Triangular pyramid and formulas for determining its area

2024 Author: Angel Austin | [email protected]. Last modified: 2023-12-17 05:14

Pyramid is a geometric spatial figure, the characteristics of which are studied in high school in the course of solid geometry. In this article, we will consider a triangular pyramid, its types, as well as formulas for calculating its surface area.

Which pyramid are we talking about?

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 under consideration should consist of an initial triangle, which is called the base of the figure, and three side triangles that 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 picture above shows an arbitrary triangular pyramid.

The figure under consideration can be oblique or straight. In the latter case, the perpendicular dropped from the top of the pyramid to its base must intersect it at the geometric center. the geometric center of anytriangle is the point of intersection 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 one. In a regular pyramid, all sides are equal to each other and are equilateral triangles.

If the height of a regular pyramid is such that its side triangles become equilateral, then it is called a tetrahedron. In a 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 side and one at the base).

The vertices are the intersection points of the three triangular sides. It is not difficult to guess that for the pyramid under consideration there are 4 of them (3 belong to the base and 1 to the top of the pyramid).

Edges can be defined as lines where two triangular sides intersect, or as lines that connect every two vertices. The number of edges corresponds to twice the number of base vertices, that is, for a triangular pyramid it is 6 (3 edges belong to the base and 3 edges are formed by the 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 vertex to each side of the triangular base,then they will be called apotems (or apothems). Thus, the triangular pyramid has one height and three apothems. The latter are equal to each other for a regular pyramid.

The base of the pyramid and its area

Since the base for the figure under consideration is generally 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 of the base is:

S_o=1/2h_oa

If the triangle of the base is equilateral, then the area of the base of the triangular pyramid is calculated using the following formula:

S_o=√3/4a²

That is, the area S_ois uniquely determined by the length of 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 show its development. She is pictured below.

The area of this sweep 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 together form the lateral area of the figure. Therefore, to determine this value, it is enough to apply the above formula for an arbitrary triangle to each of them, and then add the three results.

If the pyramid is correct, then the calculationlateral surface area is facilitated, since all lateral faces are identical equilateral triangles. Denote h_bthe length of the apothem, then the area of the lateral surface S_b can be determined as follows:

S_b=3/2ah_b

This formula follows from the general expression for the area of a triangle. The number 3 appeared in the numerators due to the fact that the pyramid has three side faces.

Apotema h_b in a regular pyramid can be calculated if the height of the figure h is known. Applying the Pythagorean theorem, we get:

h_b=√(h²+ a²/12)

Obviously, the total area S of the figure's surface is equal to the sum of its side and base areas:

S=S_o+ S_b

For a regular pyramid, substituting all known values, we get the formula:

S=√3/4a²+ 3/2a√(h²+ a ²/12)

The area of a triangular pyramid depends only on the length of the side of its base and on the height.

Example problem

It is known that the side edge of a triangular pyramid is 7 cm, and the side of the base is 5 cm. You need to find the surface area of the figure if you know that the pyramid is regular.

Use a general equality:

S=S_o+ S_b

Area S_ois equal to:

S_o=√3/4a² =√3/45^{2 ≈10, 825cm}2.

To determine the lateral surface area, you need to find the apotema. It is not difficult to show that through the length of the side edge a_b it is determined by the formula:

h_b=√(a_b²- a^{2 /4)=√(7} 2^{- 5}2^{/4) ≈ 6.538 cm.}

Then the area of S_b is:

S_b=3/2ah_b=3/256, 538=49.035 cm ².

The total area of the pyramid is:

S=S_o+ S_b=10.825 + 49.035=59.86cm².

Note that when solving the problem, we did not use the value of the height of the pyramid in the calculations.