The Rectangular Prism Calculator converts length, width and height into volume, surface area and space diagonal for a box with six rectangular faces. Opposite faces are congruent. The page redraws a labelled figure with all three edge lengths marked.
Enter three positive edge lengths. A cube is the special case where all three are equal. Related pages cover the Triangular Prism Calculator when the cross section is a triangle, the Pyramid Calculator when the solid tapers to an apex, and the Surface Area Calculator for a multi-shape catch-all.
Calculate the volume of a rectangular prism
Volume of a rectangular prism is the product of the three edge lengths. The Rectangular Prism Calculator multiplies length, width and height. All three must be positive. Units on volume are cube units of the shared length unit.
V = l * w * hVolume scales linearly with each edge. Doubling any one edge doubles the volume. When two edges and volume are known, the third edge recovers by dividing volume by the product of the known edges.
For the reference box 3 by 4 by 5:
V = 3 * 4 * 5 = 60A cube with edge 4 has volume 64. Swapping which edge is called length, width or height does not change volume; only the product matters. Packaging and room-volume problems use the same product once units are consistent (all feet, or all centimeters).
Oblique parallelepipeds with sheared angles need base area times perpendicular height rather than a simple edge product. This page models the right rectangular case with edges meeting at right angles.
Calculate the surface area
Surface area of a closed rectangular prism is the sum of the areas of six faces: two of each pairing of edges. The Rectangular Prism Calculator expands that as twice the sum of the three distinct face areas.
SA = 2 * (l*w + l*h + w*h)Open boxes that omit a lid subtract the corresponding face. Paint estimates for closed crates use the full formula. Face areas use square units of the edge length unit.
For 3 by 4 by 5:
SA = 2 * (3*4 + 3*5 + 4*5) = 2 * (12 + 15 + 20) = 2 * 47 = 94A cube of side a simplifies to SA = 6 * a^2. For a = 4, SA = 96. The 3-4-5 box is not a cube; its faces are 12, 15 and 20 in area before the factor of two.
Calculate the space diagonal
Space diagonal is the longest segment inside the box, running from one corner to the opposite corner through the interior. The Rectangular Prism Calculator applies the three-dimensional Pythagorean relation.
d = sqrt(l^2 + w^2 + h^2)Face diagonals on a single face use only two edges. Space diagonal uses all three. The same formula appears in 3D distance between opposite corners of a coordinate-aligned box.
For 3 by 4 by 5:
d = sqrt(9 + 16 + 25) = sqrt(50) = 5 * sqrt(2) approx 7.07Note that 3-4-5 is a face right triangle on the 3-by-4 face (diagonal 5), but the space diagonal through the box is longer than 5. Confusing face diagonal with space diagonal is a common worksheet error.
Solve a prism 3 by 4 by 5
Take length 3, width 4 and height 5 as the reference rectangular prism. Volume, surface area and space diagonal follow from the formulas above. 1. Volume
V = 3 * 4 * 5 = 602. Surface area
SA = 2 * (12 + 15 + 20) = 943. Face diagonal on the 3-by-4 face
sqrt(3^2 + 4^2) = 54. Space diagonal
d = sqrt(3^2 + 4^2 + 5^2) = sqrt(50) approx 7.07A triangular prism with a 3-4-5 right triangular base and length 10 has volume 60 as well, but that volume comes from base area 6 times length 10, not from a 3-by-4-by-5 product. Cross-link rather than reuse the triangular page's surface breakdown.
Face diagonals help packing problems that only care about one face. The 3-by-4 face diagonal is 5; the 3-by-5 face diagonal is sqrt(34) approx 5.83; the 4-by-5 face diagonal is sqrt(41) approx 6.40. None of those equals the space diagonal sqrt(50). Label which diagonal a worksheet asks for before substituting.
When one edge is unknown but volume is known, divide volume by the product of the other two edges. When surface area is known with two edges, the third edge solves from a linear rearrangement of SA = 2(lw + lh + wh), though that algebra is messier than the volume route and usually appears only on challenge sheets.
Frequently asked questions
What is rectangular prism volume?
Volume is V = l * w * h. For 3 by 4 by 5, V = 60.
What is surface area of a rectangular prism?
Surface area is SA = 2 * (l*w + l*h + w*h). For 3 by 4 by 5, SA = 94.
What is the space diagonal?
Space diagonal is d = sqrt(l^2 + w^2 + h^2). For 3 by 4 by 5, d = sqrt(50) approx 7.07.
Is a cube a rectangular prism?
Yes. A cube is a rectangular prism with l = w = h. Volume is a^3 and surface area is 6 * a^2.
Can the Rectangular Prism Calculator omit one face?
The closed formula includes all six faces. For an open box, subtract the omitted face area from SA.
How does this differ from a triangular prism?
A triangular prism has triangular ends. Volume is triangle area times length. Use the Triangular Prism Calculator for that solid.
What units does the Rectangular Prism Calculator use?
All three edges share one length unit. Volume uses cube units; surface area uses square units; diagonal uses length units.
Why is the space diagonal not 5 for a 3-4-5 box?
The value 5 is the face diagonal on the 3-by-4 face. The space diagonal adds the third edge and equals sqrt(50).
How do I find a missing edge from volume?
Divide volume by the product of the two known edges. For V = 60 with edges 3 and 4, height = 60 / 12 = 5.
What is the surface area of a cube with edge 4?
Surface area is 6 * 4^2 = 96. Volume is 64. That cube is not the 3-by-4-by-5 reference box.
Summary
Rectangular Prism Calculator returns volume, surface area and space diagonal from three edge lengths, with substituted arithmetic and a labelled figure. Worked numbers match the 3-by-4-by-5 box: V = 60, SA = 94 and space diagonal sqrt(50).
Face diagonals on the three faces are 5, sqrt(34) and sqrt(41), which must not be confused with the space diagonal.
Use the Triangular Prism Calculator for triangular ends and the Pyramid Calculator when the solid tapers.