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Total Surface Area of Rectangular Pyramid
The total surface area of a rectangular pyramid is the total area of all of the sides and faces of a rectangular pyramid. Basically, a rectangular pyramid is a 3D object having a rectangle for a base and also having a triangular face corresponding to each side of the base. A rectangular pyramid has a point on top of the base of the pyramid, which is known as the apex. A rectangular pyramid can be a right pyramid or an oblique pyramid. A rectangular pyramid has a total of five faces, five vertices, and eight edges.
Total Surface Area of Rectangular Pyramid Formula
The surface area of any threedimensional geometrical shape is the sum of the areas of all of the faces or surfaces of that enclosed solid. The total surface area of a rectangular pyramid is the sum of the areas of its base and its lateral faces. A rectangular pyramid has four triangular faces and one rectangular face. Thus, the total surface area of a rectangular pyramid is calculated by adding up the area of all rectangular and triangular faces. The surface area of the rectangular pyramid is:
T.S.A. = lw + l √[(w/2)^{2} + h^{2}] + w√[(l/2)^{2} + h^{2}]
Slant height = √[(w/2)^{2} + h^{2}]
Area of a front and back triangle = l √[(w/2)^{2} + h^{2}]
Area of the side triangles = w√[(l/2)^{2} + h^{2}]
where,
 l = Length of the rectangular base.
 w = Width of the rectangular base.
 h = Height of the pyramid.
 l × w = Area of the rectangular base
How to Calculate Total Surface Area of Rectangular Pyramid?
The total surface area of a rectangular pyramid can be calculated by representing the 3D shape into a 2D net, to make the shapes easier to see. After expanding the 3D figure into 2D, we will get four triangles and one rectangle.
The following steps are used to calculate the total surface area of a rectangular pyramid:
 To find the area of the rectangular base
The area of the rectangular base is the product of the length of the rectangular base with the width of the rectangular base.
 To find area of front and back identical triangles
The area of the two front and back triangles is [2 × 1/2(base)(height)], which will become:
\(l \sqrt{\left(\frac{\text{width of rectangular base}}{2}\right)^{2}+(\text {height of pyramid})^{2}}\).
 To find the slant height of the triangles
The slant height of the triangles is calculated by:
\(\sqrt{\left(\frac{\text{width of rectangular base}}{2}\right)^{2}+({\text {height of pyramid}})^{2}}\)
 To find the area of the side triangles
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 \(\sqrt{\left(\frac{\text{length of rectangular side}}{2}\right)^{2}+({\text {height of pyramid}})^{2}}\).
 Thus, the total surface area of a rectangular pyramid is calculated by adding all the areas together which become:
A = lw +l √[(w/2)^{2} + h^{2}] + w√[(l/2)^{2} + h^{2}]square units
What Is 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 nonbase faces only. The lateral area of a rectangular pyramid can be calculated by removing the rectangular area from the formula of the total surface area of a Rectangular Pyramid. Thus, the lateral surface area of a right triangular prism is l √[(w/2)^{2} + h^{2}] + w√[(l/2)^{2} + h^{2}]
where,
 l = Length of the rectangular base
 w = Width of the rectangular base
 h = Height of the pyramid
Examples on Total Surface Area of A Rectangular Pyramid

Example 1: Find the total surface area of a rectangular pyramid whose base length and width are 10 and 8 units. Also, the height of the pyramid is 5 units.
Solution:
Base length, l = 10 units
Height of the pyramid, h = 5 units
Width of a base, w = 8 units
The total surface area of a rectangular pyramid is A = lw +l √[(w/2)^{2} + h^{2}] + w√[(l/2)^{2} + h^{2}]
On putting the values, we get
T.S.A. = 10 × 8 + 10 \(\sqrt{\left(\frac{8}{2}\right)^{2}+{5}^{2}}+ 8 \sqrt{\left(\frac{10}{2}\right)^{2}+{5}^{2}}\).
T.S.A. = 80 + 64.03 + 56.56
T.S.A. = 200.59 square units.
Answer: The total surface area of a rectangular pyramid is 200.59 sq units.

Example 2: Find the total surface area of a rectangular pyramid whose area of the base rectangle is 30 square units, and lateral surface area is 25 square units.
Solution:
Area of the base rectangle = 30 square units
Lateral surface area = 25 square units
Total surface area of a rectangular pyramid = Area of the base rectangle + Lateral surface area of a pyramid.
Putting the values together,
The surface area of a right rectangular prism = 30 + 25
= 55 square units.
Answer: The total surface area of a rectangular pyramid is 55 sq units.
Practice Questions on Total Surface Area of A Rectangular Pyramid
FAQs on Total Surface Area of a Rectangular Pyramid
What Do You Mean By Total Surface Area of a Rectangular Pyramid?
The total surface area of a rectangular pyramid refers to the total area covered by all the surfaces of a rectangular pyramid. It is the sum of the area of the base and lateral faces.
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} + h^{2}] + w√[(l/2)^{2} + h^{2}], where, l is length of the rectangular base, w is width of the rectangular base and h is height.
How Do You Find the Total Surface Area of a Rectangular Pyramid?
The formula for the total surface area of a rectangular pyramid is calculated by adding up the area of all rectangular and triangular faces of a pyramid, which is T.S.A. = lw + l√[(w/2)^{2} + h^{2}] + w√[(l/2)^{2} + h^{2}],
where,
 l = Length of the rectangular base.
 w = Width of the rectangular base.
 h = Height of the pyramid.
What Is the Lateral Surface of a 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 nonbase faces only. The lateral surface of a rectangular pyramid is calculated by l √[(w/2)^{2} + h^{2}] + w√[(l/2)^{2} + h^{2}], where, l is length of the rectangular base, w is width of the rectangular base, and h is height.
What Is the Unit Used to Express the Total Surface Area of a Rectangular Pyramid?
The total surface area of a rectangular pyramid is expressed in square units, using units like in^{2}, m^{2}, cm^{2}, ft^{2}, etc.
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