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Triangular Pyramid — Definition, Formula & Examples

A triangular pyramid is a three-dimensional solid with a triangular base and three additional triangular faces that meet at a single point called the apex. When all four faces are equilateral triangles, it is called a regular tetrahedron.

A triangular pyramid, or tetrahedron, is a polyhedron composed of exactly four triangular faces, six edges, and four vertices. It is classified as a pyramid whose base is a triangle, making every face — including the base — a triangle.

Key Formula

V=13⋅B⋅hV = \frac{1}{3} \cdot B \cdot h
Where:
  • VV = Volume of the triangular pyramid
  • BB = Area of the triangular base
  • hh = Perpendicular height from the base to the apex

How It Works

To find the volume of a triangular pyramid, you need the area of its triangular base and the perpendicular height from the base to the apex. Multiply the base area by the height, then divide by 3. To find the surface area, calculate the area of each of the four triangular faces and add them together. Since the faces can all be different triangles, you may need to compute each one separately unless the pyramid is regular.

Worked Example

Problem: A triangular pyramid has a base that is a right triangle with legs of 6 cm and 8 cm. The perpendicular height of the pyramid is 9 cm. Find its volume.
Step 1: Find the area of the triangular base. The base is a right triangle, so use the leg lengths.
B=12×6×8=24 cm2B = \frac{1}{2} \times 6 \times 8 = 24 \text{ cm}^2
Step 2: Apply the pyramid volume formula using the base area and the height.
V=13×24×9V = \frac{1}{3} \times 24 \times 9
Step 3: Calculate the result.
V=2163=72 cm3V = \frac{216}{3} = 72 \text{ cm}^3
Answer: The volume of the triangular pyramid is 72 cm³.

Another Example

Problem: Find the total surface area of a regular tetrahedron (all edges equal) with an edge length of 10 cm.
Step 1: A regular tetrahedron has 4 identical equilateral triangular faces. Find the area of one equilateral triangle with side length 10 cm.
Aface=34×102=10034=253 cm2A_{\text{face}} = \frac{\sqrt{3}}{4} \times 10^2 = \frac{100\sqrt{3}}{4} = 25\sqrt{3} \text{ cm}^2
Step 2: Multiply by 4 to get the total surface area.
SA=4×253=1003≈173.2 cm2SA = 4 \times 25\sqrt{3} = 100\sqrt{3} \approx 173.2 \text{ cm}^2
Answer: The total surface area is 1003≈173.2100\sqrt{3} \approx 173.2 cm².

Why It Matters

Triangular pyramids appear throughout middle-school and high-school geometry courses when studying 3D solids, surface area, and volume. Engineers and architects use tetrahedral shapes in structural design because triangles distribute force efficiently. Understanding this shape also builds a foundation for later topics like Cavalieri's Principle and solid geometry proofs.

Common Mistakes

Mistake: Forgetting the one-third factor and computing volume as base area times height.
Correction: A pyramid's volume is always one-third of the corresponding prism with the same base and height. Always divide by 3.
Mistake: Confusing the slant height of a face with the perpendicular height of the pyramid.
Correction: The height hh in the volume formula is the perpendicular distance from the base to the apex, measured at a right angle to the base — not a length along a face.

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