Definition of a pyramid
Pyramid is a polyhedron, the base of which is a polygon, and its faces are triangles.
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It is worth dwelling on the definition of some components of the pyramid.
She, like other polyhedra, has ribs. They converge to one point called top pyramids. It can be based on an arbitrary polygon. Edge called geometric figure, formed by one of the sides of the base and two nearest ribs. In our case it is a triangle. Height pyramid is the distance from the plane in which its base lies to the top of the polyhedron. For regular pyramid there is also a concept apothems- this is a perpendicular descended 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.
Formula for surface area of a pyramid
To find the total surface area of the pyramid, you need to add the area of the lateral surface and the area of the base.
The simplest case is the case of a regular pyramid, so we will deal with it. Let us 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
Ll 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 basic
S side S_(\text(side)) S side
- area of the lateral surface of the pyramid;
S main S_(\text(basic)) S basic
- area of the base of the pyramid.
An example of solving a problem.
ExampleFind the total area of a triangular pyramid if its apothem is 8 (cm), and at the base there is an equilateral triangle with side 3 (cm)
Solution
L = 8 l=8 l =8
a = 3 a=3 a =3
Let's find the perimeter of the base. Since the base is an equilateral triangle with side a a a, then its perimeter p p p(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 lateral area of the pyramid is:
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 let's 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 using the formula:
S main = 3 ⋅ a 2 4 S_(\text(basic))=\frac(\sqrt(3)\cdot a^2)(4)S basic = 4 3 ⋅ a 2
A a a- side of the triangle.
We get:
S main = 3 ⋅ a 2 4 = 3 ⋅ 3 2 4 ≈ 3.9 S_(\text(basic))=\frac(\sqrt(3)\cdot a^2)(4)=\frac(\sqrt(3 )\cdot 3^2)(4)\approx3.9S basic = 4 3 ⋅ a 2 = 4 3 ⋅ 3 2 ≈ 3 . 9 (see sq.)
Total area:
S = S side + S main ≈ 36 + 3.9 = 39.9 S=S_(\text(side))+S_(\text(main))\approx36+3.9=39.9S=S side + S basic ≈ 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 (cm2). The apothem of a polyhedron is 3 times the side of the base a a a. Find the total surface area of this figure.
Solution
S quad = 36 S_(\text(quad))=36S quad
=
3
6
l = 3 ⋅ a l=3\cdot a l =3
⋅
a
Let's 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
Let's 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(basic))S quad = S basic
All that remains is to find the area of the lateral surface. 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.)
Total area:
S = S side + S main = 216 + 36 = 252 S=S_(\text(side))+S_(\text(main))=216+36=252
Answer: 252 cm sq.
A cylinder is a geometric body bounded by two parallel planes and a cylindrical surface. In the article we will talk about how to find the area of a cylinder and, using the formula, we will solve several problems as an example.
A cylinder has three surfaces: a top, a base, and a side surface.
The top and base of a cylinder are circles and are easy to identify.
It is known that the area of a circle is equal to πr 2. Therefore, the formula for the area of two circles (the top and base of the cylinder) will be πr 2 + πr 2 = 2πr 2.
The third, side surface of the cylinder, is the curved wall of the cylinder. In order to better imagine this surface, let's try to transform it to get a recognizable shape. Imagine that the cylinder is ordinary tin, which does not have a top cover or bottom. Let's make a vertical cut on the side wall from the top to the base of the can (Step 1 in the figure) and try to open (straighten) the resulting figure as much as possible (Step 2).
After the resulting jar is fully opened, we will see a familiar figure (Step 3), this is a rectangle. The area of a rectangle is easy to calculate. But before that, let's return for a moment to the original cylinder. The vertex of the original cylinder is a circle, and we know that the circumference is calculated by the formula: L = 2πr. It is marked in red in the figure.
When the side wall of the cylinder is fully opened, we see that the circumference becomes the length of the resulting rectangle. The sides of this rectangle will be the circumference (L = 2πr) and the height of the cylinder (h). The area of a rectangle is equal to the product of its sides - S = length x width = L x h = 2πr x h = 2πrh. As a result, we received a formula for calculating the area of the lateral surface of the cylinder.
