Rectangular Prism Calculator
Find the volume and surface of a box from length, width, and height, or of a cube from one side, or recover a missing edge from a known volume. The formulas are V = LWh, total surface S = 2(LW + Lh + Wh), and space diagonal d = √(L² + W² + h²).
Lateral 96 · total S = 168
- The base is a rectangle of area LW = 6 × 6 = 36.
- Volume V = LWh = 36 × 4 = 144.
- Lateral = 2h(L + W) = 96. Total S = 2(LW + Lh + Wh) = 168. Space diagonal d = √(L² + W² + h²) = 9.38083151965.
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How to Use the Rectangular Prism Calculator
From length, width, and height grows a box. From one side grows a cube. From a known volume and two edges recovers the third.
- Choose From L, W, and h; From cube side; or From V, L, and W.
- Enter non-negative lengths. When solving for height from volume, length and width must be positive.
- Read the volume (or the recovered height), the lateral area, the total surface, and the space diagonal.
A tent with the same rectangular base and height belongs on the Pyramid Calculator (V = (1/3)LWh, one-third of this box). A circular can belongs on the Cylinder Calculator (V = πr²h). A triangular base belongs on the Triangular Prism Calculator (V = Bh). Converting cubic units belongs on the Volume Converter.
Volume and Surface
A rectangular prism is a box: six rectangular faces, right angles, opposite faces equal. Volume is the base rectangle times the height, V = LWh. Unfolding the net gives two lids of area LW, two walls of area Lh, and two walls of area Wh, so S = 2(LW + Lh + Wh). The four walls alone are the lateral area 2h(L + W).
The space diagonal runs from one corner through the interior to the opposite corner. Two applications of Pythagoras give d = √(L² + W² + h²). A cube is the special box with L = W = h = a, so V = a³, S = 6a², and d = a√3. Solving for a missing height is a division: h = V / (LW).
Rectangular Prism Calculator Examples
6 by 6 by 4
V = 144. Lateral = 96. Total S = 168. Space diagonal √88 ≈ 9.38083151965. The square pyramid of the same base and height, on the Pyramid Calculator, is 48 — one-third of this box.
Cube of side 6
V = 216 and S = 216. Volume and surface match numerically (different units) only at this side length. Space diagonal = 6√3 ≈ 10.3923048454.
3 by 4 by 5
V = 60. S = 94. The 3-4 face is a 3-4-5 right triangle, so that face diagonal is 5. The space diagonal is √(9 + 16 + 25) = √50 ≈ 7.07106781187.
Volume 144, base 6 by 6
h = 144 / 36 = 4, the first box recovered from its volume.
Height 0
V = 0. The walls vanish. Total surface is two coincident lids: 2LW. A box crushed flat is a rectangle, not a tank.
A negative length
An edge cannot be negative. The calculator rejects −1.
The Cube a = 6
Set a³ = 6a² and cancel a² (for a ≠ 0): a = 6. Only that side makes the volume number equal the surface-area number. The units are still different — cubic versus square — so they are not the same quantity. The sphere has the same coincidence at r = 3; here it is a cube at a = 6.
A pyramid of the same square base and height as this default box is one-third the volume, 48 versus 144. That is Cavalieri’s one-third, the same rule as a cone versus a cylinder.
Common Rectangular Prism Mistakes
Using (1/3)LWh for a box
That is a pyramid, on the Pyramid Calculator. A prism is the full LWh.
Using πr²h for a box
That is a cylinder, on the Cylinder Calculator. A box has rectangular faces, not circular lids.
Reporting 2h(L + W) as the total surface
That is the walls. Add the two lids 2LW, or use 2(LW + Lh + Wh).
Treating V = S at a = 6 as the same measurement
216 cubic units is not 216 square units. Only the numerals match.
Using a face diagonal as the space diagonal
A face diagonal lives on one wall. The space diagonal uses all three edges: √(L² + W² + h²).
Converting liters on this page
V is already in the cube of your input unit. Use the Volume Converter for liters and gallons.
Rectangular Prism Calculator FAQs
What is the volume of a rectangular prism?
V = LWh. For 6 × 6 × 4, V = 144 cubic units.
What is the surface area of a rectangular prism?
S = 2(LW + Lh + Wh). The four walls alone are 2h(L + W). For 6 × 6 × 4 those are 168 and 96.
What is the volume of a cube?
V = a³. For side 6, V = 216. Surface is 6a² = 216, a numerical coincidence at this side only.
How do you find a missing edge from the volume?
h = V / (LW). Length and width must be positive.
Is a box the same as a pyramid or a cylinder?
No. A pyramid of the same base and height is one-third this volume. A cylinder has circular lids. This page is a right rectangular box.
Are my numbers sent to a server?
No. The calculation runs locally in your browser.
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Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
Sources
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Disclaimer
This calculator is intended for general educational and practical use. Review important results before relying on them in professional work.