How to find the base area of a pyramid. Lateral surface area of a regular quadrangular pyramid: formulas and examples of problems
Instruction
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 \u003d (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 \u003d (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 \u003d (a * b * c) / 4 * R, where R is the radius of the triangle described around the circle;
S \u003d (a * b) / 2 \u003d r² + 2 * r * R (if the triangle is right-angled);
S = S = (a²*√3)/4 (if the triangle is equilateral).
In fact, these are just the most basic known formulas to find the area of a triangle.
Having calculated, using the above formulas, the areas of all triangles that are the faces of the pyramid, we can begin to calculate the area of \u200b\u200bthis 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 in a formula like this:
Sp = ΣSi, where Sp is the lateral area, 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: a regular pyramid is given, 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 of the lateral surface are 17 cm. Therefore, in order to calculate the area of \u200b\u200bany 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, we 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 the base, and the vertex is projected into the center of this polygon), then to calculate the entire side surface, it is enough to multiply the perimeter of the base (that is, the sum of the lengths of all sides of the polygon that lies at the base pyramid) by the height of the side face (otherwise called 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, then you will have to separately calculate the areas of all 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 are calculated, it remains only to add them up to get the area of the side surface of the pyramid.
Then you need to calculate the area of \u200b\u200bthe 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 whose all sides 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 formed by a pyramid and its section parallel to the base. Finding the area of the lateral surface 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 the apothem. Consider an example of calculating the area of the lateral surface of a truncated pyramid. Suppose we are given a regular quadrangular 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.
A pyramid is a polyhedron, one of whose faces (base) is an arbitrary polygon, and the other faces (sides) are triangles having a common vertex. According to the number of corners 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 remaining faces are triangles with a common vertex. The apothem is the height of the side face correct pyramid, which is drawn from its top.
In a regular triangular pyramid SABC R- middle of the rib AB, S- top.
It is known that SR = 6, and the lateral surface area is 36
.
Find the length of the segment BC.
Let's make a prank. In a regular pyramid, the side faces are isosceles triangles.
Line segment SR- the median lowered to the base, and hence the height of the side face.
The lateral surface area of a regular triangular pyramid is equal to the sum of the areas
three equal sides S side = 3 S ABS. From here S ABS = 36: 3 = 12- face area.
The area of a triangle is half the product of its base times its height.
S ABS = 0.5 AB SR. Knowing the area and height, we find the side of the base AB = BC.
12 = 0.5 AB 6
12 = 3 AB
AB = 4
Answer: 4
You can approach the problem from the other end. Let the side of the base AB = BC = a.
Then the area of the face S ABS = 0.5 AB SR = 0.5 a 6 = 3a.
The area of each of the three faces is 3a, the area of three faces is 9a.
According to the condition of the problem, the area of the lateral surface of the pyramid is 36.
S side = 9a = 36.
From here a = 4.
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 faces are combined.
In order to calculate the area of the pyramid, it is worth determining that its lateral surface consists of several triangles. And we can easily find their areas using
various formulas. Depending on what data of triangles we know, we are looking 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 the sides of the triangle a , b , and the angle between them is β .
- S = (r*(a + b + c))/2 . Here 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 if the triangle is a right triangle.
- S = (a²*√3)/4 . We apply this formula to an equilateral triangle.
Only after we calculate the areas of all the triangles that are the faces of our pyramid, can we calculate the area of \u200b\u200bits lateral surface. To do 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 this with the formula:
Sp = ΣSi
Here Si is the area of the first triangle, and S P is the area of the lateral surface of the pyramid.
Let's look at an example. Given a regular pyramid, its lateral faces are formed by several equilateral triangles,
« Geometry is the most powerful tool for the refinement of 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's find the area of the lateral surface of this pyramid.
We reason like this: we know that the faces of the pyramid are triangles, they are equilateral. We also know what is the length of the edge of this pyramid. It follows that all triangles have equal 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 lateral surface of the pyramid can be easily calculated using the following formula: 125.137 cm² * 4 = 500.548 cm²
Our answer is the following: 500.548 cm² - this is the area of the lateral surface of this pyramid.
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 side area pyramids, you will need a formula consisting of the sum of identical monomials. The number of terms is determined by the number of sides of the base.
Square 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.
Solution. You need to start by calculating the perimeter of the base. Because it right 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 \u003d 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.
Solution. 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. The correct quadrangular pyramid you need to calculate the area. In it, the side of the square is 6 cm and the height is 4 cm.
Solution. 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?
Solution. 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.
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 (the 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.
Definition. Truncated pyramid (pyramidal prism)- This is a polyhedron that is located between the base of the pyramid and a section plane parallel to the base. Thus the pyramid has a large base and a smaller base which is similar to the larger one. The side faces are trapezoids. Definition. triangular pyramid(tetrahedron)- this is a pyramid in which three faces and the base are arbitrary triangles.
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. inclined pyramid is a pyramid in which one of the edges forms an obtuse angle (β) with the base. Definition. Rectangular pyramid is a pyramid in which one of the side faces is perpendicular to the base.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. He 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 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 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.
Definition. Bipyramid- a polyhedron consisting of two different pyramids (pyramids can also be cut) having common ground, and the vertices lie on opposite sides of the base plane.