Formula for the lateral surface area of a cylinder
S side = 2πrh
Total surface area of a cylinder
Finally, if we add the area of all three surfaces, we get the formula for the total surface area of a cylinder. The surface area of a cylinder is equal to the area of the top of the cylinder + the area of the base of the cylinder + the area of the side surface of the cylinder or S = πr 2 + πr 2 + 2πrh = 2πr 2 + 2πrh. Sometimes this expression is written identical to the formula 2πr (r + h).
Formula for the total surface area of a cylinder
S = 2πr 2 + 2πrh = 2πr(r + h)
r – radius of the cylinder, h – height of the cylinder
Examples of calculating the surface area of a cylinder
To understand the above formulas, let’s try to calculate the surface area of a cylinder using examples.
1. The radius of the base of the cylinder is 2, the height is 3. Determine the area of the lateral surface of the cylinder.
The total surface area is calculated using the formula: S side. = 2πrh
S side = 2 * 3.14 * 2 * 34.6. Total ratings received: 990.
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 united.
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 using
various formulas. Depending on what data we know about the triangles, we look for their area.
We list some formulas that can be used to 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 a circumscribed circle around a triangle is R .
- S = (a*b)/2 = r² + 2*r*R . This formula should only be applied when the triangle is right-angled.
- 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 its lateral surface. To do this, we will use the above formulas.
In order to calculate the area of the lateral surface of a 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 - area of the lateral surface of the pyramid.
Let's look at an example. Given a regular pyramid, its side faces formed by several equilateral triangles,
« Geometry is the most powerful tool for sharpening our mental abilities».
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 reason like this: we know that the faces of the pyramid are triangles, they are equilateral. We also know what the edge length of this pyramid is. It follows that all triangles have equal sides and 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
So, 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 lateral surface area 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.
When preparing for the Unified State Exam in mathematics, students have to systematize their knowledge of algebra and geometry. I would like to combine all known information, for example, on how to calculate the area of a pyramid. Moreover, starting from the base and side edges to the entire surface area. If the situation with the side faces is clear, since they are triangles, then the base is always different.
How to find 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 irregular one. In the Unified State Exam tasks that interest schoolchildren, there are only tasks with correct figures at the base. Therefore, we will talk only about them.
Regular triangle
That is, equilateral. The one in which all sides are equal and are designated by the letter “a”. In this case, the area of the 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 again the side:
Arbitrary regular n-gon
The side of a polygon has the same notation. For the number of angles, the Latin letter n is used.
S = (n * a 2) / (4 * tg (180º/n)).
What to do when calculating the lateral and total surface area?
Since the base is a regular figure, all faces of the pyramid are equal. Moreover, each of them is an isosceles triangle, since lateral ribs are equal. Then, in order to calculate the lateral area of the pyramid, 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.
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 = 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 apex (α) are given. Then you need to use the following formula to calculate the lateral area of the pyramid:
S = n/2 * in 2 sin α .
Task No. 1
Condition. Find total area pyramid, if its base has 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. Since this is a regular triangle, then P = 3*4 = 12 cm. Since the apothem is known, we can immediately calculate the area of the entire lateral surface: ½*12*√3 = 6√3 cm 2.
For the triangle at the base, you get the following area value: (4 2 *√3) / 4 = 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 cm 2.
Problem No. 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 find out its surface area.
Solution. Since the polyhedron is quadrangular and regular, its base is a square. Once you know the area of the base and side faces, you will be able to calculate the area of the pyramid. The formula for the square is given above. And for 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 the following 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 = 267.576 mm 2.
Answer. The desired value is 267.576 mm 2.
Problem No. 3
Condition. For a regular quadrangular pyramid, you need to calculate the area. The side of the square is known to be 6 cm and the height is 4 cm.
Solution. The easiest way is to use the formula with the product of perimeter and apothem. The first value is easy to find. The second one is a little more complicated.
We will 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 required apothem (hypotenuse of a right triangle) is equal to √(3 2 + 4 2) = 5 (cm).
