In this explainer, we will learn how to calculate the lateral and total surface areas of pyramids using their formulas.

### Definition: Pyramids

Pyramids are 3D geometric shapes, or solid objects, where the base is a polygon (triangle, square, rectangle, pentagon, etc.) and all other sides are triangles that meet at the apex or vertex.

A right pyramid is a pyramid whose apex lies above the centroid of the base.

A regular pyramid is a right pyramid whose base is a regular polygon: all the sides of the base are of equal length, and all the pyramidβs lateral edges are of equal length.

### Definition: Lateral and Total Surface Areas

The lateral surface area of a pyramid is the total surface area of only the lateral sides of the pyramid; that is, of the triangular faces that meet at the vertex.

The total surface area of a pyramid is the total surface area of the pyramid; that is, the sum of the areas of its lateral sides plus the area of the base.

Drawing the net of a pyramid helps us visualize all the faces. Here, the net of a regular square pyramid is shown. The five faces are a square and four triangles. The slant height is the altitude, or height, of the triangle comprising the face. If we know the square side or, more generally, all the sides of the base and the slant height of each face, it is then possible to work out the area of the triangular face as

In this case of a right square pyramid, the lateral surface area is and the total surface area is

### Example 1: Finding the Lateral Surface Area of a Square Pyramid

If the given figure was folded into a square pyramid, determine its lateral surface area.

### Answer

In this question, we have the net of a regular square pyramid, and we are given the squareβs side, 14 cm, and slant height, 15 cm.

The area of each triangular face is

The lateral surface area is then 4 times the area of each lateral face; that is, .

### Example 2: Finding the Total Surface Area of a Square Pyramid

Determine the surface area of the given square pyramid, given that all of its triangular faces are congruent.

### Answer

It is said here that all triangular faces are congruent, so this is a regular pyramid. Hence, the total surface area of the pyramid is

The base is a square of side 37 in, and its area is given by squaring its side :

Let us find the area of one lateral face. Each face is made of a triangle whose base, , is 37 in and whose height, , is 44 in. Its area is

The total area is then

### Example 3: Finding the Total Surface Area of a Square Pyramid given the Squareβs Side and the Lateral Edge

Find the total surface area of the given net to the nearest hundredth.

### Answer

We have here the net of a regular square pyramid. We know the side of the square, 2 cm, and the side of the triangle that is not shared with the square, 3.1 cm.

To find the total surface area, we need to find the slant height of the pyramid; that is, the height of the triangular lateral faces.

Since the lateral face is an isosceles triangle, its height cuts the triangle in two congruent right triangles. We can apply the Pythagorean theorem in one of these right triangles, with s.h. the slant height of the pyramid and the other leg being half of the squareβs side (that is, 1 cm): Taking away 1 form each side, we get Taking the square root of each side, we get

The area of each triangular face is

The area of a square is its side squared, so the area of the base (the square of side 2 cm) is

The total surface area is

### Example 4: Finding the Total Surface Area of a Regular Triangular Pyramid

Determine the total area of the following net approximated to the nearest hundredth.

### Answer

We have here a net of a regular pyramid: all the lateral faces are equilateral triangles. Indeed, the whole net is an equilateral triangle, so its anglesβ measure is , and the lateral faces are isosceles triangles with an angle of , meaning their two other anglesβ measure is half of (that is, as well): they are equilateral triangles.

So far, we do not know what type of triangle the base is. However, as all lateral triangles are congruent equilateral triangles, the triangle formed by their three bases is an equilateral triangle congruent to the lateral triangles.

To find the total surface area of this pyramid, we can either find the area of one of these equilateral triangles and multiply it by 4 or find directly the area of the total net, which is an enlargement of the smaller equilateral triangle by a scale factor of 2.

Letβs look at the bigger triangle (the whole net). We know it is an equilateral triangle of height 12 cm (double the height of the smaller triangle). We need to find its height.

Let be the side of one smaller triangle; we can apply the Pythagorean theorem in the right triangle shown in the diagram: Taking away from each side, we get Dividing each side by 3, we get Taking the square root of each side, we have We have found that the base of the bigger triangle is

Its area is, therefore,

### Example 5: Finding the Total Surface Area of a Regular Square Pyramid given Its Lateral Surface Area and Height

A square pyramid has a lateral surface area of
42 yd^{2}. If its slant height is
3 yd, determine its total surface area.

### Answer

We have here a regular square pyramid, so each face is an isosceles triangle whose base,
, is a side of the square base of the pyramid. The area of each triangular
face is given by
where is the height of the triangular face; that is, the slant height of
the pyramid. There are four triangular lateral faces, so we have
which gives, replacing by
42 yd^{2},

We can now find the value of , which is the side of the square base and, in turn, find the area of the square base that we need to add to the lateral area to find the total surface area.

Let us start by finding . Replacing by and plugging in the value of (3 yd) into the above equation, we get Multiplying both sides by 2, we find and dividing both sides by 3, we get

We can now find the area of the base of the pyramid:

The total surface area of the pyramid is

The total surface area of the square pyramid is 91 yd^{2}.

### Key Points

- Pyramids are 3D geometric shapes, or solid objects, where the base is a polygon (triangle, square, rectangle, pentagon, etc.) and all other sides are triangles that meet at the apex or vertex.
- A right pyramid is a pyramid whose apex lies above the centroid of the base. A regular pyramid is a right pyramid whose base is a regular polygon: all the sides of the base are of equal length, and all the pyramidβs lateral edges are of equal length.
- The lateral surface area of a pyramid is the total surface area of only the lateral sides of the pyramid; that is, of the triangular faces that meet at the vertex.
- The total surface area of a pyramid is the total surface area of the pyramid; that is, the sum of the areas of its lateral sides plus the area of the base.
- Drawing the net of a pyramid helps us visualize all the faces so that working out the area of each of them is easier.