Square lateral software. How to find the side surface of the pyramid
When preparing for the exam in mathematics, students have to systematize knowledge of algebra and geometry. I want to combine all the well-known information, for example, how to calculate the pyramid area. Moreover, starting on the base and side faces to the area of \u200b\u200bthe entire surface. If the situation is clear with the side faces, as they are triangles, then the base is always different.
How to be when finding the area of \u200b\u200bthe base of the pyramid?
It can be completely any figure: from an arbitrary triangle to n-square. And this basis, in addition to the difference in the number of angles, may be the right figure or incorrect. In the tasks of interest to schoolchildren, only tasks with the right figures are found at the base. Therefore, it will only be about them.
Right triangle
That is, equilateral. The one in which all parties are equal and indicated by the letter "A". In this case, the area of \u200b\u200bthe base of the pyramid is calculated by the formula:
S \u003d (A 2 * √3) / 4.
Square
The formula for calculating its area is the simplest, here "A" - again the side:
Arbitrary correct n-square
The Polygon side has the same designation. For the number of angles, the Latin letter n is used.
S \u003d (N * A 2) / (4 * TG (180º / N)).
How to do when calculating the side of the side and complete surface?
Because at the base lies proper FigureAll the faces of the pyramids turn out to be equal. Moreover, each of them is an equally traded triangle, since the side ribs are equal. Then in order to calculate the side area of \u200b\u200bthe pyramid, a formula will be required, consisting of the amount of the same single-wing. The number of components is determined by the number of bases.
Area equal triangle It is calculated by the formula in which half of the product of the base is multiplied by height. This height in the pyramid is called apophey. Her designation is "A". The general formula for the side surface area looks like this:
S \u003d ½ P * A, where p is the perimeter of the base of the pyramid.
There are situations where the sides of the base are not known, but the side edges (c) and the flat angle are given at its vertex (α). Then it should be used to use such a formula to calculate the side area of \u200b\u200bthe pyramid:
S \u003d N / 2 * B 2 SIN α .
Task number 1.
Condition. Find the total area of \u200b\u200bthe pyramid if it lies with a side of 4 cm in its foundation, and the apophem is √3 cm.
Decision. It is necessary to start with the calculation of the perimeter of the foundation. Since this is a regular triangle, then p \u003d 3 * 4 \u003d 12 cm. Since the apophem is known, you can immediately calculate the area of \u200b\u200bthe entire side surface: ½ * 12 * √3 \u003d 6√3 cm 2.
For a triangle, at the base, such a value of the area will be obtained: (4 2 * √3) / 4 \u003d 4√3 cm 2.
To determine the entire area, it will be necessary to fold two of the resulting values: 6√3 + 4√3 \u003d 10√3 cm 2.
Answer. 10√3 cm 2.
Task number 2.
Condition. There is a regular quadrangular pyramid. The length of the base side is 7 mm, the side edge is 16 mm. It is necessary to know its surface area.
Decision. Since the polyhedron is quadrangular and correct, then in its base there is a square. Upon learning of the area of \u200b\u200bthe base and side faces, it will be possible to count the area of \u200b\u200bthe pyramid. The formula for the square is given above. And the side faces are known for all sides of the triangle. Therefore, it is possible to use the Geron formula to calculate their areas.
The first calculations are simple and lead to this number: 49 mm 2. For the second value, it will be necessary to calculate the half-meter: (7 + 16 * 2): 2 \u003d 19.5 mm. Now you can calculate the area of \u200b\u200ban equifiable triangle: √ (19.5 * (19.5-7) * (19.5-16) 2) \u003d √2985.9375 \u003d 54.644 mm 2. There are only four such triangles, so when calculating the final number it will be necessary 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 number 3.
Condition. In the right quadrangular pyramid, it is necessary to calculate the area. It knows the side of the square - 6 cm and height - 4 cm.
Decision. The easiest way to use the formula with the work of the perimeter and the aponemy. The first value is simple. The second is more complicated.
We will have to remember the theorem of Pythagora and consider it formed by the height of the pyramid and apophey, which is hypotenuse. The second catat is equal to half the side of the square, since the height of the polyhedron falls in its middle.
The desired apophem (the hypotenus of the rectangular triangle) is √ (3 2 + 4 2) \u003d 5 (cm).
Now you can calculate the desired value: ½ * (4 * 6) * 5 + 6 2 \u003d 96 (cm 2).
Answer. 96 cm 2.
Task number 4.
Condition. The right side of its base is 22 mm, side ribs are 61 mm. What is the side surface area of \u200b\u200bthis polyhedron?
Decision. The arguments in it are 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.
The first thing is calculated the base area according to the above formula: (6 * 22 2) / (4 * TG (180º / 6)) \u003d 726 / (TG30º) \u003d 726√3 cm 2.
Now it is necessary to find out the half-version of an equilibrium triangle, which is a side face. (22 + 61 * 2): 2 \u003d 72 cm. It remains on the formula of Heron to count the area of \u200b\u200beach such triangle, and then multiply it on six and folded with the one that happened to base.
Calculations according to the Geron formula: √ (72 * (72-22) * (72-61) 2) \u003d √435600 \u003d 660 cm 2. Calculations that will give the side surface area: 660 * 6 \u003d 3960 cm 2. It remains to be folded to find out the entire surface: 5217.47≈5217 cm 2.
Answer. The bases are 726√3 cm 2, the side surface - 3960 cm 2, the entire area is 5217 cm 2.
In the right triangular pyramid SABC R. - Mid-rib AU, S. - Top.
It is known that Sr \u003d 6., and the side surface area is equal 36
.
Find the length of the cut BC..
Let's draw a drawing. In the right pyramid, the side faces are an equifiable triangles.
Section Sr. - Median, lowered to the base, and therefore, the height of the side face.
The side surface area of \u200b\u200bthe correct triangular pyramid is equal to the sum of the area
three equal side faces S side. \u003d 3 · s ABS. From here S ABS \u003d 36: 3 \u003d 12 - Square of the face.
The area of \u200b\u200bthe triangle is equal to half the work of its base to height
S ABS \u003d 0,5 · AB · SR. Knowing area and height, find the side of the base AB \u003d Sun..
12 \u003d 0.5 · AV · 6
12 \u003d 3 · av
AB \u003d 4.
Answer: 4
You can approach the task and from the other end. Let the base side Av \u003d sun \u003d a.
Then the area of \u200b\u200bthe face S ABS \u003d 0.5 · AB · Sr \u003d 0.5 · A · 6 \u003d 3A.
The area of \u200b\u200beach of the three faces is equal 3A., the area of \u200b\u200bthree faces is equal 9A..
Under the condition of the task, the area of \u200b\u200bthe side surface of the pyramid is equal to 36.
S side. \u003d 9a \u003d 36.
From here a \u003d 4..
What kind of figure we call the pyramid? First, this is a polyhedron. Secondly, at the base of this polyhedron there is an arbitrary polygon, and the sides of the pyramid (side faces) necessarily have the form of triangles converging in one total vertex. Now, having understood with the term, find out how to find the surface area of \u200b\u200bthe pyramid.
It is clear that the surface area of \u200b\u200bsuch a geometric body will be made up of the sum of the base area and its entire side surface.
Calculation of the base area of \u200b\u200bthe pyramid
The choice of the calculated formula depends on the form of a polygon underlying in the foundation of our pyramid. It can be correct, that is, with the sides of the same length, or incorrect. Consider both options.
Based on the right polygon
From the school course it is known:
- the square of the square will be equal to the length of its side, erected into the square;
- the equilateral triangle area is equal to the square of it, divided by 4 and multiplied by square root From three.
But there is a general formula for calculating the area of \u200b\u200bany correct polygon (SN): it is necessary to multiply the perimeter value of this polygon (P) to the radius inscribed in it (R), and then split the resulting result into two: Sn \u003d 1 / 2p * R .
