Pyramid Formula

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A pyramid is a polyhedron that has a polygonal base and triangles for sides. The three main parts of any pyramid are the apex, face, and base. The base of a pyramid may be of any shape such as triangle, square, pentagon. Faces are in the shape of an isosceles triangle. Apex is a point at the top of the pyramid where all the triangle of the pyramid meets. Let us now discuss the different pyramid formula in detail.

What is a Pyramid?

A 3-dimensional shape formed by joining all the corners of a polygon to a central point or apex is named as a pyramid. The slant height is the diagonal height from the center of one of the base edges to the apex denoted by l. Height is the perpendicular distance from the apex to the base. Height is denoted by h.

Pyramid Formula

1] Surface area of the Pyramid

The lateral surface area of a regular pyramid is the sum of the areas of the lateral faces of the pyramid.

2] Total Surface Area of a Regular Pyramid

It is the sum of the areas of lateral faces of the pyramid and base of the pyramid.

Types of Pyramid

1] Square Pyramid

A square pyramid has a square base, triangular faces as 4 and an apex. The Square Pyramid formulas are,

  • Base Area of a Square Pyramid = b2
  • Surface Area of a Square Pyramid = 2 × b × s +b2

2] Triangular Pyramid

A triangular pyramid is a pyramid, which has triangular faces, and a triangular base. The Triangular Pyramid formulas are,

3] Pentagonal Pyramid

The Pentagonal pyramid has a pentagonal base, triangular faces as 5 and an apex. The Pentagonal Pyramid Formulas are,

Solved Example for Pyramid Formula

Q.1: Find the volume of a regular square pyramid with base sides 10 cm and altitude 18 cm.

Solution: Volume of a pyramid is,

Therefore, the volume of the regular square pyramid is 600 cm3

Q.2: Find the lateral surface area of a regular pyramid with a triangular base if each edge of the base measures 88 cm and the slant height is 55 cm.

Solution: The perimeter of the base is the sum of the sides.

Perimeter of the base = 3 × (8) = 24 cm

Pyramid Formula

A polyhedron that has a polygonal base and triangles for sides, is a pyramid. The three main parts of any pyramid’s: apex, face and base. The base of a pyramid may be of any shape. Faces usually take the shape of an isosceles triangle. All the triangles meet at a point on the top of the pyramid that is called “Apex”.

The formula for finding the volume and surface area of the pyramid is given as,

Square Pyramid

A Pyramid with a square base, 4 triangular faces and an apex is a square pyramid.

The Square Pyramid formulas are,

Where,
b – base length of the square pyramid
s – slant height of the square pyramid
h – height of the square pyramid

Triangular Pyramid

A triangular pyramid is a type of pyramid with triangular faces and a triangular base.

Where,
a – apothem length of the triangular pyramid
b – base length of the triangular pyramid
s – slant height of the triangular pyramid
h – height of the triangular pyramid

Pentagonal Pyramid

This pyramid has pentagonal base, with 5 sides , triangular faces and an apex.

Where,
a – apothem length of the pentagonal pyramid
b – base length of the pentagonal pyramid
s – slant height of the pentagonal pyramid
h – height of the pentagonal pyramid

Hexagonal Pyramid

This pyramid has a hexagonal base with six sides, six triangular faces and an apex.

Where,
a – apothem length of the hexagonal pyramid
b – base length of the hexagonal pyramid
s – slant height of the hexagonal pyramid
h – height of the hexagonal pyramid

Surface Area of Pyramid

The surface area of a pyramid is obtained by adding the area of all its faces. A pyramid is a three-dimensional shape whose base is a polygon and whose side faces (that are triangles) meet at a point which is called the apex (or) vertex. The perpendicular distance from the apex to the center of the base is called the altitude or height of the pyramid. The length of the perpendicular drawn from the apex to the base of a triangle (side face) is called the ‘slant height’. Let us learn more about the surface area of a pyramid along with its formula, a few solved examples, and practice questions.

What is the Surface Area of Pyramid?

The surface area of a pyramid is a measure of the total area that is occupied by all its faces. Observe the pyramid given below to see all its faces and the other parts like the apex, the altitude, the slant height, and the base.

