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Surface of Pyramids

- Calculus
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The Great Pyramid of Giza is the largest pyramid located in Egypt. It houses the tomb of Khufu, the Fourth Dynasty Pharaoh. This ancient monument was constructed using three vital forms of material: 5.5 million tones of limestone, 8,000 tonnes of granite and 500,000 tonnes of mortar. Can you imagine the size of such a building? Given the extent of the number of textiles used to build such a structure, have you ever wondered how much surface this ancient monument takes up? Is there a way for us to calculate such a quantity?

As a matter of fact, there is! We can use the formula for finding the total surface area of a pyramid, which is exactly what we will be discussing throughout this article. Let's get started!

The phrase* surface area* is associated with 3-dimensional figures. Such objects are called **solids**. The pyramid falls under this category of objects.

To find the surface area of a pyramid you would need to add surfaces of all its sides together. The **base **of a pyramid is made up of a polygon. Recall that a polygon is a closed plane figure bounded by straight lines connected end to end. Every side of a pyramid meets at a point called the **apex**.

In some textbooks, the apex may be referred to as the vertex.

The distance perpendicular from the centre of the base of the pyramid to the apex is known as the **altitude **(or height). Meanwhile, the slant distance from the base of the pyramid to its apex is called the **slant height. **

Pyramids are categorised explicitly by the shape of their bases. For example, we could have a pyramid with a square base, a rectangular base or even a triangular one. However, we will look at pyramids with square bases to better cover this topic. As the base suggests, these are called the **square pyramids**. Here is a diagram that illustrates all these mentioned components.

Components of a square-based pyramid

Now, let us break down this pyramid and observe each of these surfaces. Essentially, we are opening up the pyramid in order to study each of its flat surfaces, also known as faces. This is called the **net of a pyramid**.

The net of a solid is a two-dimensional shape that can be folded up to form a three-dimensional solid. Laying out the faces of a solid allows us to determine the polygons that make up the object.

The net of a square-based pyramid is made up of one square and four triangles. This is shown below.

Net of a pyramid

Calculating the surface area of a pyramid means we will have to add the area of each face of the pyramid as seen above.

The form of properties we are going to study at this level is only for regular pyramids. Hence, below are the properties of regular pyramids.

A** regular pyramid** is a right pyramid where its base is a regular polygon.

Its base is a regular polygon.

All the lateral edges are congruent.

All the lateral faces are congruent isosceles triangles.

The altitudes meet the base at its center.

As mentioned earlier, the total surface area of a pyramid can be calculated by **adding the areas of all its faces**. The surfaces of pyramids are measured in **square units** like meters square, or centimeters square, depending on the units of the measurements. There are specific mathematical formulas used to find the total surface area of pyramids as long as the shape is regular.

We usually sort the area of a pyramid into two kinds; the lateral surface area and the total surface area. The lateral surface area of the pyramid is the sum of the side faces of the pyramid only disregarding the base whilst the total surface area of the pyramid is the lateral surface area including the base area. Mathematically, these are expressed as below;

The **Lateral Surface Are****a** (LSA) of a pyramid = The sum of areas of the side faces of the pyramid

\[LSA=\frac{1}{2}\times P\times I\]

The **Total Surface Area** (TSA) of a pyramid = LSA of pyramid + base area

\[TSA=\frac{1}{2}\times P\times I+B\]

where

P = Perimeter of the pyramid;

l = Height of each triangular face;

B = Area of the base of the pyramid.

In this section, we will talk about finding a pyramid's surface area depending on its **base's shape**. Here, we will explore three different types of pyramids, namely

triangular base pyramid;

square base pyramid;

hexagonal base pyramid.

Each section will explore a general formula used to find the surface area of such pyramids followed by an example to aid visual representation.

A **triangular base pyramid** is a pyramid that only consists of triangles as its faces.

It is made up of a triangular base and three triangular lateral faces. In order to deduce the total surface area of such a pyramid, you would simply add up the areas of all four of these triangular faces.

There are three types of triangular base pyramids, namely

**Regular Triangular Base Pyramid****:**Such pyramids have all their faces as equilateral triangles. This is also known as a tetrahedron.- Right Triangular Base Pyramid: Such pyramids have their base as equilateral triangles. The base is an equilateral triangle while the other faces are isosceles triangles.
- Irregular Triangular Base Pyramid
**-**a scalene or isosceles triangle forms the base.

