Lesson Explainer: Surface Areas of Pyramids
Mathematics • Second Year of Secondary School
In this explainer, we will learn how to calculate the lateral and total surface areas of pyramids using their formulas.
Pyramids are 3D geometric shapes, or solid objects, where the base is a polygon and all other faces, called lateral faces, are triangles that meet at the apex, or vertex.
A pyramid can also have other polygons as its base, like a pentagon or a hexagon, but we will focus on pyramids with triangular or rectangular bases. Additionally, we will assume the following symmetry in our pyramids.
Definition: Right and Regular Pyramids
A right pyramid is a pyramid whose perpendicular height meets the
base at its center (or centroid). In other words, the apex of a right pyramid is located directly above the center of the base. A regular pyramid is a right pyramid whose base is a regular polygon.
We will assume that all pyramids appearing in this explainer are right pyramids. This also means that a square pyramid is assumed to be a regular pyramid since a square is a regular polygon. A regular pyramid has a special property that simplifies the task of finding the surface area.
Property: Regular Pyramids
All lateral faces of a regular pyramid are congruent triangles.
This property is a statement of rotational symmetries of a regular pyramid. Using the perpendicular height of the pyramid as the rotational axis, we see that this property tells us that a regular pyramid has a rotation symmetry where the order of the symmetry is equal to the number of vertices of the base.
To find the total surface area of a pyramid, we need to sum the area of the base and the areas of the lateral faces. Often, it is helpful to visualize these areas by drawing the net of a pyramid. For instance, the net of a square pyramid is given below.
From this net, we can find a useful formula for the area of a lateral face.
Definition: Lateral and Total Surface Areas
The area of each lateral face is given by
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 apex.
The total surface area of a pyramid is the sum of the areas of its lateral sides and the area of its base.
In our first example, we will compute the lateral surface area of a square pyramid.
Example 1: Finding the Lateral Surface Area of a Pyramid
If the given figure was folded into a square pyramid, determine its lateral surface area.
Answer
In this example, we are given the net of a square pyramid, and we are given the base length, 14 cm, and slant height, 15 cm. The lateral faces of a pyramid are the triangular faces of the pyramid that meet at the apex. In the given net, these are the four triangles surrounding the square base. Therefore, the lateral surface area of this pyramid is the total area of the four triangles.
Recall that the lateral faces of a (regular) square pyramid are congruent. Hence, we can first compute the area of one of these triangles and then multiply this area by 4 to obtain the total area of the four triangles. The area of each triangular face is given by
The lateral surface area is then 4 times the area of each lateral face—that is,
In the previous example, we computed the lateral surface area of a regular square pyramid by using the property that the lateral faces are congruent. We can compute the total surface area of the square pyramid by adding the area of the base to the lateral area. This means
In our next example, we will find the total surface area of a square pyramid by drawing a net.
Example 2: Finding the Total Surface Area of a Square Pyramid given the Side Length of the Base and the Slant Height
Find the surface area of the given square pyramid.
Answer
The surface area of a pyramid is the sum of the area of the square base and the areas of the lateral faces, which are the triangular faces meeting at the apex. We recall that the lateral faces of a (regular) square pyramid are congruent triangles. We can better visualize the surface area by drawing a net of the given pyramid.
From the net, we can note that
Since the four lateral faces are congruent, we can obtain the lateral area by multiplying the area of one face by 4:
The base of this pyramid is a square with side length 22 m, so
Summing these two areas, we obtain
Hence, the surface area of the given square pyramid is 1 540 m2.
In the previous example, we computed the surface area of a regular square pyramid from the given base length and slant height. We can also consider the surface area of a regular triangular pyramid, but the area of the base will be more difficult to find in this case. To find the area of the base from a given base length, we need to first find the height of the triangle by using the Pythagorean theorem.
In the next example, we will compute the surface area of a regular triangular pyramid using this method.
Example 3: Finding the Total Surface Area of a Regular Triangular Pyramid given the Side Length of the Base and the Slant Height
Find the total surface area of the regular pyramid in the given figure, approximating the result to the nearest hundredth.
Answer
The surface area of a pyramid is the sum of the area of the triangular base and the areas of the lateral faces, which are the triangular faces meeting at the apex. We recall that the lateral faces of a regular pyramid are congruent triangles. We can better visualize the surface area by drawing a net of the given pyramid.
From the net, we can note that
Since the three lateral faces are congruent, we can obtain the lateral area by multiplying the area of one face by 3:
Next, let us find the area of the base. The base is the equilateral triangle in the next figure whose side length is 33.5 cm. To find the area of the base, we first need to find the height of the equilateral triangle. We can form a right triangle by drawing the height of the base triangle.
