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Square Pyramid — Definition, Formula & Examples

A square pyramid is a three-dimensional shape with a square base and four triangular faces that meet at a single point called the apex. It has 5 faces, 8 edges, and 5 vertices.

A square pyramid is a polyhedron whose base is a square and whose four lateral faces are triangles sharing a common vertex (the apex) not coplanar with the base. When the apex lies directly above the center of the base, the pyramid is called a right square pyramid, and all four lateral faces are congruent isosceles triangles.

Key Formula

V=13s2hSA=s2+2sℓV = \frac{1}{3}s^2 h \qquad SA = s^2 + 2s\ell
Where:
  • VV = Volume of the square pyramid
  • SASA = Total surface area of the square pyramid
  • ss = Side length of the square base
  • hh = Height (perpendicular distance from apex to base)
  • ℓ\ell = Slant height (distance from apex to the midpoint of a base edge)

How It Works

To work with a square pyramid, you need to know its base side length and its height (the perpendicular distance from the apex to the base). The volume formula is one-third the area of the base times the height. For surface area, you add the area of the square base to the combined area of the four triangular faces. Each triangular face has a base equal to one side of the square and a height called the slant height, which runs from the apex down the middle of that triangular face.

Worked Example

Problem: A square pyramid has a base side length of 6 cm and a height of 4 cm. Find its volume and total surface area.
Step 1: Find the volume using the formula.
V=13s2h=13(6)2(4)=13(144)=48 cm3V = \frac{1}{3}s^2 h = \frac{1}{3}(6)^2(4) = \frac{1}{3}(144) = 48 \text{ cm}^3
Step 2: Find the slant height. The slant height runs from the apex to the midpoint of a base edge. Use the Pythagorean theorem with the pyramid's height (4 cm) and half the base side (3 cm).
ℓ=h2+(s2)2=42+32=16+9=25=5 cm\ell = \sqrt{h^2 + \left(\frac{s}{2}\right)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ cm}
Step 3: Find the total surface area by adding the base area and the four triangular faces.
SA=s2+2sℓ=(6)2+2(6)(5)=36+60=96 cm2SA = s^2 + 2s\ell = (6)^2 + 2(6)(5) = 36 + 60 = 96 \text{ cm}^2
Answer: The volume is 48 cm³ and the total surface area is 96 cm².

Another Example

Problem: A right square pyramid has a base side length of 10 m and a slant height of 13 m. Find its volume.
Step 1: Find the pyramid's height using the slant height and half the base side.
h=ℓ2−(s2)2=132−52=169−25=144=12 mh = \sqrt{\ell^2 - \left(\frac{s}{2}\right)^2} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \text{ m}
Step 2: Calculate the volume.
V=13(10)2(12)=13(1200)=400 m3V = \frac{1}{3}(10)^2(12) = \frac{1}{3}(1200) = 400 \text{ m}^3
Answer: The volume is 400 m³.

Why It Matters

Square pyramids appear throughout geometry courses in middle school and high school, especially in units on surface area and volume of solids. Architects and engineers encounter pyramidal shapes in roof design, monuments (like the Great Pyramid of Giza), and packaging. Mastering this shape also builds your skill with the Pythagorean theorem, since finding the slant height requires it.

Common Mistakes

Mistake: Forgetting the one-third factor in the volume formula and computing the volume as the full base area times height.
Correction: A pyramid's volume is always one-third the volume of a prism with the same base and height. Always multiply by 1/3.
Mistake: Confusing the pyramid's height with its slant height when calculating surface area or volume.
Correction: The height is the vertical distance from apex to base center; the slant height runs along a triangular face. Use the Pythagorean theorem to convert between them: the relationship is ℓ² = h² + (s/2)².

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