Is there a general formula? No, in general, no. You just need to find 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 still, it is much easier in each case to add up all the areas than to memorize additional formulas. For example, let's calculate the total surface of a regular hexagonal prism.
All side faces are rectangles. Means.
This has already been taken into account when calculating the volume.
So we get:
Surface area of the pyramid
For the pyramid, the general rule also applies:
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. I need to find and.
Recall now that
This is the area of a right triangle.
And let's remember how to find this area. We use the area formula:
We have "" - this, and "" - this too, eh.
Now let's find.
Using the basic area formula and the Pythagorean theorem, we find
Attention: if you have a regular tetrahedron (i.e.), then the formula is:
Surface area of a regular quadrangular pyramid
Let the side of the base be equal, and the side edge equal.
At the base is a square, and therefore.
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 searched for the area of a regular triangle when calculating the surface area of \u200b\u200ba regular triangular pyramid, here we use the found formula.
Well, we have already searched for the area of the side face twice already
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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.
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 quadrangle, 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.
Didn't get an answer to your question? Suggest a topic to the authors.
When preparing for the exam in mathematics, students have to systematize their knowledge of algebra and geometry. I would like to combine all known information, for example, how to calculate the area of a pyramid. Moreover, starting from the base and side faces to the entire surface area. If the situation is clear with the side faces, since they are triangles, then the base is always different.
What to do when finding the area of the base of the pyramid?
It can be absolutely any figure: from an arbitrary triangle to an n-gon. And this base, in addition to the difference in the number of angles, can be a regular figure or an incorrect one. In the USE tasks of interest to schoolchildren, there are only tasks with the correct figures at the base. Therefore, we will only talk about them.
right triangle
That is equilateral. One in which all sides are equal and denoted by the letter "a". In this case, the area of \u200b\u200bthe base of the pyramid is calculated by the formula:
S = (a 2 * √3) / 4.
Square
The formula for calculating its area is the simplest, here "a" is the side again:
Arbitrary regular n-gon
The side of a polygon has the same designation. For the number of corners, the Latin letter n is used.
S = (n * a 2) / (4 * tg (180º/n)).
How to proceed when calculating the lateral and total surface area?
Since the base is a regular figure, all the faces of the pyramid are equal. Moreover, each of them is an isosceles triangle, since the side edges are equal. Then, in order to calculate the lateral area of \u200b\u200bthe pyramid, you need a formula consisting of the sum of identical monomials. The number of terms is determined by the number of sides of the base.
The area of an isosceles triangle is calculated by the formula in which half the product of the base is multiplied by the height. This height in the pyramid is called apothem. Its designation is "A". The general formula for lateral surface area is:
S \u003d ½ P * A, where P is the perimeter of the base of the pyramid.
There are situations when the sides of the base are not known, but the side edges (c) and the flat angle at its vertex (α) are given. Then it is supposed to use such a formula to calculate the lateral area of \u200b\u200bthe pyramid:
S = n/2 * in 2 sin α .
Task #1
Condition. Find the total area of the pyramid if its base lies with a side of 4 cm, and the apothem has a value of √3 cm.
Decision. You need to start by calculating the perimeter of the base. Since this is a regular triangle, then P \u003d 3 * 4 \u003d 12 cm. Since the apothem is known, you can immediately calculate the area of \u200b\u200bthe entire lateral surface: ½ * 12 * √3 = 6√3 cm 2.
For a triangle at the base, the following area value will be obtained: (4 2 * √3) / 4 \u003d 4√3 cm 2.
To determine the entire area, you will need to add the two resulting values: 6√3 + 4√3 = 10√3 cm 2.
Answer. 10√3 cm2.
Task #2
Condition. There is a regular quadrangular pyramid. The length of the side of the base is 7 mm, the side edge is 16 mm. You need to know its surface area.
Decision. Since the polyhedron is quadrangular and regular, then its base is a square. Having learned the areas of the base and side faces, it will be possible to calculate the area of \u200b\u200bthe pyramid. The formula for the square is given above. And at the side faces, all sides of the triangle are known. Therefore, you can use Heron's formula to calculate their areas.
The first calculations are simple and lead to this number: 49 mm 2. For the second value, you will need to calculate the semi-perimeter: (7 + 16 * 2): 2 = 19.5 mm. Now you can calculate the area of an isosceles triangle: √ (19.5 * (19.5-7) * (19.5-16) 2) = √2985.9375 = 54.644 mm 2. There are only four such triangles, so when calculating the final number, you will need to multiply it by 4.
It turns out: 49 + 4 * 54.644 \u003d 267.576 mm 2.
Answer. The desired value is 267.576 mm 2.
Task #3
Condition. For a regular quadrangular pyramid, you need to calculate the area. In it, the side of the square is 6 cm and the height is 4 cm.
Decision. The easiest way is to use the formula with the product of the perimeter and the apothem. The first value is easy to find. The second is a little more difficult.
We'll have to remember the Pythagorean theorem and consider It is formed by the height of the pyramid and the apothem, which is the hypotenuse. The second leg is equal to half the side of the square, since the height of the polyhedron falls into its middle.
The desired apothem (the hypotenuse of a right triangle) is √(3 2 + 4 2) = 5 (cm).
Now you can calculate the desired value: ½ * (4 * 6) * 5 + 6 2 \u003d 96 (cm 2).
Answer. 96 cm2.
