The Triangular Pyramid Surface Area Calculator calculates the total surface area of a pyramid with an equilateral triangle base by adding the base triangle to the three sloping triangular faces.
Triangular Pyramid Surface Area Calculator
Surface area of a regular tetrahedron or a triangular pyramid with an equilateral base, from slant height or height.
For a regular tetrahedron (all four faces equal) the formula is A = √3 × a². For a pyramid with a different height, the calculator uses A = (√3 ÷ 4)a² + 3 × a × l ÷ 2, working from either the slant height or the vertical height.
Formula
Regular tetrahedron:
A = √3 × a² and V = a³ ÷ (6√2)
Triangular pyramid with an equilateral base:
A = (√3 ÷ 4) × a² + (3 × a × l) ÷ 2
Slant height from vertical height:
l = √(h² + (a ÷ (2√3))²)
Where:
- A = total surface area; V = volume
- a = side of the base triangle (edge length for a tetrahedron)
- l = slant height of each side face (apex to the middle of a base edge)
- h = vertical height (apex straight down to the base center)
How the Calculation Works
Step 1: Find the base area of the equilateral triangle: (√3 ÷ 4) × a².
Step 2: Find the area of one side face: ½ × a × l. Multiply by 3 for the three side faces (the lateral area).
Step 3: Add the base and the lateral area. If you entered the vertical height, the calculator first converts it to slant height using the distance from the base center to the edge, a ÷ (2√3).
Variables Explained
| Variable | Meaning | Unit |
|---|---|---|
| a | Base side / edge length | cm, m, in, ft |
| l | Slant height of a side face | cm, m, in, ft |
| h | Vertical height | cm, m, in, ft |
| A | Total surface area | cm², m², in², ft² |
Example Calculation
Regular tetrahedron
Given: edge = 6 cm
Calculation: A = 1.73205 × 6² = 1.73205 × 36
Result: A ≈ 62.35 cm² (volume ≈ 25.46 cm³)
From base side and slant height
Given: base side = 6 cm, slant height = 5 cm
Base: (1.73205 ÷ 4) × 36 ≈ 15.59 cm²; sides: 3 × 6 × 5 ÷ 2 = 45 cm²
Result: A ≈ 60.59 cm²
From base side and vertical height
Given: base side = 6 cm, height = 4 cm
Slant height: l = √(4² + 1.732²) = √19 ≈ 4.359 cm
Result: A ≈ 15.59 + 39.23 = 54.82 cm²
Common Area Conversion Table
| Unit | Equivalent |
|---|---|
| 1 m² | 10,000 cm² |
| 1 ft² | 144 in² |
| 1 m² | 10.7639 ft² |
Shape Properties
A triangular pyramid has 4 triangular faces, 6 edges and 4 vertices. In a regular tetrahedron all four faces are identical equilateral triangles, so any face can serve as the base.
Dimensions Explained
Slant height vs edge length
The slant height runs down the middle of a side face, from the apex to the midpoint of a base edge. The lateral edge runs from the apex to a base corner and is longer. Use the slant height in this calculator.
Irregular triangular bases
This calculator assumes an equilateral base with three identical side faces. For a pyramid with different faces, calculate each triangle separately (½ × base × height) and add them.
Frequently Asked Questions
How do you find the surface area of a triangular pyramid?
Add the base triangle to the three side triangles: A = base area + 3 × (½ × a × l).
What is the surface area of a regular tetrahedron?
A = √3 × a². With 6 cm edges the surface area is about 62.35 cm².
How do you find the slant height of a triangular pyramid?
If you know the vertical height h and base side a, use l = √(h² + (a ÷ (2√3))²).
How many faces does a triangular pyramid have?
Four triangular faces: one base and three sides.
What is the lateral area of a triangular pyramid?
The three side faces only: 3 × a × l ÷ 2.