A pyramid is a geometric spatial figure whose characteristics are studied in high school with a course on stereometry. In this article, we consider a triangular pyramid, its types, as well as formulas for calculating its surface area.
What kind of pyramid will be discussed?
A triangular pyramid is a figure that can be obtained by connecting all the vertices of an arbitrary triangle with one single point that does not lie in the plane of this triangle. According to this definition, the pyramid in question should consist of an initial triangle, which is called the base of the figure, and three side triangles, which have one common side with the base and are connected to each other at a point. The latter is called the top of the pyramid.
The figure above shows an arbitrary triangular pyramid.
The figure in question may be inclined or straight. In the latter case, the perpendicular, lowered from the top of the pyramid to its base, must intersect it in the geometric center. The geometric center of any triangle is the intersection point of its medians. The geometric center coincides with the center of mass of the figure in physics.
If a regular (equilateral) triangle lies at the base of a straight pyramid, then it is called a regular triangular. In a regular pyramid, all sides are equal to each other and are equilateral triangles.
If the height of the regular pyramid is such that its side triangles become equilateral, then it is called a tetrahedron. In the tetrahedron, all four faces are equal to each other, so each of them can be considered a base.
Pyramid elements
These elements include the faces or sides of a figure, its edges, vertices, height, and apothems.
As shown, all sides of a triangular pyramid are triangles. Their number is 4 (3 lateral and one at the base).
Vertices are the intersection points of three triangular sides. It is not difficult to guess that for the pyramid in question there are 4 of them (3 belong to the base and 1 is the top of the pyramid).
Edges can be defined as lines of intersection of two triangular sides, or as lines that connect every two vertices. The number of edges corresponds to twice the number of vertices of the base, i.e. for a triangular pyramid it is 6 (3 edges belong to the base and 3 edges are formed by side faces).
Height, as noted above, is the length of the perpendicular drawn from the top of the pyramid to its base. If we draw heights from this peak to each side of the triangular base, then they will be called apothemes (or apothemes). Thus, the triangular pyramid has one height and three apothems. The latter are equal to each other for the correct pyramid.
The base of the pyramid and its area
Since the base for the figure in general is a triangle, to calculate its area it is enough to find its height h o and the length of the side of the base a, on which it is lowered. The formula for the area S o the base has the form:
S o = 1/2 * h o * a
If the base triangle is equilateral, then the base area of the triangular pyramid is calculated by the following formula:
S o = √3 / 4 * a 2
That is, the area S o is uniquely determined by the length of the side a of the triangular base.
Side and total area of the figure
Before considering the area of a triangular pyramid, it is useful to bring its development. She is depicted in the figure below.
The area of this scan, formed by four triangles, is the total area of the pyramid. One of the triangles corresponds to the base, the formula for the considered value of which was written above. Three lateral triangular faces in total form the lateral area of the figure. Therefore, to determine this value, it is enough to apply the formula written above for an arbitrary triangle to each of them, and then add the three results obtained.
If the pyramid is correct, then the calculation of the lateral surface area is facilitated, since all the side faces are the same equilateral triangles. Denote by h b the length of the apotheme, then the lateral surface area S b can be determined as follows:
S b = 3/2 * a * h b
This formula follows from the general expression for the area of a triangle. Number 3 appeared in the numerators due to the fact that the pyramid has three side faces.
The apothem h b in the regular pyramid can be calculated if the height of the figure h is known. Applying the Pythagorean theorem, we obtain:
h b = √ (h 2 + a 2/12)
Obviously, the total surface area S of the figure is equal to the sum of its areas of the side surface and the base:
S = S o + S b
For the correct pyramid, substituting all known quantities, we obtain the formula:
S = √3 / 4 * a 2 + 3/2 * a * √ (h 2 + a 2/12)
The area of the triangular pyramid depends only on the length of the side of its base and on the height.
Task example
It is known that the lateral edge of a triangular pyramid is 7 cm, and the side of the base is 5 cm. It is necessary to find the surface area of the figure, if it is known that the pyramid is correct.
We use the general equality:
S = S o + S b
Area S o equal to:
S o = √3 / 4 * a 2 = √3 / 4 * 5 2 ≈ 10.825 cm 2 .
To determine the area of the lateral surface, it is necessary to find the apotheme. It is not difficult to show that through the length of the side rib a b it is determined by the formula:
h b = √ (a b 2 - a 2/4) = √ (7 2 - 5 2/4) ≈ 6.538 cm.
Then the area S b is equal to:
S b = 3/2 * a * h b = 3/2 * 5 * 6.538 = 49.035 cm 2 .
The total area of the pyramid is:
S = S o + S b = 10.825 + 49.035 = 59.86 cm 2 .
Note that in solving the problem we did not use the value of the height of the pyramid in the calculations.