The surface area of a pyramid is the sum of areas of its faces and hence it is measured in square units such as m2, cm2, in2, ft2, etc. A pyramid has two types of surface areas, one is the Lateral Surface Area (LSA) and the other is the Total Surface Area (TSA).

  • The Lateral Surface Area (LSA) of a pyramid = The sum of areas of the side faces (triangles) of the pyramid.
  • The Total Surface Area (TSA) of a pyramid = LSA of pyramid + Base area

In general, the surface area of a pyramid without any specifications refers to the total surface area of the pyramid.

Surface Area of Pyramid Formula

The surface area of a pyramid can be calculated by finding the areas of each of its faces and adding them. If the pyramid is regular (i.e., a pyramid whose base is a regular polygon and whose altitude passes through the center of the base), there are some specific formulas to find the lateral surface area and total surface area. Consider a regular pyramid whose base perimeter is ‘P’, the base area is ‘B’, and slant height (the height of each triangle) is ‘l’. Then,

  • The Lateral Surface Area of pyramid (LSA) = (1/2) Pl
  • The Total Surface Area of pyramid (TSA) = LSA + base area = (1/2) Pl + B

Note that we will use the formulas for the area of polygons to calculate the base areas here. Now, let us see how to derive the formulas of the surface area of a pyramid.

Proof of Surface Area of Pyramid Formula

The surface area of a pyramid involves the perimeter and slant height. Let us understand the formulas of LSA and TSA of a pyramid by taking a specific pyramid as an example. Let us consider a square pyramid whose base length is ‘a’ and whose slant height is ‘l’.

Then,

  • The base area (area of square) of the pyramid is, B = a2
  • The base perimeter (perimeter of square) of the pyramid is, P = 4a
  • The area of each of the side faces (area of triangle) = (1/2) × base × height = (1/2) × (a) × l

Therefore, the sum of all side faces (sum of all 4 triangular faces) = 4 [(1/2) × (a) × l] = (1/2) × (4a) × l = (1/2) Pl. (Here, we replaced 4a with P which represents its perimeter.)

Hence, the Lateral Surface Area of the pyramid (LSA) = (1/2) Pl

We know that the Total Surface Area of a pyramid (TSA) is obtained by adding the base and lateral surface areas. Thus,

The total surface area of pyramid (TSA) = LSA + base area = (1/2) Pl + B

Using these two formulas, we can derive the surface area formulas of different types of pyramids.

Surface Area of Pyramid with Altitude

The surface area of a pyramid can be calculated if its altitude is given. Observe the figure given below which shows that the triangle formed by half the side length of the base (a/2), the slant height (l), and the altitude (h) is a right-angled triangle. Hence, we can apply the Pythagoras theorem and find out the slant height if the altitude and base length is given. Thus, l2 = h2 + (a/2)2

So, we can calculate the slant height using the formula, l2 = h2 + (a/2)2. Now that we have the slant height, the base length, and the height, we can find the surface area of the pyramid using the formula, Total surface area of pyramid (TSA) = LSA + base area = (1/2) Pl + B

Volume of Pyramid

The volume of pyramid is space occupied by it (or) it is defined as the number of unit cubes that can be fit into it. A pyramid is a polyhedron as its faces are made up of polygons. There are different types of pyramids such as a triangular pyramid, square pyramid, rectangular pyramid, pentagonal pyramid, etc that are named after their base, i.e., if the base of a pyramid is a square, it is called a square pyramid. All the side faces of a pyramid are triangles where one side of each triangle merges with a side of the base. Let us explore more about the volume of pyramid along with its formula, proof, and a few solved examples.

What is Volume of Pyramid?

The volume of a pyramid refers to the space enclosed between its faces. The volume of any pyramid is always one-third of the volume of a prism where the bases of the prism and pyramid are congruent and the heights of the pyramid and prism are also the same, i.e., three identical pyramids of any type can be arranged to form a prism of the same type such that the heights of the pyramid and the prism are the same and their bases are congruent, i.e., three rectangular pyramids can be arranged to form a rectangular prism.

We can understand this by the following activity. Take a rectangular pyramid full of sand and take an empty rectangular prism whose base and height are as same as that of the pyramid. Pour the sand from the pyramid into the prism, we can see that the prism is exactly one-third full.