Based on the wrong polygon
The scheme of finding its area is to first divide the entire polygon on triangles, calculate the area of \u200b\u200beach of them by the formula: 1 / 2a * h (where A is the base of the triangle, H is the height-lowered to this base), fold all the results.
Square side surface of the pyramid
Now we calculate the area of \u200b\u200bthe side surface of the pyramid, i.e. The sum of the squares of all its side. There are also 2 options here.
- Let us have an arbitrary pyramid, i.e. Such, at the base of which - an irregular polygon. Then the area of \u200b\u200beach face should be calculated and folded the results. Since only triangles can be the side sides of the pyramid, then the calculation is based on the formula mentioned above: S \u003d 1 / 2a * h.
- Let our pyramid be correct, i.e. In its foundation lies the right polygon, and the projection of the peak of the pyramid turns out to be in its center. Then, to calculate the side surface area (SB), it is sufficient to find half the work of the perimeter of the polygon base (P) to the height (H) of the side (the same for all edges): Sb \u003d 1/2 p * h. The perimeter of the polygon is determined by the addition of the lengths of all of its sides.
The total surface area of \u200b\u200bthe right pyramid is due to the summation of the area of \u200b\u200bits base with the area of \u200b\u200bthe entire side surface.
Examples
For example, calculate the algebraically surface area of \u200b\u200bseveral pyramids.
Surface Surface Triangular Pyramid
Based on such a pyramid - a triangle. According to the formula SO \u003d 1 / 2a * h we find the base area. The same formula is used to find the area of \u200b\u200beach face of the pyramid, also having a triangular shape, and we obtain 3 areas: S1, S2 and S3. The area of \u200b\u200bthe side surface of the pyramid is the sum of all areas: Sb \u003d S1 + S2 + S3. After folding the side of the side and base, we obtain the full surface area of \u200b\u200bthe desired pyramid: SP \u003d SO + SB.
The surface area of \u200b\u200bthe quadrangular pyramid
The side surface area is the sum of 4-exharicted: Sb \u003d S1 + S2 + S3 + S4, each of which is calculated by the formula of the triangle area. And the base area will have to search, depending on the form of a quadrangle - correct or incorrect. The area of \u200b\u200bthe full surface of the pyramid will again result in the addition of the base area and the full surface area of \u200b\u200bthe predetermined pyramid.
Definition. Side - This is a triangle, in whom one corner lies in the top of the pyramid, and the party opposed to him coincides with the base side (polygon).
Definition. Side edges - These are common side of the side faces. The pyramid has so many ribs how many corners have a polygon.
Definition. Height of the pyramid - This is a perpendicular, lowered from the top to the base of the pyramid.
Definition. Apothem - This is a 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 cross section of a pyramid with a plane passing through the top of the pyramid and the base diagonal.
Definition. Right pyramid - This is a pyramid in which the basis is the right polygon, and the height falls into the center of the base.
The volume and surface area of \u200b\u200bthe pyramid
Formula. Pyramid volume Through the base area and height:
Properties of the pyramid
If all the side ribs are equal, then around the base of the pyramid can be described, and the center of the base coincides with the center of the circle. Also, perpendicular, lowered from the top passes through the center of the base (circle).
If all side ribs are equal, then they are tilted to the base plane at the same angles.
The side ribs are equal when they form with the plane of the base equal angles or if the circle can be described around the base of the pyramid.
If the side faces are tilted to the base plane at one angle, then in the base of the pyramid you can enter the circle, and the peak of the pyramid is designed to its center.
If the side faces are tilted to the base plane at one angle, then the apophems of the side faces are equal.
Properties of the right pyramid
1. The vertex of the pyramid is equidistant from all corners of the base.
2. All side ribs are equal.
3. All side edges are tilted under the same corners to the base.
4. Apofims of all side faces are equal.
5. Areas of all side faces are equal.
6. All faces have the same dihedral (flat) angles.
7. Around the pyramid you can describe the sphere. The center of the described sphere is the intersection point of perpendiculars, which pass through the middle of the ribs.
