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Surface Area of a Triangular Pyramid

We see enormous shapes in our surrounding every day. Some are two-dimensional, while others are three-dimensional. Square, triangle, rectangle, circle, rhombus, parallelogram, kite are common two-dimensional shapes and cube, cuboid, cone, cylinder, sphere, a pyramid is the most common three-dimensional geometrical shapes.

Let's talk about the pyramids which are solid shapes with a polygonal base and triangular lateral faces. A vertex of all triangular lateral faces meets one another. It is referred as an "apex". The pyramids are named on the basis of the kind of shape of its base; such as - a pyramid with a square base is called square pyramid, that with a triangular base is known as a triangular pyramid and so on. In this article, we shall throw light on triangular pyramids. We are going to focus especially on the formula and calculation of the surface area of triangular pyramids.

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Definition

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A triangular pyramid is one in which the base is triangular in shape. In this pyramid, all the surfaces are triangular. In other words, a triangular pyramid is a pyramid composed of four triangles. It is shown in the figure below.

Surface Area of a Triangular Pyramid

A triangular pyramid is also termed a "tetrahedron". A regular tetrahedron is referred to a kind of triangular pyramid in which all the sides are equal, i.e. all the triangles are congruent equilateral triangles.

The surface area of a pyramid is said to be the area of all its surfaces. It is known as total surface area. When we are required to find only the area of lateral surfaces, it is called lateral surface area which does not include the area of the base. While total surface area includes the base area as well.

Formula

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Lateral Surface Area

It can be calculated using the following formula :
LSA = $\frac{1}{2}$ P l

Where LSA stands for the lateral surface area, P denotes perimeter of the base and l represents slant height which is the height of the lateral triangle. Have a look at the following diagram to understand it better.

Lateral Surface Area Formula
Total Surface Area

This is found using the formula written below

TSA = LSA + A

Where A denotes area of the base.

Have a look at the image below.

Total Surface Area Formula

Here, the triangular base is said to have one side (base) "b" and height (perpendicular drawn from thevertex of a triangle to the base b) "a".

So, area of base = $\frac{1}{2}$ base $\times$ height = $\frac{1}{2}$ a b

Hence,
TSA = $\frac{1}{2}$ P l + $\frac{1}{2}$ a b

 In the case of a regular tetrahedron, all four triangles are equilateral. So, we may use the formula of area of an equailateral triangle and multiply it by 4.

Therefore,

TSA of a regular tetrahedron = $4\ \times$ $\frac{\sqrt{3}}{4}$$a^{2}$ = $\sqrt{3} a^{2}$

Where, "a" is the measure of side of triangle.

Calculate Surface Area of a Triangular Pyramid

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The surface area of a triangular pyramid can be found in the following simple steps.

Step 1 : Calculate the perimeter of the base by adding all the sides of the triangle.

Step 2 : Determine slant height of pyramid and substitute the values in formula

LSA = $\frac{1}{2}$ Perimeter of base $\times$ slant height.

Step 3 : Find the area of base using the formula, A = $\frac{1}{2}$ base $\times$ height. In case of equilateral triangle, use formula A = $\frac{\sqrt{3}}{4}$$a^{2}$.

Step 4 : Add the area of the base in the lateral surface area which will give us required total surface area.

Examples

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The examples based on this concept are given below.
Example 1 : Find the surface area of a triangular pyramid whose slant height is 10 cm and base is an equilateral triangle of side length 6 cm.

Solution :

Slant height, l = 10 cm

Perimeter of base, P = 3 $\times$ 10 = 30 cm

Area of base, A = $\frac{\sqrt{3}}{4}$ $a^{2}$

A = $\frac{\sqrt{3}}{4}$ $\times\ 6^{2}$

= $\frac{\sqrt{3}}{4}$ $\times\ 36$

= $\sqrt{3} \times\ 9$

$\approx$ 15.6 cm$^{2}$

Plugging all these values in -

Total surface area = $\frac{1}{2}$ P l + A

Total surface area = $\frac{1}{2}$ $\times$ 30 $\times$ 10 + 15.6

= 150 + 15.6 = 165.6 cm$^{2}$
Example 2 : Calculate the surface area of a triangular pyramid shown in the figure below. All measures are given in the image.

Surface Area of a Triangular Pyramid Examples
Solution : Given that

Slant height, l = 13.9 unit

The side of the triangular base is 15, 17 and 8 units.

Perimeter, P = 15 + 17 + 8 = 40 unit

Also, given that the base triangle is a right triangle.

Hence, we may take base, b = 15 unit and height, a = 8 unit

Area of triangle = $\frac{1}{2}$ a b

A = $\frac{1}{2}$ $\times$ 8 $\times$ 15 = 4 $\times$ 15 = 60 unit$^{2}$

Substituting all the values in the formula -

Total surface area = $\frac{1}{2}$ P l + A

TSA = $\frac{1}{2}$ $\times$ 40 $\times$ 13.9 + 60

TSA = 20 $\times$ 13.9 + 60

TSA = 278 + 60 = 338 unit$^{2}$
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