Rectangular Pyramid

Introduction

Rectangular Pyramid is a three-dimensional solid figure with a rectangular base and four triangular flat surfaces. There are many types of pyramids, such as the square pyramid, triangular pyramid, pentagonal pyramid, and hexagonal pyramid concerning the shapes of the base and the number of triangle surfaces.

The surfaces of the pyramid are plane figures with straight lines, which can also be called polygons.

If a solid figure is made up of polygons then it is called a polyhedron. Rectangular pyramids, or pyramids in general, are Polyhedrons. One of the common types of the pyramid is the rectangular pyramid. The Pyramids in Egypt can be given as a living example of rectangular pyramids. The Egyptians were the first ones to measure the Volume of a pyramid.

In this tutorial, we are going to learn about calculating the surface area and the Volume of rectangular pyramids in different scenarios.

Rectangular Pyramid

A solid figure with five points and eight straight lines forming four triangular surfaces and one rectangular base is called a Rectangular Pyramid.

• The pointy notch or say, the point at the top joining all the triangular surfaces is known to be the apex of the rectangular pyramid.

• The remaining points or say corners acting as the intersection points of the other ends of the edges are the vertices.

• The straight lines formed by the intersection of the sides are the edges.

• The base and other surfaces of the figure are the faces.

The Formula for the Surface Area

When we explore the surface area of a rectangular pyramid. We need to consider the total surface area and the lateral surface area.

Total Surface Area of the Rectangular Pyramid

The sum of the lateral surface area and the base area is the total Surface Area Or the T. S. A of the Rectangular Pyramid.

The formula is given by,

$$\mathrm{T.S.A\:=\:lw\:+\:l\sqrt{[(\frac{W}{2})^{2}\:+\:h^{2}]}\:+\:w\sqrt{[(\frac{l}{2})^{2}}\:+\:h^{2}]}$$

Where l is the base length

w is the base width

h is the height of the pyramid

The Lateral Surface Area of the Rectangular Pyramid

The sum of all areas of the lateral triangular faces excluding the base area is the Lateral surface area Or L. S. A of the Rectangular Pyramid.

The formula is given by, $$\mathrm{L.S.A\:=\:l\sqrt{[(\frac{W}{2})^{2}\:+\:h^{2}]}\:+\:w\sqrt{[(\frac{l}{2})^{2}}\:+\:h^{2}]}$$

Where l is the base length

w is the base width

h is the height of the pyramid

The formula for the Volume of the Rectangular Pyramid

The Volume of the rectangular pyramid is measured by multiplying the base area B by the perpendicular height h of the pyramid.

The formula is given by

$\mathrm{volume\:V\:=\:\frac{1}{3}\times\:base\:are\:\times\:perpendicular\:height}$

$\mathrm{V\:=\:\frac{1}{3}\:Bh}$

$\mathrm{V\:=\:\frac{1}{3}\times\:l\times\:w\times\:h}$

Where base area B is lw

l is the base length

w is the base width

h is the perpendicular height of the pyramid.

Solved Examples

1)In a clerk's table there is a paperweight in the shape of a rectangular pyramid with height 5 cm, base width 4 cm and base length 6 cm. Find the total surface area of the paper weight.

Answer −

The total surface area of the rectangular pyramid

$$\mathrm{=\:lw\:+\:l\sqrt{[(\frac{W}{2})^{2}\:+\:h^{2}]}\:+\:w\sqrt{[(\frac{l}{2})^{2}}\:+\:h^{2}]}$$

The total surface area of the paperweight

$$\mathrm{=\:6\times\:4\:+\:6\sqrt{[(\frac{4}{2})^{2}\:+\:5^{2}]}\:+\:4\sqrt{[(\frac{l}{2})^{2}}\:+\:5^{2}]\:=\:79.63\:cm^{2}}$$

2) Suppose a wood is carved as a rectangular pyramid with the same height, width and length measuring 7 cm. Find the lateral surface area of the wood.

