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Right Square Prism

Right Square Prism

A box with at least one pair of opposite faces that are squares. It can also be described as a right prism with square bases. A right square prism which has square lateral surfaces is a cube.

 

Right Square Prism

Volume = x2h
Lateral Surface Area = 4xh
Surface Area = 2x2 + 4xh

3D right square prism with square base side labeled x and length labeled h.

 

See also

Prism , right regular prism

Key Formula

V=x2hAlateral=4xhAsurface=2x2+4xhV = x^2 h \qquad A_{\text{lateral}} = 4xh \qquad A_{\text{surface}} = 2x^2 + 4xh
Where:
  • xx = Side length of the square base
  • hh = Height of the prism (distance between the two square bases)
  • VV = Volume of the prism
  • AlateralA_{\text{lateral}} = Lateral surface area (area of the four rectangular side faces)
  • AsurfaceA_{\text{surface}} = Total surface area (lateral area plus the two square bases)

Worked Example

Problem: A right square prism has a square base with side length 5 cm and a height of 12 cm. Find its volume, lateral surface area, and total surface area.
Step 1: Identify the values: the base side length is x = 5 cm and the height is h = 12 cm.
x=5,h=12x = 5, \quad h = 12
Step 2: Calculate the volume by squaring the base side and multiplying by the height.
V=x2h=52×12=25×12=300 cm3V = x^2 h = 5^2 \times 12 = 25 \times 12 = 300 \text{ cm}^3
Step 3: Calculate the lateral surface area. There are four rectangular faces, each with width x and height h.
Alateral=4xh=4×5×12=240 cm2A_{\text{lateral}} = 4xh = 4 \times 5 \times 12 = 240 \text{ cm}^2
Step 4: Calculate the total surface area by adding the areas of the two square bases to the lateral area.
Asurface=2x2+4xh=2(25)+240=50+240=290 cm2A_{\text{surface}} = 2x^2 + 4xh = 2(25) + 240 = 50 + 240 = 290 \text{ cm}^2
Answer: The volume is 300 cm³, the lateral surface area is 240 cm², and the total surface area is 290 cm².

Another Example

Problem: A shipping box is shaped like a right square prism. Its square base has a side length of 8 inches, and you need it to hold exactly 1,024 cubic inches. What height should the box be, and what is its total surface area?
Step 1: Use the volume formula and solve for h.
V=x2h  ⟹  1024=82×h=64hV = x^2 h \implies 1024 = 8^2 \times h = 64h
Step 2: Divide both sides by 64 to find the height.
h=102464=16 inchesh = \frac{1024}{64} = 16 \text{ inches}
Step 3: Now compute the total surface area with x = 8 and h = 16.
Asurface=2(82)+4(8)(16)=128+512=640 in2A_{\text{surface}} = 2(8^2) + 4(8)(16) = 128 + 512 = 640 \text{ in}^2
Answer: The box needs a height of 16 inches and has a total surface area of 640 in².

Frequently Asked Questions

Is a right square prism the same as a cube?
Not necessarily. A cube is a special case of a right square prism where the height equals the base side length, making all six faces squares. If the height differs from the base side length, the four lateral faces are non-square rectangles, and the solid is not a cube.
What is the difference between a right square prism and a rectangular prism?
A rectangular prism (box) can have any rectangle as its base, so the length and width may differ. A right square prism specifically requires the base to be a square, meaning the base's length and width are equal. Every right square prism is a rectangular prism, but not every rectangular prism is a right square prism.

Right Square Prism vs. Cube

Both have square bases and lateral faces perpendicular to those bases. The key distinction is that a cube requires all six faces to be congruent squares, so height must equal the base side length. A right square prism allows the height to be any positive value, so the lateral faces can be non-square rectangles. A cube is therefore a special right square prism where h=xh = x.

Why It Matters

Right square prisms appear constantly in everyday life—cardboard boxes, structural columns, storage containers, and building blocks are common examples. Understanding their formulas lets you calculate how much material is needed to wrap or construct a box and how much space it encloses. These calculations form the foundation for more advanced work in engineering, architecture, and packaging design.

Common Mistakes

Mistake: Confusing lateral surface area with total surface area by forgetting to include the two square bases.
Correction: Lateral surface area (4xh4xh) counts only the four side faces. Total surface area adds the two base areas: 2x2+4xh2x^2 + 4xh. Always check whether a problem asks for lateral or total surface area.
Mistake: Assuming every right square prism is a cube.
Correction: A right square prism becomes a cube only when h=xh = x. If the height and base side length differ, the solid has rectangular lateral faces and is not a cube.

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