What is the volume of regular pyramid below?

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A = total surface area

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This online calculator will calculate the various properties of a square pyramid given 2 known variables. The square pyramid is a special case of a pyramid where the base is square. It is a regular pyramid with a square base.

Units: Note that units are shown for convenience but do not affect the calculations. The units are in place to give an indication of the order of the results such as ft, ft2 or ft3. For example, if you are starting with mm and you know r and h in mm, your calculations will result with s in mm, V in mm3, L in mm2, B in mm2 and A in mm2.

NAN: means not a number. This will show as a result if you are using values that just do not make sense as reasonable values for a pyramid.

Below are the standard formulas for a pyramid. Calculations are based on algebraic manipulation of these standard formulas.

Square Pyramid Formulas derived in terms of side length a and height h:

• Volume of a square pyramid:
• Slant Height of a square pyramid:
  • By the pythagorean theorem we know that
  • s2 = r2 + h2
  • since r = a/2
  • s2 = (1/4)a2 + h2, and
  • s = √(h2 + (1/4)a2)
  • This is also the height of a triangle side
• Lateral Surface Area of a square pyramid (4 isosceles triangles):
  • For the isosceles triangle Area = (1/2)Base x Height. Our base is side length a and for this calculation our height for the triangle is slant height s. With 4 sides we need to multiply by 4.
  • L = 4 x (1/2)as = 2as = 2a√(h2 + (1/4)a2)
  • Squaring the 2 to get it back inside the radical,
  • L = a√(a2 + 4h2)
• Base Surface Area of a square pyramid (square):
• Total Surface Area of a square pyramid:
  • A = L + B = a2 + a√(a2 + 4h2))
  • A = a(a + √(a2 + 4h2))

Square Pyramid Calculations:

Other formulas for calculations are derived from the formulas above.

References

Weisstein, Eric W. "Square Pyramid." From MathWorld--A Wolfram Web Resource. Square Pyramid.

What do we mean by the volume of a rectangular pyramid and how do we calculate it? Volume is nothing but the space an object occupies. So, the volume of a rectangular pyramid will be the space occupied by the rectangular pyramid. Volume of a rectangular pyramid can also be termed as the capacity of the rectangular pyramid.

Let's learn how to find the volume of a rectangular pyramid with the help of a few solved examples and practice questions.

What Is the Volume of a Rectangular Pyramid?

The volume of the rectangular pyramid is defined as the capacity of the rectangular pyramid. In geometry, a rectangular pyramid is a three-dimensional geometric shape that has a rectangular base and four triangular faces that are joined at a vertex. A rectangular pyramid is a polyhedron (pentahedron) with five faces. The top point of the pyramid is called the apex. The bottom rectangle is called the base. The image below shows the shape of a rectangular pyramid.

The volume of a rectangular pyramid is the number of unit cubes that can fit into it. The unit of volume is "cubic units". For example, it can be expressed as m3, cm3, in3, etc. depending upon the given units.

Did you know that one of the oldest pyramid structures known to man is the "Great Pyramid of Giza?" It was constructed around 2550 BC, in Egypt. They are considered among the seven wonders of the world. They are pyramids, alright, but are they rectangular pyramids as well.

We can distinguish the rectangular pyramids on the basis of the lengths of their edges, position of the apex, and so on. Below are given the two main types of rectangular pyramids:

• Right Rectangular Pyramid
• Oblique Rectangular Pyramid

Volume of a Rectangular Pyramid Formula

The formula to determine the volume of a rectangular pyramid is:

\(\text{Volume}=\dfrac{1}{3} \times \text{Base Area} \times \text{h}\)

Here 'h' is the perpendicular height and the rectangular base area = L × W.

• If the apex of the rectangular pyramid is right above the center of the base, it forms a perpendicular to the base, which marks its height. Such a rectangular pyramid is called the right rectangular pyramid. We will mention the right rectangular pyramid as a simply rectangular pyramid going forward.
• If the apex of the rectangular pyramid is not aligned right above the center of the base, the pyramid is called an oblique rectangular pyramid. This type of pyramid appears to have tilted. Thus, in the case of an oblique rectangular pyramid, height is taken as the length of the perpendicular drawn apex to the base of the pyramid.

