The lateral area of a quadrangular pyramid. Lateral surface area of a regular quadrangular pyramid: formulas and example problems
Whether there is a general formula? No, in general, no. You just need to look for the areas of the side faces and sum them up.
The formula can be written for straight prism:
Where is the perimeter of the base.
But it’s still much easier to add up all the areas in each specific case than to memorize additional formulas. For example, let's calculate full surface regular hexagonal prism.
All side faces are rectangles. Means.
This was already shown when calculating the volume.
So we get:
Surface area of the pyramid
The general rule also applies to the pyramid:
Now let's calculate the surface area of the most popular pyramids.
Surface area of a regular triangular pyramid
Let the side of the base be equal and the side edge equal. We need to find and.
Let us now remember that
This is the area of a regular triangle.
And let’s remember how to look for this area. We use the area formula:
For us, “ ” is this, and “ ” is also this, eh.
Now let's find it.
Using the basic area formula and the Pythagorean theorem, we find
Attention: if you have a regular tetrahedron (i.e.), then the formula turns out like this:
Surface area of a regular quadrangular pyramid
Let the side of the base be equal and the side edge equal.
The base is a square, and that's why.
It remains to find the area of the side face
Surface area of a regular hexagonal pyramid.
Let the side of the base be equal and the side edge.
How to find? A hexagon consists of exactly six identical regular triangles. We have already looked for the area of a regular triangle when calculating the surface area of a regular triangular pyramid; here we use the formula we found.
Well, we’ve already looked for the area of the side face twice
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Instructions
First of all, it is worth understanding that the lateral 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 right-angled);
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, you can begin to calculate the area of this pyramid. This is done extremely simply: you need to add up the areas of all the triangles that form the side surface of the pyramid. This can be expressed by the formula:
Sp = ΣSi, where Sp is the area of the lateral surface, Si is the area of the i-th triangle, which is part of its lateral surface.
For greater clarity, we can consider a small example: given a regular pyramid, the side faces of which are formed by equilateral triangles, and at its base 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 on the lateral surface are equal to 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 at the base of the pyramid lies a square. Thus, it is clear that there are four given equilateral triangles. Then the area of the lateral surface of the pyramid is calculated as follows:
125.137 cm² * 4 = 500.548 cm²
Answer: The lateral surface area of the pyramid is 500.548 cm²
First, let's calculate the area of the lateral surface of the pyramid. The lateral surface is the sum of the areas of all lateral faces. If you are dealing with a regular pyramid (that is, one that has a regular polygon at its base, and the vertex is projected into the center of this polygon), then to calculate the entire lateral surface it is enough to multiply the perimeter of the base (that is, the sum of the lengths of all sides of the polygon lying at the base pyramid) by the height of the side face (otherwise called the apothem) and divide the resulting value by 2: Sb = 1/2P*h, where Sb is the area of the side surface, P is the perimeter of the base, h is the height of the side face (apothem).
If you have an arbitrary pyramid in front of you, you will have to separately calculate the areas of all the faces and then add them up. Since the side faces 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 up to get the area of the lateral surface of the pyramid.
Then you need to calculate the area of the base of the pyramid. The choice of formula for calculation depends on which polygon lies at the base of the pyramid: regular (that is, one with all sides of the same length) or irregular. The area of a regular polygon can be calculated by multiplying the perimeter by the radius of the inscribed circle 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 inscribed circle in the polygon .
A truncated pyramid is a polyhedron that is formed by a pyramid and its cross section parallel to the base. Finding the lateral surface area of the pyramid is not difficult at all. Its very simple: the area is equal to the product of half the sum of the bases by . Let's consider an example of calculating the lateral surface area. Suppose we are given a regular pyramid. The lengths of the base are b = 5 cm, c = 3 cm. Apothem a = 4 cm. To find the area of the lateral 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. Therefore, the area will be equal to: s=1/2(20+12 )*4=32/2*4=64 cm.
The total area of the lateral surface of a pyramid consists of the sum of the areas of its lateral faces.