In the same way, we can see that in a cube, there are three square pyramids arranged invisibly.

Volume of Pyramid Formula

Let us consider a pyramid and prism each of which has a base area ‘B’ and height ‘h’. We know that the volume of a prism is obtained by multiplying its base by its height. i.e., the volume of the prism is Bh. In the earlier section, we have seen that the volume of pyramid is one-third of the volume of the corresponding prism (i.e., their bases and heights are congruent). Thus,

Volume of pyramid = (1/3) (Bh), where

  • B = Area of the base of the pyramid
  • h = Height of the pyramid (which is also called “altitude”)

Note: The triangle formed by the slant height (s), the altitude (h), and half the side length of the base (x/2) is a right-angled triangle and hence we can apply the Pythagoras theorem for this. Thus, (x/2)2 + h2 = s2. We can use this while solving the problems of finding the volume of the pyramid given its slant height.

Volume Formulas of Different Types of Pyramids

From the earlier section, we have learned that the volume of a pyramid is (1/3) × (area of the base) × (height of the pyramid). Thus, to calculate the volume of a pyramid, we can use the areas of polygons formulas (as we know that the base of a pyramid is a polygon) to calculate the area of the base, and then by simply applying the above formula, we can calculate the volume of pyramid. Here, you can see the volume formulas of different types of pyramids such as the triangular pyramid, square pyramid, rectangular pyramid, pentagonal pyramid, and hexagonal pyramid and how they are derived.

Solved Examples on Volume of Pyramid​​​​​​

  • Example 2: A pyramid has a regular hexagon of side length 6 cm and height 9 cm. Find its volume.

    Solution:

    The side length of the base (regular hexagon) is, a = 6.

    The base area (area of regular hexagon) is,

    B = (3√3/2) × a2

    B = (3√3/2) × 62 ≈ 93.53 cm2.

    The height of the pyramid is h = 9 cm.

    The volume of the hexagonal pyramid is,

    V = (1/3) (Bh)

    V = (1/3) × 93.53 × 9

    V = 280.59 cm3

    Answer: The volume of the pyramid is 280.59 cm3.

  • Example 3: Tim built a rectangular tent (that is of the shape of a rectangular pyramid) for his night camp. The base of the tent is a rectangle of side 6 units × 10 units and the height is 3 units. What is the volume of the tent?

    Solution:

    The base area (area of rectangle) of the tent is, B = 6 × 10 = 60 square units.

    The height of the tent is h = 3 units.

    The volume of the tent using the volume of pyramid formula is,

    V = (1/3) (Bh)

    V = (1/3) × 60 × 3

    V = 60 cubic units.

    Answer: The volume of the tent = 60 cubic units.

FAQs on Volume of Pyramid

What Is Meant By Volume of Pyramid?

The volume of a pyramid is the space that a pyramid occupies. The volume of a pyramid whose base area is ‘B’ and whose height is ‘h’ is (1/3) (Bh) cubic units.

What Is the Volume of Pyramid With a Square Base?

If ‘B’ is the base area and ‘h’ is the height of a pyramid, then its volume is V = (1/3) (Bh) cubic units. Consider a square pyramid whose base is a square of length ‘x’. Then the base area is B = x2 and hence the volume of the pyramid with a square base is (1/3)(x2h) cubic units.

What Is the Volume of Pyramid With a Triangular Base?

To find the volume of a pyramid with a triangular base, first, we need to find its base area ‘B’ which can be found by applying a suitable area of triangle formula. If ‘h’ is the height of the pyramid, its volume is found using the formula V =(1/3) (Bh).

What Is the Volume of Pyramid With a Rectangular Base?

A pyramid whose base is a rectangle is a rectangular pyramid. Its base area ‘B’ is found by applying the area of the rectangle formula. i.e., if ‘l’ and ‘w’ are the dimensions of the base (rectangle), then its area is B = lw. If ‘h’ is the height of the pyramid, then its volume is V =(1/3) (Bh) = (1/3) lwh cubic units.

What Is the Formula To Find the Volume of Pyramid?