Answer − The lateral surface area of a rectangular pyramid

$$\mathrm{=\:l\sqrt{[(\frac{W}{2})^{2}\:+\:h^{2}]}\:+\:w\sqrt{[(\frac{l}{2})^{2}}\:+\:h^{2}]}$$

The lateral surface area of the wood

$$\mathrm{=\:7\sqrt{[(\frac{7}{2})^{2}\:+\:7^{2}]}\:+\:7\sqrt{[(\frac{l}{2})^{2}}\:+\:7^{2}]}$$

$$\mathrm{=\:109.57\:cm^{@}}$$

3)A group of 26 friends had a campfire. They built a big tent in the shape of a rectangular pyramid with base area 300 𝒇𝒕𝟐and perpendicular height 30 ft. Find the Volume of the tent.

Answer −

$\mathrm{volume\:of\:a\:rectangular\:pyramid\:=\:\frac{1}{3}\times\:base\:area\:\times\:perpendicular\:height}$

$\mathrm{volume\:of\:the\:tent\:=\:\frac{1}{3}\times\:300\times\:30}$

$\mathrm{=\:3000\:ft^{3}}$

4)X built the roof of his pet dog's kennel in the shape of a rectangular pyramid with perpendicular height 20 cm and the base area 1800 𝒄𝒎𝟐 . Find the Volume of the roof.

Answer −

The Volume of the rectangular pyramid $\mathrm{=\:\frac{1}{3}\:Bh}$

The Volume of the tent $\mathrm{=\:\frac{1}{3}\times\:1800\times\:20}$

$\mathrm{=\:1.2\times\:10^{5}\:cm^{3}}$

5)Suppose the Volume of a rectangular pyramid is 2000 𝒄𝒎𝟑 and the base area is 240 𝒄𝒎𝟐. What is the perpendicular height of the pyramid?

Answer −

Volume of a rectangular pyramid $\mathrm{=\:\frac{1}{3}\times\:base\:area\times\:perpendicular\:height}$

$\mathrm{2000\:=\:\frac{1}{3}\times\:240\times\:h}$

$\mathrm{600\:=\:24\times\:240\times\:h}$

$\mathrm{100\:=\:4\times\:h}$

$\mathrm{h\:=\:25\:cm}$

6)Suppose the height of the rectangular pyramid is 12 cm and the Volume is 1800 𝒄𝒎𝟑. What is the base area of the pyramid?

Answer −

Volume of a rectangular pyramid $\mathrm{=\:\frac{1}{3}\times\:base\:area\:\times\:perpendicular\:height}$

$\mathrm{1800\:=\:\frac{1}{3}\:\times\:B\times\:12}$

$\mathrm{5400\:=\:B\times\:12}$

$\mathrm{1800\:=\:B\times\:4}$

$\mathrm{Base\:area\:B\:=\:450\:cm^{2}}$

Conclusion

• A solid with four triangular faces and one rectangular base is known as a rectangular pyramid.

• The base area of the rectangular pyramid is the product of its length and width.

• The total surface area of the rectangular pyramid is calculated by multiplying the lateral surface area and the base area.

• The lateral surface area of the rectangular pyramid is measured by adding all the areas of the triangular faces

• The Volume of the rectangular pyramid is found by multiplying the base area with its perpendicular height.

FAQs

1. What is a pyramid? Give one real-life example?

A solid figure formed by joining the base with the lateral triangular faces is called a Pyramid

The Word Pyramid is a Greek word meaning wheat cake since Greek used to compare the Egyptian buildings to pointy wheat cakes.

The Great Pyramid of Giza in Egypt is a suitable example.

2. What is a Square Pyramid?

. A solid with four triangular faces and one square base is called a Square Pyramid. It is also called a Pentahedron since it has five faces.

3. What is the difference between a square and a rectangular pyramid?

The difference between square and rectangular pyramids is the base area but the square pyramid can be called a rectangular pyramid because it has the same surface area and Volume as the rectangular pyramid but it cannot be called vice versa.

4. What is a triangular pyramid?

A geometric solid figure that has three triangular faces and a triangular base is called a Triangular Pyramid.

5. How will you measure the Volume of a square pyramid?

The Volume of the square pyramid is the same as the rectangular pyramid, it is calculated by one-third of the product of the base area and the pyramid height