Greetings adventurer, to the right rectangular pyramid calculator. Today we will learn how to find the volume of a rectangular pyramid, along with its surface area. As the name suggests, we deal with pyramids that have rectangles as their bases, while the "right" part means that the apex is directly above the center of the base. This calculator will find V, A, A_l, and A_b for any base and height you enter.
We use the following notation:
- V — volume of the pyramid;
- A — total surface area of the pyramid;
- A_l — lateral surface area;
- A_b — base area; and
- H — pyramid height.
Remember that every square is a rectangle (but the opposite is not true!), so this calculator can also deal with problems related to right square pyramids, such as how to find the surface area of a square pyramid.
Example: Volume of a square pyramid formula
Let's begin with a question that may have bugged you for years: what is the volume of the Great Pyramid of Giza? Let's take a look at the square pyramid volume formula to see what we're dealing with:
We need the base area first. The Great Pyramid of Giza's base is a square, so we use the well-known formula area = a × a, where a is the base side length — roughly 230 meters. The height of the pyramid is 147 meters. Now we're left with the most fun part — counting!
… or roughly 3,390,339 cubic yards if you prefer imperial. It seems we weren't so far from the actual answer with our initial "probably a lot."
Right rectangular pyramid calc: find A_l
When we asked how to find the volume above, we had to know the base area, A_b. Let us now take care of the other faces of the pyramid and find the lateral area, A_l.
At first glance you might think there is a problem, since the lateral faces are triangles whose area is area = (a × h) / 2, where a is the base length and h is the height projected onto that side — the slant of the pyramid. We usually know a well enough, but h can get a little tricky. Worry not! As long as we know the base side and the pyramid height, we can calculate the slant using the Pythagorean theorem:
This way, we arrive at the not-so-scary-anymore formula:
where the indexed A-s denote the surface area of consecutive lateral faces. A scrupulous eye will notice that, among the four, we have two pairs of equal areas, and if the base is a square, all four of them are equal. In terms of the base sides and slants, this simplifies to A_l = a·l_a + b·l_b.
Right rectangular pyramid calc: find A_b and A
We've seen how to find the lateral surface area, so let's now find A_b and A, the base area and the total surface area, respectively.
The general formula for the base of a rectangular based pyramid is just as easy as for a square:
where a and b denote the lengths of the sides of the base. The last step is given by:
or A = A_b + A_l for short, with the notation we introduced earlier.
Example: Use the right rectangular pyramid calc to find V and A
Let's work through the default values — a base of a = 6 cm by b = 4 cm and a height of H = 9 cm:
V = ⅓ · 24 · 9 = 72 cm³
l_a = √(9² + 2²) ≈ 9.22 cm, l_b = √(9² + 3²) ≈ 9.49 cm
A_l = 6 · 9.22 + 4 · 9.49 ≈ 93.30 cm²
A = 24 + 93.30 ≈ 117.30 cm²
How to find the volume of a non-rectangular pyramid
The volume rule V = ⅓ · base_area · height works for any pyramid, no matter the shape of its base — triangular, pentagonal, hexagonal, and so on. All you have to do is swap in the correct base area. For a triangular base you would use base_area = ½ · base · height, for a regular polygon base_area = ½ · perimeter · apothem, and then plug the result into the same one-third rule. The surface area, however, does depend on the base shape, since each type of pyramid has a different set of lateral faces.
How to use the right rectangular pyramid calculator
- Pick your unit system — metric (cm) or US/Imperial (in).
- Enter the base length a and base width b. Set them equal if you are working with a square pyramid.
- Enter the pyramid height H. The calculator instantly returns V, A, A_l, and A_b, together with the slant heights and lateral edge.
Unit Systems
Metric System
Enter the dimensions in centimeters (cm). Areas are shown in cm² and the volume is given in cm³, mL, L, and m³.
US / Imperial System
Enter the dimensions in inches (in). Areas are shown in in² and the volume is given in in³, ft³, yd³, and US gallons.
FAQs
What is a right rectangular pyramid?
A pyramid with a rectangular base whose apex sits directly above the center of that base. Because a square is a special rectangle, a right square pyramid is just a right rectangular pyramid with equal base sides.
How do I find the volume of a rectangular pyramid?
Multiply the base area (a × b) by the height H and divide by three: V = ⅓ · a · b · H.
Why are there two different slant heights?
A rectangular base has two pairs of sides of different lengths, so the triangular faces come in two sizes. Each pair has its own slant height: l_a = √(H² + (b/2)²) and l_b = √(H² + (a/2)²). For a square base they are equal.
What is the difference between lateral and total surface area?
The lateral area A_l covers only the four triangular faces. The total area A = A_l + A_b also includes the rectangular base A_b.