The surface area of a triangular pyramid is the area of its triangular base plus the areas of its three triangular side faces.
Triangular Pyramid Surface Area Formulas
| Case | Formula | Use When |
|---|---|---|
| Any triangular pyramid | SA = B + A1 + A2 + A3 | Side faces have different sizes |
| Regular triangular pyramid | SA = B + 1/2 Pl | One slant height works for all side faces |
| Regular tetrahedron | SA = sqrt(3)a2 | All 6 edges are equal |
What the Variables Mean
A1, A2, A3 = the three lateral triangle areas
P = perimeter of the triangular base
l = slant height of a side face
a = edge length for a regular tetrahedron
Step 1: Find the Base Area
For any triangular base, use B = 1/2 x base x height. For an equilateral base with side a, use B = sqrt(3)/4 x a2.
Step 2: Find the Lateral Area
If all side faces share the same slant height, use Lateral area = 1/2 x P x l. If the side faces differ, calculate each triangular face separately with 1/2 x base x height.
Worked Example: Regular Triangular Pyramid
Base area = 1/2 x 8 x 6.93 = 27.72 cm2
Base perimeter = 8 + 8 + 8 = 24 cm
Lateral area = 1/2 x 24 x 10 = 120 cm2
Total SA = 147.72 cm2
Worked Example: Regular Tetrahedron
SA = sqrt(3)a2
SA = sqrt(3) x 36
SA = 62.35 cm2
Slant Height vs Vertical Height
Slant height is measured along a triangular side face. Vertical height is measured straight from the apex down to the base. Surface area formulas use slant height for lateral faces, not vertical height.
1/2Pl. First find the slant height from the right triangle inside the pyramid.Common Mistakes
- Using pyramid height as slant height: lateral faces need face height.
- Forgetting the base: total surface area includes the base unless the pyramid is open.
- Using the tetrahedron formula for every triangular pyramid:
sqrt(3)a2only works when all edges are equal. - Leaving units linear: answers should be cm2, m2, in2, or ft2.
Use the Calculator
Use the Pyramid Surface Area Calculator for triangular and other pyramid bases. For related shapes, compare the Triangular Prism Calculator, composite surface area, and square-unit conversion.