8. In the pyramid you can enter the sphere. The center of the inscribed sphere will be the intersection point of bisector emanating from the corner between the edge and the base.
9. If the center of the inscribed sphere coincides with the center of the described sphere, then the sum of flat corners at the top is equal to π or vice versa, one angle is π / n, where N is the number of angles at the base of the pyramid.
Pyramid connection with sphere
Around the pyramid, you can describe the sphere when at the base of the pyramid lies a polyhedron around which you can describe the circle (necessary and sufficient condition). The center of the sphere is the intersection point of the planes passing perpendicular through the middle of the side ribs of the pyramids.
Around any triangular or correct pyramid can always be described by the sphere.
In the pyramid, you can enter the sphere if the biss-sector planes of the internal dwarfrani corners of the pyramids intersect at one point (necessary and sufficient condition). This point will be the center of the sphere.
Pyramid connection with cone
The cone is called inscribed in the pyramid if their vertices coincide, and the base of the cone is inscribed in the base of the pyramid.
The cone can be entered into the pyramid if the apophems of the pyramids are equal to each other.
The cone is called the pyramid described around, if their vertices coincide, and the base of the cone is described around the base of the pyramid.
The cone can be described around the pyramid if all the side ribs of the pyramid are equal to each other.
Pyramid connection with cylinder
The pyramid is called inscribed in the cylinder if the top of the pyramid lies on one basis of the cylinder, and the base of the pyramid is written to another base of the cylinder.
The cylinder can be described around the pyramid if around the base of the pyramid you can describe the circle.
The four-edged four faces and four vertices and six ribs, where any two ribs do not have common vertices but not come into contact.
Each peak consists of three faces and ribs that form three corner.
The segment connecting the vertex of the tetrahedron with the center of the opposite face is called median tetrahedron (GM).
Bimedian It is called a segment connecting the mid-opposite ribs that do not come into contact (KL).
All bimedians and medians of the tetrahedral intersect at one point (s). At the same time, the bimedians are divided by half, and medians in respect of 3: 1 starting from the vertex.
Definition. Acreditated pyramid - This is a pyramid in which the apophem is more than half the length of the base side of the base.
Definition. Stupid pyramid - This is a pyramid in which the apophem is less than half the length of the base side.
Definition. Right tetrahedron - A tetrahedron who has all four faces - equilateral triangles. It is one of the five right polygons. In the right tetrahedron, all dumarted angles (between the edges) and triangular angles (at the top) are equal.
Definition. Rectangular tetrahedron A tetrahedron is called a straight angle between three ribs at the top (ribs perpendicular). Three faces form rectangular triangular corner And the faces are rectangular triangles, and the basis of an arbitrary triangle. Apothem of any face is equal to half the side of the foundation that the apophem falls.
Definition. A washing tetrahedron The tetrahedron is called the lateral facets are equal to each other, and the base is the right triangle. Such a tetrahedron serves an isolated triangles.
Definition. Orthocentric tetrahedron A tetrahedron is called all heights (perpendicular), which is omitted from the top to the opposite face, intersect at one point.
Definition. Star pyramid The polyhedron is called the base is the star.
Typical geometric tasks on the plane and in three-dimensional space are the problems of determining the areas of surfaces of different figures. In this article, we give the formula of the side surface area of \u200b\u200bthe correct pyramid of quadrangular.
What is the pyramid?
Let us give a strict geometric definition of the pyramid. Suppose there is some polygon with N sides and with n angles. Choose an arbitrary point of space that will not be in the plane of the specified N-carbon, and connect it from each pin of the polygon. We will get a figure having some volume called an N-coal pyramid. For example, we will show in the figure below how the pentagonal pyramid looks like.
Two important elements of any pyramid is its base (N-square) and therapy. These elements are connected to each other n triangles, which are generally not equal to each other. Perpendicular, lowered from the top to the base, is called the height of the figure. If it crosses the base in the geometric center (coincides with the center of the masses of the polygon), then such a pyramid is called straight. If, in addition to this condition, the base is the right polygon, then the entire pyramid is called proper. The figure below shows how the right pyramids with triangular, quadrangular, pentagonal and hexagonal base look like.