Now you can calculate the required value: ½*(4*6)*5+6 2 = 96 (cm 2).
Answer. 96 cm 2.
Problem No. 4
Condition. Dana correct side its bases are 22 mm, side ribs are 61 mm. What is the lateral surface area of this polyhedron?
Solution. The reasoning in it is the same as that described in task No. 2. Only there was given a pyramid with a square at the base, and now it is a hexagon.
First of all, the base area is calculated using the above formula: (6*22 2) / (4*tg (180º/6)) = 726/(tg30º) = 726√3 cm 2.
Now you need to find out the semi-perimeter of an isosceles triangle, which is the side face. (22+61*2):2 = 72 cm. All that remains is to use Heron’s formula to calculate the area of each such triangle, and then multiply it by six and add it to the one obtained for the base.
Calculations using Heron's formula: √(72*(72-22)*(72-61) 2)=√435600=660 cm 2. Calculations that will give the lateral surface area: 660 * 6 = 3960 cm 2. It remains to add them up to find out the entire surface: 5217.47≈5217 cm 2.
Answer. The base is 726√3 cm 2, the side surface is 3960 cm 2, the entire area is 5217 cm 2.
What figure do we call a pyramid? Firstly, it 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 shape of triangles converging at one common vertex. Now, having understood the term, let’s find out how to find the surface area of the pyramid.
It is clear that the surface area of such a geometric body is made up of the sum of the areas of the base and its entire lateral surface.
Calculating the area of the base of a pyramid
The choice of calculation formula depends on the shape of the polygon underlying our pyramid. It can be regular, that is, with sides of the same length, or irregular. Let's consider both options.
At the base is a regular polygon
From the school course we know:
- the area of the square will be equal to the length of its side squared;
- The area of an equilateral triangle is equal to the square of its side divided by 4 and multiplied by Square root out of three.
But there is also a general formula for calculating the area of any regular polygon (Sn): you need to multiply the perimeter of this polygon (P) by the radius of the circle inscribed in it (r), and then divide the result by two: Sn=1/2P*r .
At the base is an irregular polygon
The scheme for finding its area is to first divide the entire polygon into triangles, calculate the area of each of them using the formula: 1/2a*h (where a is the base of the triangle, h is the height lowered to this base), add up all the results.
Lateral surface area of the pyramid
Now let’s calculate the area of the lateral surface of the pyramid, i.e. the sum of the areas of all its lateral sides. There are also 2 options here.
- Let us have an arbitrary pyramid, i.e. one with an irregular polygon at its base. Then you should calculate the area of each face separately and add the results. Since the sides of a pyramid, by definition, can only be triangles, the calculation is carried out using the above-mentioned formula: S=1/2a*h.
- Let our pyramid be correct, i.e. at its base lies a regular polygon, and the projection of the top of the pyramid is at its center. Then, to calculate the area of the lateral surface (Sb), it is enough to find half the product of the perimeter of the base polygon (P) and the height (h) of the lateral side (the same for all faces): Sb = 1/2 P*h. The perimeter of a polygon is determined by adding the lengths of all its sides.
The total surface area of a regular pyramid is found by summing the area of its base with the area of the entire lateral surface.
Examples
For example, let's algebraically calculate the surface areas of several pyramids.
Surface area of a triangular pyramid
At the base of such a pyramid is a triangle. Using the formula So=1/2a*h we find the area of the base. We use the same formula to find the area of each face of the pyramid, which also has a triangular shape, and we get 3 areas: S1, S2 and S3. The area of the lateral surface of the pyramid is the sum of all areas: Sb = S1+ S2+ S3. By adding up the areas of the sides and base, we obtain the total surface area of the desired pyramid: Sp= So+ Sb.
Surface area of a quadrangular pyramid
The area of the lateral surface is the sum of 4 terms: Sb = S1+ S2+ S3+ S4, each of which is calculated using the formula for the area of a triangle. And the area of the base will have to be looked for, depending on the shape of the quadrilateral - regular or irregular. The total surface area of the pyramid is again obtained by adding the area of the base and the total surface area of the given pyramid.