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Free Rectangular Pyramid Volume Calculator

Enter base length, width, and height to calculate

What Is a Rectangular Pyramid?

A rectangular pyramid is a three‑dimensional polyhedron that has a rectangular base and four triangular faces that meet at a single point (the apex) above the base. This shape appears in famous structures such as ancient Egyptian pyramids and the modern Louvre Pyramid, and it is frequently encountered in geometry problems and architecture.

Rectangular Pyramid Volume Formula

The volume (or capacity) of a rectangular pyramid is obtained using the rectangular pyramid volume formula:

V=l×w×h3V = \dfrac{l \times w \times h}{3}

where:

  • ll = length of the rectangular base
  • ww = width of the rectangular base
  • hh = perpendicular height from the apex to the base plane

This formula works for any pyramid with a rectangular base—calculate the base area by multiplying ll by ww, multiply that by the height, and then divide by three.

Surface Area of a Rectangular Pyramid

To find the total surface area, add the area of the rectangular base to the sum of the areas of the four triangular lateral faces:

A=lw+l⋅(w2)2+h2+w⋅(l2)2+h2A = lw + l \cdot \sqrt{\left(\dfrac{w}{2}\right)^2 + h^2} + w \cdot \sqrt{\left(\dfrac{l}{2}\right)^2 + h^2}

Here, lwlw is the base area; the two square‑root terms account for the lateral faces. This formula directly gives the surface area of a rectangular pyramid in one step.

How to Use the Free Rectangular Pyramid Calculator Online

This free rectangular pyramid calculator online is built for ease of use. Follow these steps:

  1. Enter the length and width of the rectangular base.
  2. Input the vertical height of the pyramid.
  3. The tool immediately displays both the volume and the total surface area.

Worked Example

Take a rectangular pyramid with:

  • Base length = 7 cm
  • Base width = 5 cm
  • Height = 10 cm

The calculator will show:

  • Volume = 7×5×103=116.67 cm3\dfrac{7 \times 5 \times 10}{3} = 116.67\ \text{cm}^3
  • Surface area = 7×5+7×(52)2+102+5×(72)2+102≈160.13 cm27 \times 5 + 7 \times \sqrt{\left(\dfrac{5}{2}\right)^2 + 10^2} + 5 \times \sqrt{\left(\dfrac{7}{2}\right)^2 + 10^2} \approx 160.13\ \text{cm}^2

Whether you need the volume of a rectangular pyramid for homework, design, or any project, this online tool gives accurate results in seconds.

FAQ

1. How do I manually calculate the volume of a rectangular pyramid?

Multiply the length and width of the base to get the base area, then multiply by the height, and finally divide by three: V = (l × w × h) / 3.

2. What is the difference between the height and the slant height in a rectangular pyramid?

The height (h) is the perpendicular distance from the apex to the base plane. The slant height is the distance along a triangular face from the apex to the midpoint of a base edge. The volume formula uses the vertical height, not the slant height.

3. Can the same calculator also compute the surface area?

Yes, the calculator simultaneously computes the total surface area using the formula A = lw + l·√((w/2)²+h²) + w·√((l/2)²+h²) and displays it alongside the volume.

4. What units should I use for the input dimensions?

Any consistent unit works (e.g., cm, m, inches). The volume will be expressed in cubic units and the surface area in square units.

How to Use

  1. Enter the base length (a) and base width (b) of your rectangular pyramid.
  2. Enter the pyramid height (H). Select the preferred length unit for each dimension.
  3. View the calculated volume instantly. Toggle surface area display to see both values. Switch the output unit to display the result in your preferred unit.