Rectangular Pyramid

A rectangular pyramid is a type of pyramid with the base shaped like a rectangle but the sides are shaped like a triangle. A pyramid usually has triangular sides but with different bases such as a square pyramid or a hexagonal pyramid. The distinguishing factor among all these pyramids is the base, the face, the edges, and vertices. A rectangular pyramid has its own properties and formulas. Let us learn more about the rectangular pyramid and solve a few examples.

A rectangular pyramid is a 3-D object with a base shaped like a rectangle and triangle-shaped faces or sides that correspond to each side of the base. Like the other types of pyramids, a rectangular pyramid also has a point on top of the base of the pyramid that is joined together by bringing the top of all the sides which is known as the apex. A rectangular pyramid has a total of 5 faces, 5 vertices, and 8 edges and is of two types a right pyramid or an oblique pyramid. The base and the sides of the pyramid are joined at the vertex. The image below interprets the shape of the pyramid.

Every pyramid has its own properties that make it different from one another. A rectangular pyramid is unique in its shape due to its properties. Let us look at what they are:

• A rectangular pyramid has 5 faces, 5 vertices, and 8 edges.
• One face is the rectangle base and 4 are in the shape of a triangle.
• All the triangular faces are congruent to the opposite face.
• One of the vertexes meet at the apex which is right above the base of the pyramid.
• The vertex is the intersection point between the base and the side and four edges are the intersection point at the base.
• 4 edges formed above the base of the pyramid, make the slope. While the other 4 edges are at the base.

There are two types of rectangular pyramids namely - the right rectangular pyramid and the oblique rectangular pyramid.

Right Rectangular Pyramid

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. When a rectangular pyramid is mentioned, it is usually referred to as the right rectangular pyramid.

Oblique Rectangular Pyramid

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.

The volume of the rectangular pyramid is defined as the capacity of the rectangular pyramid and 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. The formula to find the volume of a rectangular pyramid is:

Volume = 1/3 × Base Area × h

Where,

• h is the perpendicular height
• base area = Length of the rectangular base × Width of the rectangular base.

The volume of the rectangular pyramid can be found by finding the length, width, and height after which the measurements can be placed in the above-mentioned formula.

The total surface area of a rectangular pyramid is the sum of the areas of its base and its lateral faces. It is calculated by adding up the area of all rectangular and triangular faces. The important aspects to remember where finding the total surface area are: area of the rectangular base, area of front and back identical triangles, slant height, and area of side triangles. The combination of all these aspects will help in finding the total surface area of a rectangular pyramid. The formula for each of these aspects and the steps are:

1. Area of the rectangular base = length of the rectangular base × width of the rectangular base.

2. Area of front and back identical triangles = \(l \sqrt{\left(\frac{\text{width of rectangular base}}{2}\right)^{2}+(\text {height of pyramid})^{2}}\). Since the area of the two front and back triangles is- [2 × 1/2(base)(height)].

3. Slant height of the triangles = \(\sqrt{\left(\frac{\text{width of rectangular base}}{2}\right)^{2}+({\text {height of pyramid}})^{2}}\)

4. The side triangles' area can be found similar to the front and back triangles, only the slant height formula is changed:

\(w \sqrt{\left(\frac{\text{length of rectangular side}}{2}\right)^{2}+({\text {height of pyramid}})^{2}}\)

where, the slant height is \(\sqrt{\left(\frac{\text{length of rectangular side}}{2}\right)^{2}+({\text {height of pyramid}})^{2}}\).

Therefore, the total surface area of a rectangular pyramid is calculated by adding all the areas together which become:

A = lw +l √[(w/2)² + h²] + w√[(l/2)² + h²] square units

Lateral Surface Area of Rectangular Pyramid

The lateral surface area of any object is calculated by removing the base area or we can say that the lateral surface area is the area of the non-base faces only. Once the rectangular area is removed from the total surface area formula we derive the lateral surface area formula. Thus, the lateral surface area of a rectangular pyramid is l √[(w/2)² + h²] + w√[(l/2)² + h²]
where,

• l = Length of the rectangular base
• w = Width of the rectangular base
• h = Height of the pyramid

The net of a rectangular pyramid consists of 4 triangular faces and 1 rectangular base. This is seen when the shape of the pyramid is opened i.e. when the object goes from 3D to 2D. The apex is opened making the triangle faces flattened. The below image shows the net of a rectangular pyramid.

Related Topics

Listed below are a few interesting topics related to the rectangular pyramid, take a look.

• Triangular Pyramid
• Rectangular Pyramid Volume Calculator
• Square Pyramid

What is a Rectangular Pyramid?

A rectangular pyramid is a 3-D object with a base shaped like a rectangle and triangle-shaped faces or sides that correspond to each side of the base. The top of the base of the pyramid that is joined together by bringing the top of all the sides is known as the apex. A rectangular pyramid has a total of 5 faces, 5 vertices, and 8 edges and is of two types a right pyramid or an oblique pyramid. The base and the sides of the pyramid are joined at the vertex.

How Many Faces, Edges, and Vertices does a Rectangular Pyramid Have?

A rectangular pyramid consists of 5 faces, 5 vertices, and 8 edges. One of the faces is the rectangular base whereas the rest 4 are the triangles. 4 edges of the rectangular pyramid that are joined by the triangular sides are considered as the slope of the rectangular pyramid.

What are the Two Types of Rectangular Pyramid?

The two types of the rectangular pyramid are:

• Right Rectangular Pyramid: If the apex of the rectangular pyramid is aligned right above the center of the base, the pyramid is called a right rectangular pyramid.
• Oblique Rectangular Pyramid: 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.

How Do You Find the Volume of a 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. Where base area = length × width of the rectangular pyramid.

What is the Formula to Calculate the Total Surface Area of a Rectangular Pyramid?

The total surface area of a rectangular pyramid formula using the base width, length, and height is given as, T.S.A. = lw + l√[(w/2)² + h²] + w√[(l/2)² + h²].

where,

• l is length of the rectangular base.
• w is width of the rectangular base.
• h is height.

What is the Formula to Calculate the Lateral Surface of a Rectangular Pyramid?

The lateral surface of a rectangular pyramid is calculated by l √[(w/2)² + h²] + w√[(l/2)² + h²].

where,

• l is length of the rectangular base.
• w is width of the rectangular base.
• h is height.

What is the Net of a Rectangular Pyramid?

The net of a rectangular pyramid is consists of 4 triangular faces and 1 rectangular base. When the pyramid is flattened out, the triangles and rectangle can be seen clearly.