The surface of the pyramid
Before switching to the question about the side surface area of \u200b\u200bthe right pyramid of quadrangular, it is necessary to dwell on the concept of the surface itself.
As mentioned above and shown in the drawings, any pyramid is formed by a set of faces or sides. One side is the basis, and N of the parties are triangles. The surface of the whole figure is the sum of the area of \u200b\u200beach side.
The surface is convenient to study on the example of the scan of the figure. The scan for the correct quadrangular pyramid is shown in the drawings below.
We see that its surface area is equal to the sum of the four squares of the same inaccessible triangles and square square.
The total area of \u200b\u200ball triangles, which form the side sides of the figure, is customary to be called the side surface area. Next, we show how to calculate it for the quadrangular pyramid correctly.
Side surface of the quadrangular correct pyramid
To calculate the side surface area of \u200b\u200bthe specified figure, turn back to the above scan. Suppose we know the side of the square foundation. Denote it with a symbol a. It can be seen that each of the four identical triangles has a base of length a. To calculate their total area, you need to know this value for one triangle. From the course of geometry, it is known that the triangle SA T is equal to the product of the base to the height, which should be divided by half. I.e:
Where H b is the height of an inaccessible triangle, conducted to the base a. For the pyramid, this height is appotable. Now it remains to multiply the obtained expression on 4 to obtain the area S B side surface for the pyramid under consideration:
S B \u003d 4 * S T \u003d 2 * H b * a.
This formula contains two parameters: apotheme and side of the base. If the latter is known in most conditions, then the first one has to calculate, knowing other values. We give formulas for calculating Apotheme H B for two cases:
- when the length of the side edge is known;
- when the height of the pyramid is known.
If you designate the length of the side of the side (side of an equilibried triangle) symbol L, then Apotheme H B is to determine the formula:
h B \u003d √ (L 2 - A 2/4).
This expression is the result of the use of the Pythagorean theorem for the side surface triangle.
If the height h of the pyramid is known, then the H B is designed as follows:
h B \u003d √ (H 2 + A 2/4).
Get this expression is also not difficult if you consider inside the pyramid right triangleformed by Cates H and A / 2 and hypotenuse H b.
We show how to apply these formulas by deciding two interesting tasks.
Task with a well-known surface area
It is known that the side surface area of \u200b\u200bthe quadrangular is 108 cm 2. It is necessary to calculate the value of its length of its Apotheme H B, if the height of the pyramid is 7 cm.
We write the surface formula s b surface of the side through the height. We have:
S B \u003d 2 * √ (H 2 + A 2/4) * a.
Here we simply substituted the appropriate formula of Apotheme into an expression for s b. Erected both parts of equality in the square:
S B 2 \u003d 4 * A 2 * H 2 + A 4.
To find the value A, we will replace the variables:
t 2 + 4 * H 2 * T - S b 2 \u003d 0.
Now we substitute the known values \u200b\u200band solve the square equation:
t 2 + 196 * T - 11664 \u003d 0.
We prescribed only a positive root of this equation. Then the bases of the base of the pyramid will be equal to:
a \u003d √t \u003d √47,8355 ≈ 6,916 cm.
To get the length of Apotheme, it is enough to use the formula:
h B \u003d √ (H 2 + A 2/4) \u003d √ (7 2 + 6.916 2/4) ≈ 7.808 cm.
Side Surface of Heops Pyramid
We define the importance of the side for the largest Egyptian pyramid. It is known that in its foundation there is a square on the side of 230,363 meters. The height of the structure was initially 146.5 meters. We substitute these numbers into the appropriate formula for s b, we get:
S B \u003d 2 * √ (H 2 + A 2/4) * a \u003d 2 * √ (146.5 2 +230.363 2/4) * 230,363 ≈ 85860 m 2.
The value found is a bit more square 17 football fields.