The formula for the total surface area of a triangular base pyramid is given by

\[TSA=\frac{1}{2}(h\times b)+\frac{3}{2}(b\times s)\]

where

\(\frac{1}{2}(h\times b)\) is the base area;

\(\frac{3}{2}(b\times s)\) is the product of the perimeter and slant height of the pyramid;

\(h\) is the perpendicular height from the base;

\(b\) is the base length;

\(s\) is the slant height.

Here is a graphical representation of a triangular base pyramid together with all the mentioned components.

Triangular base pyramid

Let us now look at a worked example that demonstrates this formula.

Given that the base length of a triangular pyramid is 16 cm, the perpendicular height from the base is 21 cm and the slant height is 19 cm, determine the total surface area.

**Solution **

Here, \(b=16\), \(h=21\) and \(s=19\). Substituting these values into our formula above for the total surface area of a triangular pyramid yields

\[TSA=\frac{1}{2}(21\times 16)+\frac{3}{2}(16\times 19)=\frac{1}{2}\times 336+\frac{3}{2}\times 304\]

Simplifying this, we obtain

\[TSA=168+456=624\]

Thus, the total surface area of this triangular base pyramid is 624 units^{2}.

A **square base pyramid** is a pyramid made up of a square base and four triangular lateral faces.

These triangular faces are isosceles and congruent. Furthermore, the base of each triangle coincides with a side of the square base. The total surface area of a square base pyramid is the sum of the area of the square base and the area of all four triangular faces.

A square base pyramid is sometimes referred to as a pentahedron, as it has five faces.

The formula for the total surface area of a square pyramid is given by

\[TSA=a^2+2al\]

where

\(a^2\) is the base area;

\(a\) is the base length;

\(l\) is the slant height.

Below is a diagram of a square base pyramid along with all these mentioned components.

Square base pyramid

We shall now look at a worked example that uses this formula.

Calculate the slant height is a square base pyramid whose total surface area is 2200 units^{2} and the base length is 22 units.

**Solution**

Based on the given information above, we have \(a=22\) and \(TSA=2200\). Given our formula for the total surface area of a square pyramid, let's rearrange this equation so that \(l\) is the subject.

\[TSA=a^2+2al\implies 2al=TSA-a^2\]

Now placing only \(l\) on the left-hand side of this equation yields

\[l=\frac{TSA-a^2}{2a}\]

Now substituting our known values, we obtain

\[l=\frac{2200-22^2}{2(22)}=\frac{1716}{44}\]

Simplifying this yields

\[l=39\]

Thus, the slant height of this pyramid is 39 units.

As the name suggests,

a **hexagonal base pyramid** is a pyramid that has a hexagonal-shaped base.

This particular base has six sides and six triangular lateral faces. Another name for such a pyramid is called a Heptahedron. The formula for the total surface area of a hexagonal pyramid is given by

\[TSA=3ab+3bs\]

where

\(3ab\) is the base area of the hexagonal pyramid;

\(a\) is the apothem of the pyramid;

\(b\) is the base length;

\(s\) is the slant height.

The **apothem** of a regular polygon is defined by a line segment from its center to the midpoint of one of its sides.

Here is an illustration of a hexagonal pyramid that indicated all these mentioned components.

Hexagonal base pyramid

Let us now look at a worked example that applies this formula.

Find the base area and total surface area of a hexagonal pyramid given the following dimensions.

Apothem length = 12 units

Base length = 17 units

Slant height = 21 units

**Solution **

To determine both these areas, we would simply use the given formula above and substitute these given numbers. The dimensions are: \(a=12\), \(b=17\) and \(s=21\). We will first find the base area of this pyramid.

\[B=3ab=3(12)(17)=612\]

Thus, the base area of this hexagonal pyramid is 612 units^{2}. Next, we shall identify the total surface area.

\[TSA=3ab+3bs=612+3(17)(21)\]

Solving this yields

\[TSA=612+1071=1683\]

Therefore the total surface area of this hexagonal pyramid is 1683 units^{2}.

In this section, we are going to explore how to find both the surface lateral and the total surface area of a pyramid. Let us take an example to aid the process.

Solve the lateral surface area of a square pyramid given the side length of the base is 14 cm and the slant height of the pyramid is 20 cm.