From the diagram above, we can see the highlighted right triangle where the height is labeled as an unknown constant . The hypotenuse of this right triangle is 33.5 cm. Recalling that the height of an equilateral triangle is the perpendicular bisector of the base, we know that the remaining side of the right triangle is exactly half of the bottom side of the equilateral triangle. Hence, the base of the right triangle has length
We can use the Pythagorean theorem to write
Rearranging this equation and taking the positive square root, we obtain
Hence, the height of the base triangle is cm. Using the given base length of 33.5 cm, we can obtain
Summing these two areas, we obtain
The surface area of the given regular pyramid is 2 420.57 cm2 to the nearest hundredth.
So far, we computed the surface area of a regular pyramid when we were given its base length and slant height. In the next example, we will compute the surface area of a square pyramid when we are given the base length and the perpendicular height.
Example 4: Finding the Total Surface Area of a Square Pyramid given the Side Length of the Base and the Height
Find the total surface area of the given regular pyramid, and approximate the result to the nearest hundredth.
Answer
The surface area of a pyramid is the sum of the area of the base and the areas of the lateral faces, which are the triangular faces meeting at the apex. We recall that the lateral faces of a regular square pyramid are congruent triangles. We can better visualize the surface area by drawing a net of the given pyramid.
The slant height, labeled in the net above, is not provided in this example, so we need to begin by finding this length. Since this is a regular pyramid, we know that the perpendicular height of the pyramid meets the base at its center. The perpendicular height is the blue-dashed line segment of length 37 cm in the given diagram. This diagram also gives another blue-dashed line from the center of the base to the midpoint of one of the sides. We can form a right triangle with these two blue-dashed lines by adding the height of this lateral face to this diagram.
We can see that the slant height is the hypotenuse of this right triangle. The perpendicular height of the pyramid, with length 37 cm, is one of the sides, and the remaining side of this right triangle is half of the given base length: . Using the Pythagorean theorem, we can write
This leads to
Now that we know the slant height, we can find the area of a lateral face:
Since the four lateral faces are congruent, we can obtain the lateral area by multiplying the area of one face by 4:
The base of this pyramid is a square with side length 32 cm, so
Summing these two areas, we obtain
Hence, the surface area of the given square pyramid is 3 603.92 cm2 rounded to the nearest hundredth.
In our final example, we will find a missing slant height when we are given the total surface area of a regular triangular pyramid.
Example 5: Finding the Slant Height of a Regular Triangular Pyramid given Its Total Area and the Side Length of the Base
The total surface area of the following triangular pyramid is 958 square centimetres, and its base is an equilateral triangle with a side length of 19 centimetres. Determine its slant height to the nearest tenth.
Answer
The surface area of a pyramid is the sum of the area of the triangular base and the areas of the lateral faces, which are the triangular faces meeting at the apex. Since the base of this pyramid is an equilateral triangle, we can assume that this is a regular pyramid. We recall that the lateral faces of a regular pyramid are congruent triangles. We can better visualize the surface area by drawing a net of the given pyramid.
In the net above, we have labeled the unknown slant height by . We will find the total surface area of this pyramid using the unknown constant , then we will find the value of in the end by setting the expression for the total area equal to the given value.
The area of one lateral triangle is given by
Since the three lateral faces are congruent, we can obtain the lateral area by multiplying the area of one face by 3:
Next, let us find the area of the base. The base is the equilateral triangle in the next figure whose side length is 19 cm. To find the area of the base, we first need to find the height of the equilateral triangle. We can form a right triangle by drawing the height of the base triangle.
From the diagram above, we can see the highlighted right triangle where the height is labeled as an unknown constant . The hypotenuse of this right triangle is 19 cm. Recalling that the height of an equilateral triangle is the perpendicular bisector of the base, we know that the remaining side of the right triangle is exactly half of the bottom side of the equilateral triangle. Hence, the base of the right triangle has length
We can use the Pythagorean theorem to write
Rearranging this equation and taking the positive square root, we obtain
Using the given base length of 19 cm, we can obtain
Summing these two areas, we obtain
Now, we can set this expression equal to the given value for the total surface area:
We can simplify this into
Hence, the slant height of the given pyramid is 28.1 cm to the nearest tenth.
Let us finish by recapping a few important concepts from this explainer.
Key Points
Pyramids are 3D geometric shapes, or solid objects, where the base is a polygon and all other faces, called lateral faces, are triangles that meet at the apex, or vertex.
A pyramid can also have other polygons as its base, like a pentagon or a hexagon.
The surface area of a pyramid is the sum of the area of the base and the lateral area. When computing the surface area of a pyramid, it is helpful to draw the net of the pyramid.
All lateral faces of a regular pyramids are congruent triangles.
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