The pyramid is a perfect geometric figure, the shape of which can be found in some objects from our lives, for example, in magic amulets. In this article, we will consider how to find the area of a regular triangular pyramid and give the corresponding formula.
Triangular pyramid or tetrahedron
In geometry, a pyramid is a geometric object that consists of n triangles and one n-gon. All triangles intersect at a point called the vertex of the figure, and the n-gon is its base. It is not difficult to guess that the name of the pyramid is determined by the number of sides of the n-gon.
In accordance with the topic of this article, we will consider a triangular pyramid. Its n-carbon base is also a triangle. Therefore, such a pyramid consists of 4 triangular faces, each of which can be considered as a base. The triangular pyramid has 4 equal vertices and 6 edges. Since the number of sides of the figure is 4, it is also called the tetrahedron. For clarity, we present the image of a triangular pyramid:
The figure shows a top view of the figure.
Regular pyramid with a triangular base and its development
In the general case, the triangle at the base may be an arbitrary shape. However, if this triangle is an equilateral figure, and the perpendicular, lowered from the top to the base, intersects the triangle in its center, then we are talking about a regular pyramid.
A regular triangular pyramid consists of an equilateral triangle, the side of which is denoted by the letter a, and three isosceles triangles that are equal to each other. With a certain ratio of the height of the figure h and the length a, isosceles triangles can become equilateral, then all four faces of the pyramid will be equal to each other.
To determine the area of a considered figure, it is easiest to scan it onto a plane. The figure below shows what this scan is.
Four triangles are shown here, of which equilateral is the base of the pyramid, and three isosceles figures make up its side surface. The sum of the areas of all triangles forms the area of a regular triangular pyramid.
Square formula
From the planimetric course, every student knows how to find the area of a triangle. To do this, make the following calculations:
S 3 = 1/2 * a * h a .
Here a is the base of the triangle, h a is its height (the index a is introduced to distinguish this value from the height of the pyramid h).
In the case of an equilateral triangle, its height is:
h a = √3 / 2 * a.
Then the formula for the base area of the regular triangular pyramid takes the form:
S 3 = 1/2 * a * √3 / 2 * a = √3 / 4 * a 2 .
That is, to determine the area of the base, it is enough to know only the length of its side.
To determine the area of the lateral surface S b , we introduce the concept of apothem of the pyramid. Apothem is the height of any of the side triangles, which is lowered from the top of the pyramid to the side of the base. All apothems in the regular pyramid are equal to each other. Denote their lengths by h b . Since the pyramid under consideration consists of three sides, the area S b is calculated by the formula:
S b = 3/2 * a * h b .
It remains to take the last step to write down the area formula of the regular triangular pyramid:
S = S 3 + S b = √3 / 4 * a 2 + 3/2 * a * h b .
Note that the surface area of the considered geometric figure is determined uniquely if you know its two linear parameters (a and h b ).
The solution of the problem
The height of the base of the regular triangular pyramid is 10 cm. The height of the figure itself is two times the length of the side of the base. What is the surface area of this pyramid?
Since we know the value of the height of the base and the relationship between the height of the figure with the side of an equilateral triangle, this information is enough to answer the question of the problem. First of all, we define the side a and the value height h, we have:
h a = √3 / 2 * a;
a = 2 * h a / √3 = 11.547 cm;
h = 2 * a = 23.094 cm.
In the paragraph above, the formula for S was given, but we cannot use it at this stage of the problem, since we do not know the apothem h b . The latter is easy to calculate if you see that it is a hypotenuse in a right-angled triangle whose legs are height h and 1/3 of the height of the base. Given the above, we get:
h b = √ (h 2 + (1/3 * h a ) 2 ) = √ (23,094 2 + (1/3 * 10) 2 ) = 23.333 cm.
Note that the value of h b is slightly larger than the height h of the figure.
We found all the parameters that are in the formula for S, now you can calculate the desired area:
S = √3 / 4 * a 2 + 3/2 * a * h b = √3 / 4 * 11.547 2 + 3/2 * 11.547 * 23.333 = 409.14 cm 2 .
The formula for S is written in this form, which allows you to separately determine the area of the base and side surface.