In a quadrangular pyramid, there are two types of faces - a quadrangle 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 formula 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 h drawn to it is known, 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 edge at the base and the acute angle opposite it at the top are given, then the area of the lateral surface can be calculated by the ratio of the square of the side a to the double cosine of half the angle α: 
Let's consider an example of calculating the surface area of a quadrangular pyramid through the side edge and the 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 showed calculations of the area of one side face for a regular pyramid. Respectively. To find the area of the entire surface, you need 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 the area must be calculated 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 the base of a quadrangular pyramid directly depends on which quadrilateral lies at the base. If the pyramid is correct, then the area of the base is calculated using the formula, if the base is a rhombus, then you will need to remember how it is located. If there is a rectangle at the base, then finding its area will be quite simple. It is enough to know the lengths of the sides of the base. Let's consider an example of calculating the area of the base of a quadrangular pyramid.
Problem: Let a pyramid be given, at the base of which lies a rectangle with sides a = 3 cm, b = 5 cm. An apothem is lowered from the top of the pyramid to each of the sides. h-a =4 cm, h-b =6 cm. The top of the pyramid lies on the same line as the point of intersection 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 let's look at the sides of the pyramid. They are identical in pairs, because the height of the pyramid intersects the point of intersection 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 let's 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 would look like this:
is a figure whose base is an arbitrary polygon, and the side faces are represented by triangles. Their vertices lie at the same point and correspond to the top of the pyramid.
The pyramid can be varied - triangular, quadrangular, hexagonal, etc. Its name can be determined depending on the number of corners adjacent to the base.
The right pyramid called a pyramid in which the sides of the base, angles, and edges are equal. Also in such a pyramid the area of the side faces will be equal.
The formula for the area of the lateral surface of a pyramid is the sum of the areas of all its faces:
That is, to calculate the area of the lateral surface of an arbitrary pyramid, you need to find the area of each individual triangle and add them together. If the pyramid is truncated, then its faces are represented by trapezoids. There is another formula for a regular pyramid. In it, the lateral surface area is calculated through the semi-perimeter of the base and the length of the apothem:
Let's consider an example of calculating the area of the lateral surface of a pyramid.
Let a regular quadrangular pyramid be given. Base side b= 6 cm, apothem a= 8 cm. Find the area of the lateral surface.
At the base of a regular quadrangular pyramid is a square. First, let's find its perimeter:
Now we can calculate the area of the lateral surface of our pyramid: 
In order to find the total area of a polyhedron, you will need to find the area of its base. The formula for the area of the base of a pyramid may differ depending on which polygon lies at the base. To do this, use the formula for the area of a triangle, area of a parallelogram etc.
Consider an example of calculating the area of the base of a pyramid given by our conditions. Since the pyramid is regular, there is a square at its base.
Square area calculated by the formula: ,
where a is the side of the square. For us it is 6 cm. This means the area of the base of the pyramid is: 
Now all that remains is to find the total area of the polyhedron. The formula for the area of a pyramid consists of the sum of the area of its base and the lateral surface.
Definition. Side edge- this is a triangle in which one angle lies at the top of the pyramid, and the opposite side coincides with the side of the base (polygon).
Definition. Side ribs- This common aspects side edges. A pyramid has as many edges as the angles of a polygon.
Definition. Pyramid height- this is a perpendicular lowered from the top to the base of the pyramid.
Definition. Apothem- this is a perpendicular to the side face of the pyramid, lowered from the top of the pyramid to the side of the base.
Definition. Diagonal section- this is a section of a pyramid by a plane passing through the top of the pyramid and the diagonal of the base.
Definition. Correct pyramid is a pyramid in which the base is a regular polygon, and the height descends to the center of the base.
Volume and surface area of the pyramid
Formula. Volume of the pyramid through base area and height:
Properties of the pyramid
If all the side edges are equal, then a circle can be drawn around the base of the pyramid, and the center of the base coincides with the center of the circle. Also, a perpendicular dropped from the top passes through the center of the base (circle).
If all the side edges are equal, then they are inclined to the plane of the base at the same angles.