The volume of a pyramid is found using the formula V = (1/3) Bh, where ‘B’ is the base area and ‘h’ is the height of the pyramid. As we know the base of a pyramid is any polygon, we can apply the area of polygons formulas to find ‘B’.

How To Find Volume of Pyramid With Slant Height?

If ‘x’ is the base length, ‘s’ is the slant height, and ‘h’ is the height of a regular pyramid, then they satisfy the equation (the Pythagoras theorem) (x/2)2 + h2 = s2. If we are given with ‘x’ and ‘s’, then we can find ‘h’ first using this equation and then apply the formula V = (1/3) Bh to find the volume of the pyramid where ‘B’ is the base area of the pyramid.

Why is There a 1/3 in the Formula for the Volume of Pyramid?

A cube of unit length can be divided into three congruent pyramids. So, the volume of pyramid is 1/3 of the volume of a cube. Hence, we have a 1/3 in the volume of pyramid.

Sample Problems

Problem 1: What is the volume of a square pyramid if the sides of a base are 6 cm each and the height of the pyramid is 10 cm?

Solution:

Given data,

Length of the side of the base of a square pyramid = 6 cm

Height of the pyramid = 10 cm.

Volume of a square pyramid (V) = 1/3 × Area of square base × Height

Area of square base = a2 = 62 = 36 cm2

V = 1/3 × (36) ×10 = 120 cm3

Hence, the volume of the given square pyramid is 120 cm3.

Problem 2: What is the volume of a triangular pyramid whose base area and height are 120 cm2  and 13 cm, respectively?

Solution:

Given data,

Area of the triangular base = 120 cm2

Height of the pyramid = 13 cm

Volume of a triangular pyramid (V) = 1/3 × Area of triangular base × Height

V = 1/3 × 120 × 13 = 520 cm3

Hence, the volume of the given triangular pyramid = 520 cm3

Problem 3: What is the volume of a triangular pyramid if the length of the base and altitude of the triangular base are 3 cm and 4.5 cm, respectively, and the height of the pyramid is 8 cm?

Solution:

Given data,

Height of the pyramid = 8 cm

Length of the base of the triangular base = 3 cm

Length of the altitude of the triangular base = 4.5 cm

Area of the triangular base (A) = 1/2 b × h = 1/2 × 3 × 4.5 = 6.75‬ cm2

Volume of a triangular pyramid (V) = 1/3 × A × H

V = 1/3 × 6.75 × 8 = 18 cm3

Hence, the volume of the given triangular pyramid is 18 cm3

Problem 4: What is the volume of a rectangular pyramid if the length and width of the rectangular base are 8 cm and 5 cm, respectively, and the height of the pyramid is 14 cm?

Solution:

Given data,

Height of the pyramid = 14 cm

Length of the rectangular base (l) = 8 cm

Width of the rectangular base (w) = 5 cm

Area of the rectangular base (A) = l‬ × w = 8 × 5 = 40 cm2

We have,

Volume of a rectangular pyramid (V) = 1/3 × A × H

V = 1/3 × 40 × 14 = 560/3 = 186.67 cm3

Hence, the volume of the given rectangular pyramid is 186.67 cm3.

Problem 5: What is the volume of a hexagonal pyramid if the sides of a base are 8 cm each and the height of the pyramid is 15 cm?

Solution:

Given data,

Height of the pyramid = 15 cm

Length of the side of the base of a hexagonal pyramid = 6 cm

Area of the hexagonal base (A) = 3√3/2 a = 3√3/2 (6)2 = 54√3 cm2

Volume of a hexagonal pyramid (V) = 1/3 × A × H

V = 1/3 × 54√3 × 15 = 270√3 cm3

Hence, the volume of the given hexagonal pyramid is  270√3 cm3.

Problem 6: What is the volume of a pentagonal pyramid if the base area is 150 cm2 and the height of the pyramid is 11 cm?

Solution:

Given data,

Area of the pentagonal base = 150 cm2

Height of the pyramid = 11 cm

Volume of a pentagonal pyramid (V) = 1/3 × Area of pentagonal base × Height

V = 1/3 × 150 × 11 = 550 cm3

Hence, the volume of the given pentagonal pyramid = 550 cm3

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