Task #4
Condition. The correct side of its base is 22 mm, the side ribs are 61 mm. What is the area of the lateral surface of this polyhedron?
Decision. The reasoning in it is the same as described in problem No. 2. Only there was given a pyramid with a square at the base, and now it is a hexagon.
First of all, the area of \u200b\u200bthe base is calculated using the above formula: (6 * 22 2) / (4 * tg (180º / 6)) \u003d 726 / (tg30º) \u003d 726√3 cm 2.
Now you need to find out the semi-perimeter of an isosceles triangle, which is a lateral face. (22 + 61 * 2): 2 = 72 cm. It remains to calculate the area of \u200b\u200beach such triangle using the Heron formula, and then multiply it by six and add it to the one that turned out for the base.
Calculations using the Heron formula: √ (72 * (72-22) * (72-61) 2) \u003d √ 435600 \u003d 660 cm 2. Calculations that will give the lateral surface area: 660 * 6 \u003d 3960 cm 2. It remains to add them up to find out the entire surface: 5217.47≈5217 cm 2.
Answer. Base - 726√3 cm 2, side surface - 3960 cm 2, entire area - 5217 cm 2.
- this is a figure, at the base of which lies an arbitrary polygon, and the side faces are represented by triangles. Their vertices lie at one 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.
Correct pyramid called a pyramid, in which the sides of the base, angles, and edges are equal. Also, in such a pyramid, the area of \u200b\u200bthe 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, it is necessary to find the area of each individual triangle and add them together. If the pyramid is truncated, then its faces are represented by trapezoids. For the correct pyramid, there is another formula. In it, the lateral surface area is calculated through the semiperimeter of the base and the length of the apothem:
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, and apothem a\u003d 8 cm. Find the area of \u200b\u200bthe lateral surface.
At the base of a regular quadrangular pyramid lies a square. First, let's find its perimeter:
Now we can calculate the area of the lateral surface of our pyramid: 
To find the total area of a polyhedron, you 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, parallelogram area etc.
Consider an example of calculating the area of \u200b\u200bthe base of the pyramid given by our conditions. Since the pyramid is regular, it has a square at its base.
square area calculated by the formula: ,
where a is the side of the square. We have it equal to 6 cm. So the area of \u200b\u200bthe base of the pyramid: 
Now it remains only to find the total area of the polyhedron. The formula for the area of a pyramid is the sum of the area of its base and its lateral surface.
Definition. Side face- this is a triangle in which one angle lies at the top of the pyramid, and the opposite side of it coincides with the side of the base (polygon).
Definition. Side ribs are the common sides of the side faces. A pyramid has as many edges as there are corners in a polygon.
Definition. pyramid height is a perpendicular dropped from the top to the base of the pyramid.
Definition. Apothem- this is the perpendicular of 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 the pyramid by a plane passing through the top of the pyramid and the diagonal of the base.
Definition. Correct pyramid- This 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. pyramid volume through base area and height:
pyramid properties
If all side edges are equal, then a circle can be circumscribed around the base of the pyramid, and the center of the base coincides with the center of the circle. Also, the perpendicular dropped from the top passes through the center of the base (circle).
If all side ribs are equal, then they are inclined to the base plane at the same angles.
The lateral ribs are equal when they form equal angles with the base plane, 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 one angle, then a circle can be inscribed in the base of the pyramid, and the top of the pyramid is projected into its center.
If the side faces are inclined to the base plane at one 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 the same angles to the base.
4. Apothems of all side 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 described sphere will be the intersection point of the perpendiculars that pass through the middle of the edges.
8. A sphere can be inscribed in a pyramid. The center of the inscribed sphere will be the intersection point 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 flat angles at the apex 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 of the pyramid with the sphere
A sphere can be described around the pyramid when at the base of the pyramid lies a polyhedron around which a circle can be described (a necessary and sufficient condition). The center of the sphere will be the point of intersection of planes passing perpendicularly through the midpoints of the side edges of the pyramid.
A sphere can always be described 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.
The connection of the pyramid with the cone
A cone is called 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.
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 side edges of the pyramid are equal to each other.
Connection of a pyramid with a cylinder
A pyramid is said to be 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 circumscribed around a pyramid if a circle can be circumscribed around the base of the pyramid.
A tetrahedron has four faces and four vertices and six edges, where any two edges have no common vertices but do not touch.
Each vertex consists of three faces and edges that form trihedral angle.
The segment connecting the vertex of the tetrahedron with the center of the opposite face is called median of the tetrahedron(GM).
Bimedian is 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 in a ratio of 3: 1 starting from the top.
Definition. Acute Angled Pyramid is a pyramid in which the apothem is more than half the length of the side of the base.
Definition. obtuse pyramid is a pyramid in which the apothem is less than half the length of the side of the base.
Definition. regular tetrahedron A tetrahedron whose four faces are equilateral triangles. It is one of five regular polygons. In a regular tetrahedron, all dihedral angles (between faces) and trihedral angles (at a vertex) are equal.
Definition. Rectangular tetrahedron a tetrahedron is called which has a right angle between three edges at the vertex (the edges are perpendicular). Three faces form rectangular trihedral angle and the faces 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 A tetrahedron is called in which the side faces are equal to each other, and the base is a regular triangle. The faces of such a tetrahedron are isosceles triangles.
Definition. Orthocentric tetrahedron a tetrahedron is called 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.