How to Find the Volume of a Rectangular Pyramid?

As we learned in the previous section, the volume of a rectangular pyramid could be found using \(\dfrac{1}{3} \times \text{Base Area} \times \text{h}\). Thus, we follow the below steps to find the volume of a rectangular pyramid.

• Step 1: Determine the base area (L × W) and the height (h) of the pyramid.
• Step 2: Find the volume using the formula:\(\dfrac{1}{3} \times \text{Base Area} \times \text{h}\)
• Step 3: Represent the final answer with cubic units.

Example:

If the height of a rectangular pyramid h = 10 units and the length of the base edges are L = 9 units and W = 5 units, respectively, then, the volume of the rectangular pyramid is:

Volume = (1/3) × L × W × h =(1/3) × 9 × 5 × 10 = (1/3) × 45 × 10

= 150 cubic units

1. Example 1:

   We already learned that the volume of a rectangular pyramid is \(\text{V}=\dfrac{1}{3} \times \text{Base Area} \times \text{h}\). Let us solve a few examples applying this formula.

   Jason has a vessel in the form of an inverted right rectangular pyramid that has to be filled with water. The altitude of the vessel is 10 inches and the area of the base is 7 × 6 inches. Determine the volume of water Julia can fill in the vessel?

   Solution:

   The height h of the vessel is 10 inches.

   The base area of the vessel is 7 × 6 inches2.

   The volume of the vessel is given by:

   \[\begin{align} \dfrac{1}{3}\text{LWh}&=\dfrac{1}{3}\times 7 \times 6 \times10\\&=14 \times 10 \\&= 140 \text{ inches}^3\end{align}\]

   Answer: Volume of the rectangular pyramid is 140 units3.

2. Example 2:

   What will be the volume of a regular rectangular pyramid with base sides 10 in and 9 in, and a height of 18 in?

   Solution:

   The formula for the volume of a pyramid is given by:

   V = 1/3 × Base Area × h

   The area of the base = Length × Width = 10 × 9 = 90  in2.

   Putting the values:

   Base area = 90 and h = 18 in the formula.

   V = 1/3 × 90 × 18 = 540 in3.

   Answer: The volume of the given rectangular pyramid is 540 in3.

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FAQs on Volume of a Rectangular Pyramid

The capacity of the rectangular pyramid is defined as the volume of the rectangular pyramid which can be calculated using the formula, \(\text{Volume}=\dfrac{1}{3} \times \text{Base Area} \times \text{h}\)

How Do You Find the Volume of a Right Rectangular Pyramid?

The volume (V) of a rectangular pyramid can be easily found out by just knowing the base area and its height and putting the values of these dimensions in the formula, V = 1/3 × Base Area × Height

What Units Are Used With the Volume of the Right Rectangular Pyramid?

In the metric system of measurement, the most common units of volume are milliliters and liters.

What Is the Formula for Finding the Volume of a Right Rectangular Pyramid?

The volume of a rectangular pyramid is found using the formula: V = (1/3) × L × W × h, where L x W represents the base area of the rectangular pyramid and h represents its total height.

How to Calculate the Volume of a Rectangular Pyramid?

To calculate the volume of a rectangular pyramid, we need to follow the steps given below:

• Step 1: Check for the given information like the length and width of the rectangular base and the height of the pyramid.
• Step 2: Put the given values in the volume of the rectangular pyramid formula.
• Step 3: Write the numerical value of volume so obtained with an appropriate unit

How to Find the Height When the Volume of a Rectangular Pyramid is Given?

In case the volume of a rectangular pyramid is given, together with the length and width of the base, then we can find the height of the rectangular pyramid,

• Step 1: Identify the given values, the volume of the rectangular pryramid, apothem length, and base length.
• Step 2: Divide the volume of the rectangular pyramid by its base area.

How to Find the Volume of a Rectangular Pyramid with Base Area and Height?

The simple formula to find the volume of a rectangular pyramid is the product of the base area of pyramid and height of the pyramid,

• base area = area of the base (which is a rectangle)
• height = height of the rectangular pyramid