The lateral edges are equal when they form equal angles with the plane of the base or if a circle can be described around the base of the pyramid.
If the side faces are inclined to the plane of the base at the same angle, then a circle can be inscribed into the base of the pyramid, and the top of the pyramid is projected at its center.
If the side faces are inclined to the plane of the base at the same angle, then the apothems of the side faces are equal.
Properties of a regular pyramid
1. The top of the pyramid is equidistant from all corners of the base.
2. All side edges are equal.
3. All side ribs are inclined at equal angles to the base.
4. The apothems of all lateral faces are equal.
5. The areas of all side faces are equal.
6. All faces have the same dihedral (flat) angles.
7. A sphere can be described around the pyramid. The center of the circumscribed sphere will be the intersection point of the perpendiculars that pass through the middle of the edges.
8. You can fit a sphere into a pyramid. The center of the inscribed sphere will be the point of intersection of the bisectors emanating from the angle between the edge and the base.
9. If the center of the inscribed sphere coincides with the center of the circumscribed sphere, then the sum of the plane angles at the vertex is equal to π or vice versa, one angle is equal to π/n, where n is the number of angles at the base of the pyramid.
The connection between the pyramid and the sphere
A sphere can be described around a pyramid when at the base of the pyramid there is a polyhedron around which a circle can be described (a necessary and sufficient condition). The center of the sphere will be the intersection point of planes passing perpendicularly through the midpoints of the side edges of the pyramid.
It is always possible to describe a sphere around any triangular or regular pyramid.
A sphere can be inscribed in a pyramid if the bisector planes of the internal dihedral angles of the pyramid intersect at one point (a necessary and sufficient condition). This point will be the center of the sphere.
Connection of a pyramid with a cone
A cone is said to be inscribed in a pyramid if their vertices coincide and the base of the cone is inscribed in the base of the pyramid.
A cone can be inscribed in a pyramid if the apothems of the pyramid are equal to each other.
A cone is said to be circumscribed around a pyramid if their vertices coincide and the base of the cone is circumscribed around the base of the pyramid.
A cone can be described around a pyramid if all the lateral edges of the pyramid are equal to each other.
Relationship between a pyramid and a cylinder
A pyramid is called inscribed in a cylinder if the top of the pyramid lies on one base of the cylinder, and the base of the pyramid is inscribed in another base of the cylinder.
A cylinder can be described around a pyramid if a circle can be described around the base of the pyramid.
A tetrahedron has four faces and four vertices and six edges, where any two edges do not have common vertices but do not touch.
Each vertex consists of three faces and edges that form triangular angle.
The segment connecting the vertex of a tetrahedron with the center of the opposite face is called median of the tetrahedron(GM).
Bimedian called a segment connecting the midpoints of opposite edges that do not touch (KL).
All bimedians and medians of a tetrahedron intersect at one point (S). In this case, the bimedians are divided in half, and the medians are divided in a ratio of 3:1 starting from the top.
Definition. Acute angled pyramid- a pyramid in which the apothem is more than half the length of the side of the base.
Definition. Obtuse pyramid- a pyramid in which the apothem is less than half the length of the side of the base.
Definition. Regular tetrahedron- a tetrahedron in which all four faces are equilateral triangles. It is one of the five regular polygons. In a regular tetrahedron, all dihedral angles (between faces) and trihedral angles (at the vertex) are equal.
Definition. Rectangular tetrahedron is called a tetrahedron in which there is a right angle between three edges at the apex (the edges are perpendicular). Three faces form rectangular triangular angle and the edges are right triangles, and the base is an arbitrary triangle. The apothem of any face is equal to half the side of the base on which the apothem falls.
Definition. Isohedral tetrahedron called a tetrahedron whose side faces are equal to each other, and the base is regular triangle. Such a tetrahedron has faces that are isosceles triangles.
Definition. Orthocentric tetrahedron is called a tetrahedron in which all the heights (perpendiculars) that are lowered from the top to the opposite face intersect at one point.
Definition. Star pyramid A polyhedron whose base is a star is called.
