ a l = 3 \ cdot a l =3
⋅
a
Find the side of the base, that is, the side of the square. Its area and side length are related:
S quad = a 2 S _ (\ text (quad)) = a ^ 2S quad
=
a 2
36 = a 2 36 = a ^ 2 3
6
=
a 2
a = 6 a = 6 a =6
Find the perimeter of the base of the pyramid (that is, the perimeter of the square):
P = a + a + a + a = 4 ⋅ a = 4 ⋅ 6 = 24 p = a + a + a + a = 4 \ cdot a = 4 \ cdot 6 = 24p =a +a +a +a =4 ⋅ a =4 ⋅ 6 = 2 4
Let's find the length of the apothem:
L = 3 ⋅ a = 3 ⋅ 6 = 18 l = 3 \ cdot a = 3 \ cdot 6 = 18l =3 ⋅ a =3 ⋅ 6 = 1 8
In our case:
S quad = S main S _ (\ text (quad)) = S _ (\ text (main))S quad = S main
It remains to find only the lateral surface area. According to the formula:
S side = 1 2 ⋅ l ⋅ p = 1 2 ⋅ 18 ⋅ 24 = 216 S _ (\ text (side)) = \ frac (1) (2) \ cdot l \ cdot p = \ frac (1) (2) \ cdot 18 \ cdot 24 = 216S side = 2 1 ⋅ l ⋅p =2 1 ⋅ 1 8 ⋅ 2 4 = 2 1 6 (see sq.)
Full area:
S = S side + S main = 216 + 36 = 252 S = S _ (\ text (side)) + S _ (\ text (main)) = 216 + 36 = 252
Answer: 252 cm. Sq.
The lateral surface area of an arbitrary pyramid is equal to the sum of the areas of its lateral faces. It makes sense to give a special formula for expressing this area in the case of a regular pyramid. So, let a regular pyramid be given, at the base of which there is a regular n-gon with side equal to a. Let h be the height of the side face, also called apothem pyramids. The area of one side face is equal to 1 / 2ah, and the entire lateral surface of the pyramid has an area equal to n / 2ha. Since na is the perimeter of the base of the pyramid, we can write the formula found in the form:
Lateral surface area of a regular pyramid is equal to the product of its apothem and half the perimeter of the base.
Concerning total surface area, then just add the base area to the side.
Inscribed and described sphere and ball... It should be noted that the center of the sphere inscribed in the pyramid lies at the intersection of the bisector planes of the inner dihedral angles of the pyramid. The center of the sphere described near the pyramid lies at the intersection of the planes passing through the midpoints of the pyramid's edges and perpendicular to them.
Truncated pyramid. If the pyramid is cut by a plane parallel to its base, then the part enclosed between the secant plane and the base is called truncated pyramid. The figure shows a pyramid, discarding its part that lies above the cutting plane, we get a truncated pyramid. It is clear that the small discarded pyramid is homothetic to the large pyramid with the center of homothety at the apex. The coefficient of similarity is equal to the ratio of heights: k = h 2 / h 1, or side edges, or other corresponding linear dimensions of both pyramids. We know that the areas of such figures are related as squares of linear dimensions; so the areas of the bases of both pyramids (i.e. the area of the bases of the truncated pyramid) are referred to as
Here S 1 is the area of the lower base, and S 2 is the area of the upper base of the truncated pyramid. The side surfaces of the pyramids are in the same relation. There is a similar rule for volumes.
Volumes of similar bodies refer as cubes to their linear dimensions; for example, the volumes of the pyramids are related as the products of their heights on the area of the bases, from where our rule is obtained immediately. It has a completely general character and directly follows from the fact that volume always has the dimension of the third power of length. Using this rule, we derive a formula expressing the volume of the truncated pyramid in terms of the height and area of the bases.
Let there be given a truncated pyramid with height h and base areas S 1 and S 2. If we imagine that it is continued to a full pyramid, then the coefficient of similarity of the full pyramid and the small pyramid can be easily found as the root of the ratio S 2 / S 1. The height of the truncated pyramid is expressed as h = h 1 - h 2 = h 1 (1 - k). Now we have for the volume of the truncated pyramid (V 1 and V 2 denote the volumes of the full and small pyramids)
truncated pyramid volume formula
Let us derive the formula for the area S of the lateral surface of a regular truncated pyramid through the perimeters P 1 and P 2 of the bases and the length of the apothem a. We argue in exactly the same way as when deriving the formula for the volume. We supplement the pyramid with the upper part, we have P 2 = kP 1, S 2 = k 2 S 1, where k is the similarity coefficient, P 1 and P 2 are the perimeters of the bases, and S 1 and S 2 are the horses of the lateral surfaces of the entire resulting pyramid and its top part respectively. For the lateral surface, we find (a 1 and a 2 are the apothems of the pyramids, a = a 1 - a 2 = a 1 (1-k))
formula for the lateral surface area of a regular truncated pyramid
