Square Pyramid - Definition, Formula, and Examples
Square Pyramid - Definition, Formula, and Examples
A square pyramid is a 3D solid with a square base and four triangular faces meeting at an apex, giving 5 faces, 8 edges, and 5 vertices. Its volume is V=(\frac{1}{3}a^2h) and its surface area is (a^2 + 2a\ell). This article covers the definition, properties, net, formulas, and worked examples.
The Shape That Has Held Its Corners For 4,500 Years
The Great Pyramid of Giza still stands because a square base spreads a huge load evenly onto the ground while four sloping faces carry that load up to a single point. Long before anyone wrote a volume formula, builders trusted that a square-based pyramid holds its shape better than almost any other solid.
What Is A Square Pyramid?
A square pyramid is a three-dimensional solid with a square base and four triangular faces that rise from the base edges and meet at a single point called the apex.
The apex is the top point where all four triangular faces meet. The base is the square at the bottom. When the apex sits directly above the centre of the base, the solid is a right square pyramid; when it leans to one side, it is an oblique square pyramid.
What Are the Faces, Edges, and Vertices of a Square Pyramid?
A square pyramid has:
- Faces: 5
- Edges: 8
- Vertices: 5
These satisfy Euler's formula for any convex polyhedron, V−E+F=2.
What Is the Net of a Square Pyramid?
The net is the flat pattern you get by unfolding the solid. For a square pyramid, the net is a central square (the base) with a triangle attached to each of its four sides.
What Are the Square Pyramid Formulas?
Every formula comes from the "square base plus four triangles" picture:
| Quantity | Formula | Where it comes from |
|---|---|---|
| Volume | (V=\frac{1}{3}a^2h) | A third of the prism (a^2h) on the same base |
| Slant height | (\ell=\sqrt{h^2 + (\frac{a}{2})^2}) | Right triangle: height and half a base side |
| Lateral surface area | (L=2a\ell) | Four triangles, each (\frac{1}{2}a\ell) |
| Total surface area | (A=a^2 + 2a\ell) | Base (a^2) plus the four triangles |
Examples of Square Pyramid
Example 1
A square pyramid has base side a = 6 cm and height h = 4 cm. Find its volume.
Using (V=\frac{1}{3}a^2h) :
(V=\frac{1}{3} \times 6^2 \times 4)
Final answer: the volume is 48 cm³.
Example 2
A square pyramid has base side a = 6 cm and height h = 4 cm. Find the correct surface area.
First find the slant height: (\ell=\sqrt{h^2 + (\frac{a}{2})^2} = 5 cm)
Then total surface area:
(A = a^2 + 2a\ell = 96 cm^2)
Final answer: the surface area is 96 cm².
Example 3
Find the slant height of a square pyramid with base side a = 10 cm and height h = 12 cm.
Final answer: the slant height is 13 cm.
Example 4
A square pyramid has base side a = 8 cm and slant height ℓ = 5 cm. Find its lateral surface area.
Final answer: the lateral surface area is 80 cm².
Example 5
A square pyramid has volume 100 cm³ and base side a = 5 cm. Find its height.
Final answer: the height is 12 cm.
Example 6
A square pyramid has base side a = 6 cm and height h = 4 cm. Find its total surface area.
Final answer: total surface area 96 cm²;
Conclusion
A square pyramid has a square base, four triangular faces, and one apex.
Volume is (V=\frac{1}{3}a^2h); total surface area is (A=a^2 + 2a\ell).
The triangular faces use the slant height (\ell), not the vertical height (h).