**Solution **

Let us note the formula for finding the lateral surface area and see what values are missing. We do not have the value for the perimeter yet. However, we can find that using the side length of the base.

\[P=4a\]

Where a = side length of the base. Then

\[P=4(14)=56\]

The slant height of the pyramid is \(l = 20\)

Now we will substitute the values into the equation

\[LSA=\frac{1}{2}\times 56\times 20\]

Simplifying this, we obtain

\[LSA=1\times 56\times 1=560\]

Thus, the lateral surface area of this pyramid is 560 cm^{2}.

We will now take an example to find the total surface area of a pyramid.

What is the total surface area of a pyramid if each edge of the base measures 16 m, the slant height of a side is 17 m and the altitude is 15 m?

**Solution **

Noting the formula for finding the total surface area of a pyramid first, we can determine the values that are not available to us. Again, we do not have the value for the perimeter, but we have the value for the length of the side.

\[P=4a\]

where a = side length of the base. Then,

\[P = 4(16)=64\]

The area of the base is s^{2}. Thus,

\[B = 16^2= 256\]

Furthermore, we are given the slant height as \(l = 17\). Now substituting these values into our formula, we obtain

\[TSA=\frac{1}{2}\times 64\times 17+256\]

Solving this yields

\[TSA=544+256=800\]

Therefore, the total surface area of a pyramid is 800 m^{2}.

Here is another example that demonstrates the use of our given formulas for the TSA and LSA.

The lateral surface area of a pyramid measures 706 m^{2} while the total surface area measures 932 m^{2}. Given these areas, identify the area of the base of this pyramid.

**Solution **

From the information above, we are given that \(LSA=706\) and \(TSA=932\).

Using the formula for finding the TSA, we have that

\[TSA=\frac{1}{2}\times P\times I+B\]

Plugging the LSA equation into the expression above yields

\[TSA=LSA+B\]

Now making \(B\) the subject, we have

\[B=TSA-LSA\]

We can now substitute our given values for the TSA and LSA into the expression above.

\[B=932-706=226\]

Thus, the area of the base of this pyramid is 226 m^{2}.

We shall end this discussion with the following final example that encapsulates everything we have learnt throughout this discussion.

Given that the total surface area of a triangular pyramid is 74 units^{2}, determine the slant height of this pyramid. Furthermore, the base area and perimeter of this pyramid are 36 units^{2} and 45 units respectively.

**Solution**

The formula for the total surface area of a triangular pyramid is given by

\[TSA=B+\frac{1}{2}(P\times I)\]

where

- B = base area of the pyramid;
- P = perimeter of the pyramid;
- I = slant height.

From our equation above, let us rearrange it to make \(I\) the subject.

\[TSA-B=\frac{1}{2}(P\times I)\implies P\times I=2(TSA-B)\]

Finally,

\[I=\frac{2(TSA-B)}{P}\]

Now substituting our known values, we find that

\[I=\frac{2(74-36)}{45}=\frac{76}{45}=1.69\]

Therefore, the slant height of this triangular pyramid is 1.69 units, correct to two decimal places.

- A pyramid is a three-dimensional figure that has a shape that has a base as a polygon and its sides meet at a point called the apex, also known as the vertex.
- The distance perpendicular from the centre of the base of the pyramid to the apex is known as the altitude or height.
- The total surface area of a pyramid can be calculated by adding the areas of all its faces.
- The formula for finding the lateral surface area of a pyramid is \[LSA=\frac{1}{2}\times P\times I\]
- The formula for finding the total surface area of a pyramid is \[TSA=\frac{1}{2}\times P\times I+B\]

The general way to find the surface area of a pyramid is to use its formulas and substitute the values available.

Total surface area = ^{1}/_{2}Pl + B

Lateral surface area = ^{1}/_{2}Pl

Where

P = Perimeter of the pyramid

l = Height of each triangular face

B = Area of the base of the pyramid

The term surface area is the surface space covered by 3-dimensional figures and objects.

The total surface area of a pyramid if each edge of the base measures 16m, the slant height of a side is 17m and the altitude is 15m.

Solution

Total surface area = ^{1}/_{2}Pl + B

TSA = ^{1}/_{2} (64) (17) + 256

TSA = 544 + 256

TSA = 800cm^{2}

The general formula is used to calculate. Surface area = A + (^{3}/_{2})bh, where

A = the area of the pyramid's base,

b = the base of one of the faces

h = height of one of the face

More about Surface of Pyramids

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