Surface Area of a Triangular Pyramid Formula

A triangular pyramid is a three-dimensional shape with four triangular faces. To calculate the surface area of a triangular pyramid, we need to find the area of each triangular face and summing them up. This process can be straightforward if you know the dimensions of the triangles involved. In simple words, A solid with a triangular base such that the triangles have a common vertex on all three lateral sides is called a triangular pyramid.

It can be interpreted as a tetrahedron with equilateral triangles on all four faces. It is a three-dimensional object with a triangle base and four triangular faces, three of which meet at one vertex. The base of a right triangular pyramid is a right-angled triangle, while the rest of the faces are isosceles triangles.

Let’s know more about Surface Area of a Triangular Pyramid with formula and derivation in detailed below.

The surface area of a triangular pyramid has two formulas, i.e. lateral surface area and total surface area. Let’s learn about both the formulae in more detail,

Formula for Lateral surface area of a triangular pyramid 

It is calculated by adding the area of triangular faces with no base, i.e., by subtracting the base area from the total surface area. It is equal to half the product of perimeter and slant height.

L = 3/2 (b × l)

Where,

b is the side of triangular base,

l is the slant height of pyramid.

Formula for Total surface area of a triangular pyramid

It is computed by adding the area of all triangular faces of the pyramid. Its formula is equal to the sum of the base area and half the product of perimeter and slant height. It is the sum of the base area and lateral surface area of the pyramid.

A = 1/2 (b × h) + 3/2 (b × l)

Where,

b is the side of triangular base,

h is the height of triangular base,

l is the slant height of pyramid.

The surface area (SASASA) of a triangular pyramid can be calculated using the following steps:

1. Identify the triangles: A triangular pyramid has one base triangle and three lateral triangles.
2. Calculate the area of the base triangle (AbaseA_{\text{base}}Abase​): Use the appropriate formula based on the type of triangle (equilateral, isosceles, or scalene).
3. Calculate the area of the three lateral triangles (Alateral1A_{\text{lateral1}}Alateral1​, Alateral2A_{\text{lateral2}}Alateral2​, Alateral3A_{\text{lateral3}}Alateral3​): These triangles may have different sizes, so calculate each area individually.
4. Sum the areas: Add the areas of the base and the three lateral triangles to get the total surface area.

Surface Area of Triangular Pyramid = Area_base + Area_{Lateral1 +} Area_{Lateral2 +} Area_Lateral3

Problem 1: Find the lateral surface area of a triangular pyramid with a base of 10 m and a slant height of 12 m. 

Solution:

We have, b = 10 and l = 12

Using the formula we have,

L = 3/2 (b × l)

= 3/2 (10 × 12)

= 360/2

= 180 sq. m

Problem 2: Find the lateral surface area of a triangular pyramid with a base of 15 m and a slant height of 20 m.

Solution:

We have, b = 15 and l = 20

Using the formula we have,

L = 3/2 (b × l)

= 3/2 (15 × 20)

= 900/2

= 450 sq. m

Problem 3: Find the base length of a triangular pyramid with a slant height of 12 m and a lateral area of 180 sq. m.

Solution:

We have, l = 15 and A = 180

Using the formula we have,

L = 3/2 (b × l)

=> 180 = 3/2 (b × 12)

=> 180 = 36b/2

=> 18b = 180

=> b = 10 m

Problem 4: Find the slant length of a triangular pyramid with a base of 6 m and a lateral area of 81 sq. m.

Solution:

We have, b = 6 and A = 81

Using the formula we have,

L = 3/2 (b × l)

=> 81 = 3/2 (6 × l)

=> 81 = 18l/2

=> 9l = 81

=> l = 9 m

Problem 5: Find the total surface area of a triangular pyramid with a base of 6 m, height of 10 m, and lateral area of 100 sq. m.

Solution:

We have, b = 6, h = 10 and L = 100

Using the formula we have,

A = 1/2 (b × h) + L

= 1/2 (6 × 10) + 100

= 30 + 100

= 130 sq. m

Problem 6: Find the total surface area of a triangular pyramid with a base of 2 m, height of 4 m, and slant height of 7 m.

Solution:

We have, b = 2, h = 4 and l = 7.

Using the formula we have,

A = 1/2 (b × h) + 3/2 (b × l)

= 1/2 (2 × 4) + 3/2 (2 × 7)

= 4 + 21

= 25 sq. m

Problem 7: Find the total surface area of a triangular pyramid with a base of 16 m, height of 11 m, and slant height of 24 m.

Solution:

We have, b = 16, h = 11 and l = 24.

Using the formula we have,

A = 1/2 (b × h) + 3/2 (b × l)

= 1/2 (16 × 11) + 3/2 (16 × 24)

= 88 + 576

= 664 sq. m