Surface area and volume of a pyramid

A pyramid is a geometric body whose surface consists of several flat surfaces. All the flat faces converge at a point above the base of the pyramid at a point called apex. The area below the top is called the base area.

We divide the pyramids into two large groups:

This is a somewhat loose description of upright and oblique pyramids. The properties of vertical pyramids will be discussed in more detail below.

As we have seen, there are several types of pyramids, and we can ascribe some common features to all of them. The properties of the pyramids will be looked at in detail below.

A pyramid is an geometric body bounded by:

The pyramid is characterized by the following elements:

In the introduction, we saw that we distinguish between upright and oblique pyramids. Oblique pyramids are too demanding for the framework of primary school, so in the following we will focus only on vertical pyramids.

A pyramid is upright if all the side edges are the same length. This, in turn, means that all the triangles that make up the mantle of the pyramid are isosceles. However, the triangles in the coat are not necessarily consistent. As we shall see below, the conformity of the side faces applies only to regular and uniform pyramids.

These features make it easier for us to calculate the area and volume of the pyramid, which we will learn about below.

The surface of a pyramid is its surface, which consists of several boundary surfaces. We calculate it by summing the areas of all its boundary surfaces. The boundary surfaces of the prism are:

The entire surface of the pyramid is arranged in several planes. To make the surface easier to imagine, the boundary surfaces are spread out on one plane.

If we extend the boundary surfaces of the pyramid to the plane, we obtain the grid of the pyramid. The entire grid represents a composite geometric figure.

The calculation of the area of the pyramid can be imagined as the calculation of the area of the composite geometric figure. In doing so:

The base of the pyramid is n-angle. Therefore, there is no single formula for calculating its area. However, we can write:

As we have already said, the lateral surface of an upright pyramid consists of isosceles triangles.

The total surface area of the pyramid is calculated by summing the areas:

The volume of a pyramid is the size of the space that the pyramid takes up.

We know several types of pyramids, which are distinguished by their properties. The properties of pyramids are the basis for their naming. Let's see what properties distinguish them.

The pyramid is named after the n-angle, which represents the base surface.

In general we can write:

Let’s focus on the edges of the pyramid. Specific examples are pyramids that have the following:

Let’s look at both groups of pyramids in more detail.

A pyramid that has consistent base edges is called a regular base pyramid.

The base edges represent the sides of the base area. A polygon with the same length of sides is correct, so we write similarly for the pyramid.

If a pyramid has all the edges (both base and lateral) of the same length, it is called a regular pyramid.

For the equilateral pyramid we can also write the following:

For special cases of pyramids, simplified forms for calculating area and volume apply. We derive them from the basic formulas taking into account the properties of each pyramid.

We will look at some examples in more detail:

vertical pyramids.

Let’s look at the three-sided pyramids first.

A regular three-sided pyramid has an equilateral triangle as its base:

Area

To facilitate the calculation of the area, we draw a grid of the pyramid:

The plane of the base plane of a right three-sided pyramid is equal to the plane of an equilateral triangle:

The lateral surface of the pyramid consists of three consistent isosceles triangles. We denote the area of the triangle in the lateral surface by and derive the equation for calculating the lateral surface area of a regular three-sided pyramid. We start from the equation:

We derive it in terms of the height :

We have obtained the equation for the lateral surface area of the regular three-sided pyramid.

We also derive the equation for the total surface area of the regular three-sided pyramid with the general equation for the area of the pyramid:

We derive it:

We obtained the equation for the area of a regular three-sided pyramid.

To calculate the slant height we can use the side height .

The side face of the pyramid is an isosceles triangle. If we halve it perpendicular to the base, we get two right triangles:

For both the left and right right triangles we can write Pythagoras theorem:

The rule applies generally to all regular upright pyramids.

Volume

We derive the equation for the volume of a regular three-sided pyramid with the general equation for the volume of the pyramid:

Write it down:

We have obtained the equation for the volume of an equilateral triangular base pyramid.

The four-sided vertical pyramid has a rectangle for the base surface:

Area

To facilitate the calculation of the area, we draw a grid of the pyramid:

The area of the base surface of a four-sided pyramid is equal to the area of the rectangle:

Let's write another equation for calculating the lateral surface area. The side faces are isosceles triangles to which:

The triangles that make up the lateral surface area of the pyramid are therefore in pairs. We denote the area of the first pair by and the second by .

To get the area of the lateral surface, we will start from the equation:

We derive the equation for the area of the lateral:

We obtained the equation for the area of the mantle of the four-sided pyramid.

Each side face of the pyramid is an isosceles triangle. These isosceles triangles can be of different sizes. The pyramid, which has a rectangle as its base, has in the lateral:

If we bisect an isosceles triangle with the base a perpendicular to the base, two right triangles are formed, for which we can write The Pythagorean Theorem, with which we calculate the side edge of the pyramid:

In the same way as above, halve the isosceles triangle with the base b. In this case, too, two right triangles are formed, for which we can write The Pythagorean Theorem:

Write down the rule for using the Pythagorean theorem in calculating the side edge of a pyramid:

We also derive the equation for the total surface area of an upright four-sided pyramid with the general equation for the area of the pyramid:

Let's derive:

We obtained the equation for the total surface area of an upright four-sided pyramid.

Volume

We also derive the equation for the volume of an upright four-sided pyramid with the general equation for the volume of the pyramid:

Let's derive:

We obtained the equation for the volume of an upright four-sided pyramid.

Forms for calculating side heights and side edge

If we cut a four-sided pyramid with the plane on which the height of the pyramid lies and two of the side heights, we get an isosceles triangle. Two cross-sections are possible:

In both cases, we divide an isosceles triangle perpendicular to the base and obtain two right triangles in which the Pythagorean theorem holds:

Let us write Pythagoras' theorem again:

The four-sided pyramid can also be cut with the plane on which the top of the diagonal of the base plane lies. On this plane lies the height of the pyramid and two of the side edges. Here, too, we obtain an isosceles triangle:

We halve the isosceles triangle perpendicular to the base. We obtain two right triangles in which Pythagoras' theorem holds:

A regular four-sided vertical pyramid has a square as its base:

Area

To facilitate the calculation of the area, we draw a grid of the pyramid:

The area of the base area of a regular four-sided pyramid is equal to the area of the square.

The lateral surface of the pyramid consists of four consistent isosceles triangles. We denote the areas of the triangles in the lateral surface by and derive the equation for the area of the lateral surface of a regular four-sided pyramid. The derivation is based on the equation:

Let's derive the equation for the lateral surface area:

We obtained the equation for the area of the lateral surface of a regular four-sided pyramid.

The side face of the pyramid is an isosceles triangle. If we halve it perpendicular to the base, we get two right triangles:

For both the left and right right triangles we can write The Pythagorean Theorem:

We also derive the equation for the total surface area of a regular four-sided pyramid with the general equation for the area of the pyramid:

Let's derive it:

We obtained the equation for the total surface area of a regular four-sided pyramid.

Volume

We also derive the equation for the volume of a regular vertical four-sided pyramid with the general equation for the volume of the pyramid:

Let's derive it:

We obtained the equation for the volume of a regular upright four-sided pyramid.

Forms for side height and side edge

The correct four-sided pyramid can be cut through the top:

In both sections we get an isosceles triangle. This isosceles triangle is halved into two right triangles, on which we can use the Pythagorean theorem. This gives us two forms to help us calculate in the tasks: