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## Volume of a Square Pyramid

What do we mean by the volume of a square pyramid and how do we define it? Volume is nothing but the space that an object occupies. A square pyramid is a three-dimensional geometric shape that has a square base and four triangular bases that are joined at a vertex. Thus, the volume of a pyramid refers to the space enclosed between its faces.

Let’s learn how to find the volume of a square pyramid here with the help of few solved examples and practice questions.

## What Is the Volume of a Square Pyramid?

The volume of a square pyramid refers to the space enclosed between its five faces. The volume of a square pyramid is one-third of the product of the area of the base and the height of the pyramid. Thus, volume = (1/3) × (Base Area) × (Height). The volume of a square pyramid is the number of unit cubes that can fit into it and is represented in “cubic units”. Commonly it’s expressed as m^{3}, cm^{3}, in^{3}, etc.

A square pyramid is a three-dimensional shape with five faces. A square pyramid is a polyhedron (pentahedron) that consists of a square base and four triangles connected to a vertex. Its base is a square and the side faces are triangles with a common vertex. A square pyramid has three components:

- The top point of the pyramid is called the apex.
- The bottom square of the pyramid is called the base.
- The triangular sides of the pyramid are called faces.

Examples of a square pyramid are the Great Pyramid of Giza, perfume bottles, etc

Let us learn more about the formula of the volume of a square pyramid.

## Volume of a Square Pyramid Formula

The volume of a square pyramid can be easily found out by just knowing the base area and its height, and is given as:

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

Now, consider a regular square pyramid made of equilateral triangles of side ‘b’.

The volume of a regular square pyramid can be given as:

Volume of a regular square pyramid = 1/3 × b^{2} × h

where,

- b is the side of the base of the square pyramid, and,
- h is the height of the square pyramid

## How To Find the Volume of a Square Pyramid?

As we learned in the previous section, the volume of a square pyramid could be found using (1/3) Base Area × Height. Thus, we follow the below steps to find the volume of a square pyramid.

**Step 1:**Note the dimensions of the pyramid, like the base area and the height of the pyramid from the given data.**Step 2:**Multiply the area of the base by height and (1/3).**Step 3:**Represent the final answer with cubic units.

Now that we have learned about the volume of a square pyramid, let us understand it better using a few solved examples.

## Examples on Volume of a Square Pyramid

**Example 1:** A sanitizer bottle is shaped like a square pyramid of side 3 inches and having a height of 9 inches. Use the volume of a square pyramid formula to find how much sanitizer can the bottle hold?

**Solution:**

We know that for a square pyramid whose side is a, and height is h the volume is:

Volume of a square pyramid = 1/3 × a^{2} × h

Substituting the value of a and h we get

Volume of a square pyramid = 1/3 × a^{2} × h

= 1/3 × 3^{2} × 9

= 27

Therefore, the volume of a sanitizer bottle is 27 inches^{3}.

**Example 2:** Sam was building a toy parachute for his science exhibition whose base area is 18 in^{2} and height is 6 inches. How much air does the parachute need to be filled completely?

**Solution:**

Given,

The base area of a toy parachute = 18 in^{2}

Height of a toy parachute = 4 in

As we know,

The volume of a square pyramid = 1/3 × Base Area × Height

Putting the values in the formula: 1/3 × 18 × 6 = 36 in^{3}

Therefore, The volume of air in the parachute is 36 in^{3}.

## FAQs on Volume of a Square Pyramid

### What Is Volume of Square Pyramid?

The volume of a square pyramid is the space enclosed by the solid shape in a three-dimensional plane. A square pyramid is a three-dimensional shape with five faces. Its base is a square and the side faces are triangles with a common vertex.

### How Do You Find the Volume of a Square Pyramid?

The volume of a square pyramid can be easily found by just knowing the base area and its height. We can directly apply the volume of a square pyramid formula given the base and height as, Volume = 1/3 × Base Area × Height.

### What Is the Volume of a Regular Square Pyramid?

Volume of square pyramid is the space or region enclosed by a regular square pyramid in a three-dimensional plane. The formula to calculate the volume of a regular square pyramid is given as, Regular square pyramid volume:1/3 × a^{2} × h, where ‘a’ is the side of the square faces and ‘h’ is the height of the pyramid.

### What Units Are Used With the Volume of the Square Pyramid?

The volume of a square pyramid is expressed in cubic units. Generally, units like cubic meters (m^{3}), cubic centimeters (cm^{3}), liters (l), etc are the most common units used with the volume of the square pyramid.

### How Do You Find the Volume of Pyramid and Prisms?

The volume of the prism can be calculated using the base area and height. The formula for the volume of a prism is equal to the base area × height of the pyramid. While the volume of a pyramid can be calculated as, 1/3 × Base Area × Height.

### What Is the Formula for Finding the Volume of a Square Pyramid?

The volume of a square pyramid is found using the formula using the base area and height given as, V = 1/3 × Base Area × Height. For a regular pyramid, we can apply the following formula, given the side of square face and height,

1/3 × a^{2} × h, where ‘a’ is the side of the square faces and ‘h’ is the height of the pyramid.

### How to Find the Height of a Square Pyramid When Given the Volume?

To find the height of a square pyramid using the volume, we can apply the volume of a pyramid formula, substitute the given values and solve for the missing height,

Volume of square pyramid = Volume = 1/3 × Base Area × Height

### How Does the Volume of a Square Pyramid Relate to the Volume of a Square Prism?

Given a prism and a pyramid with congruent bases and the same height, if we put the pyramid inside the prism, their bases overlap exactly. Since both the shapes have the same height, the top of the pyramid will touch the top of the prism. Thus, the pyramid fits completely in the given pyramid. The relation between the volume of a pyramid and a prism can be given as,

Volume of a square pyramid = (1/3) Volume of a square prism

## Volume of a Right Square Pyramid

The volume of a right square pyramid is the space occupied by the right square pyramid. A right square pyramid is a three-dimensional geometric shape that has a right square base and four triangular faces that are joined at a vertex. Let’s learn how to find the volume of a right square pyramid with the help of a few solved examples and practice questions.

## What is Volume of a Right Square Pyramid?

The volume of a right square pyramid is the number of unit cubes that can fit into it. A right square pyramid is a three-dimensional shape that has a right square base and four triangular faces that are joined at a vertex. A right square pyramid is a polyhedron (pentahedron) with five faces. The unit of volume is “cubic units”. For example, it can be expressed as m^{3}, cm^{3}, in^{3}, etc depending upon the given units.

A right square pyramid has three components.

- The top point of the pyramid is called the apex.
- The bottom right square is called the base.
- The triangular sides are called faces.

## Formula of Volume of a Right Square Pyramid

The formula to determine the volume of a right square pyramid is V = 1/3 × b^{2} × h where “b” is the length of the base and *“*h” is the perpendicular height. The relation between slant height, perpendicular height, and the base is given by using Pythagoras Theorem s^{2} = h^{2} + (b/2)^{2} where “s”, “h” and “b” are slant height, the height of perpendicular, and base length of the right square pyramid, respectively. Thus, the volume of the right square pyramid is given by replacing the given dimensions in the formula V = 1/3 × b^{2} × h.

## How to Find the Volume of a Right Square Pyramid?

As we learned in the previous section, the volume of a right square pyramid could be found using

. Thus, we follow the below steps to find the volume of a right square pyramid.

**Step 1:**Determine the base area (b^{2}) and the height (h) of the pyramid.**Step 2:**Find the volume using the formula 1/3 × b^{2}× h**Step 3:**Represent the final answer with cubic units.

**Example: **Find the volume of the right square pyramid having height and length of the base edge of 9 units and 5 units respectively.

**Solution:** Given that h = 9 units and b = 5 units.

Then, the volume of the right square pyramid is V = 1/3 × b^{2} × h

⇒ V = 1/3 × 5^{2} × 9

⇒ V = 5^{2} × 3 = 75 cubic units

**Answer:** The volume of the right square pyramid is 75 cubic units.

## Solved Examples on Volume of a Right Square Pyramid

**Example 2:** What will be the volume of a regular square pyramid with base sides 10 cm and a height of 18 cm?

**Solution: **Given that b = 10 cm and h = 18 cm

The formula for the volume of a square pyramid is given by V = 1/3 × b^{2} × h

The area of the base = The square of base length = 10^{2} or 100 cm^{2}.

Putting the values b^{2} = 100 and h = 18 in the formula.

V = 1/3 × 100 × 18 = 600 cm^{3}.

**Answer:** The volume of the right square pyramid is 600 cm^{3}.

## FAQs on Volume of a Right Square Pyramid

### What is the Volume of a Right Square Pyramid?

The volume of a right square pyramid is defined as the number of unit cubes that can fit into a right square pyramid. A right square pyramid is a three-dimensional geometric shape that has a right square base and four triangular faces that are joined at a vertex.

### What Units Are Used With the Volume of the Right Square Pyramid?

The unit used with the volume of the right square pyramid is given in cubic units. For example, m^{3}, cm^{3}, in^{3}, etc depending upon the given units.

### What is the Formula of the Volume of the Right Square Pyramid?

The formula of the volume of the right square pyramid is given as V = 1/3 × Base Area × Height = 1/3 × b^{2} × h where b^{2} shows the base area and h shows the height of the right square pyramid.

### How to Find the Volume of a Right Square Pyramid?

The volume (V) of a right square pyramid can be found by using the following steps:

**Step 1:**Find the base area (b^{2}) and the height (h) of the pyramid.**Step 2:**Determine the volume using the formula 1/3 × b^{2}× h**Step 3:**Now, write the final answer with cubic units.

### How to Find the Volume of a Right Square Pyramid with Slant Height?

The volume (V) of a right square pyramid can be found by using the following steps:

**Step 1:**Identify the given dimensions of the right square pyramid.**Step 2:**Find the missing dimension using the formula s^{2}= h^{2}+ (b/2)^{2}where “s”, “h”, and “b” are slant height, the height of perpendicular, and base length of the right square pyramid, respectively.**Step 2:**Determine the volume using the formula V = 1/3 × b^{2}× h**Step 3:**Now, write the final answer with cubic units.

### What Happens to the Volume of a Right Square Pyramid If the Height of the Pyramid is Doubled?

The volume of the right square pyramid is doubled if the height of the pyramid is doubled as “h”. In the formula, V = 1/3 × b^{2} × h, substitute height = 2h, V = 1/3 × b^{2} × (2h) = 2 × (1/3 × b^{2} × h) which gives double the volume of the original volume of the right square pyramid.

### What Happens to the Volume of a Right Square Pyramid If the Base of the Pyramid is Halved?

The volume of the right square pyramid becomes one-fourth the original volume if the base of the pyramid is halved as “b/2”. In the formula, V = 1/3 × b^{2} × h, substitute base edge = b/2, V = 1/3 × (b/2)^{2} × h = (1/4) × (1/3 × b^{2} × h) which gives one fourth the volume of the original volume of the right square pyramid.

### How do you calculate the volume of a square pyramid?

Let’s say we have a small pyramid with a 6-inch × 6-inch square base and a height of 10 inches. To calculate its volume:

- First, find the area of its base,
**6 in × 6 in = 36 in²**. - Then, multiply this area by the pyramid’s height,
**36 in² × 10 in = 360 in³**. - Finally, divide this product by 3 to get the volume,
**360 in³ / 3 = 120 in³**.

### What is the volume of a regular square pyramid formula?

You can find the volume of a regular square pyramid using the formula **volume = base area × height / 3**. Take note that the **base area** equals the square of the pyramid’s base edge length. On the other hand, the **height** is the perpendicular distance between the pyramid’s base and the pyramid’s vertex.

### What is the volume of the Great Pyramid of Giza?

The Great Pyramid of Giza’s volume was around **2.6 million m³** (92 million ft³). Originally, including its smooth limestone casing, the Great Pyramid had a **height of 146.7 m** (481.4 ft) and a **base edge length of 230.6 m** (756.4 ft). Using these values and the square pyramid volume, we obtained its volume, V = 230.6² × 146.7 / 3 = 2600324.004 m³ ≈ 2.6 million m³.

## What is a Square Pyramid?

A three-dimensional geometric shape having a square base and four triangular faces/sides that meet at a single point (called vertex) is called a square pyramid. It is called a pentahedron, due to its five faces. A square pyramid has:

- An apex: top vertex or point of the pyramid
- A Base in square shape
- Four triangular faces

If all the triangular faces have equal edges, then this pyramid is said to be an **equilateral square pyramid**.

If the apex of the pyramid is right above the centre of its base, it forms a perpendicular with the base and such a square pyramid is known as the **right square pyramid**.

All the pyramids are categorized based on their bases, such as:

- Rectangular pyramid
- Triangular pyramid
- Square pyramid
- Pentagonal pyramid
- Hexagonal pyramid

## Properties of Square Pyramid

The properties of a square pyramid are:

- It has 5 Faces
- The 4 Side Faces are Triangles
- The Base is a Square
- It has 5 Vertices (corner points)
- It has 8 Edges

## Representation of Square Pyramid

A square pyramid is shown by the Wheel graph W_{5}. A wheel graph is one formed by connecting a single universal vertex to all vertices of a cycle.

## Types of Square Pyramid

The different types of square pyramids are :

- Equilateral Square Pyramid
- Right Square Pyramid
- Oblique Square Pyramid

### Equilateral Square Pyramid

If all the edges are of equal length, then the sides form equilateral triangles, and the pyramid is an equilateral square pyramid.

The Johnson square pyramid can be classified by a single edge-length parameter l. The height h, the surface area A, and the volume V of such a pyramid are:

- h = (1/√2)l
- A = (1+√3)l
^{2} - V = (√2/6)l
^{3}

### Right Square Pyramid

In a right square pyramid, all the lateral edges are of the same length, and the sides other than the base are congruent isosceles triangles.

A right square pyramid with base length l and height h has the following formula for surface area and volume:

### Oblique Square Pyramid

If the apex of the square pyramid is not aligned with the center of the square base, then it is called an oblique square pyramid.

## Square Pyramid Formula

There are formulas for square pyramid based on volume, surface area, height and base area. Find the formulas for:

- Volume of square pyramid
- Surface Area of square pyramid
- Lateral edge length
- Height of square pyramid

### Volume of Square Pyramid

Where a is the base edge length.

h is the height.

### Base Area of square pyramid

The base of a square pyramid is a square. Therefore, here we can find the base area by finding the square of its edge-length.

Base area = side × side = edge^{2}

### Surface Area of a Square Pyramid

## Net of a Square Pyramid

A net of a 3D shape is a pattern reached when the surface of a three-dimensional figure is laid out horizontally showing each face of the figure. A solid may have different nets.

The net of the square pyramid will give a flattened view, showing each and every face. When the net of the square pyramid is folded back, it will result in the original 3d shape.

Steps to determine whether net forms a solid are as follows:

- Solid and the net should have the same number of faces. Also, the shapes of the faces of the solid should match with the shapes of the corresponding faces in the net.
- Imagine how the net is to be folded to create the solid and assure that all the sides fit collectively properly.

Nets of the square pyramid are of use when we need to find its surface area.

In the above net of a square pyramid, we can see, the shape is formed by one square and four triangles, attached with the four sides of the square. Hence, the surface area of the square pyramid will be the sum of the surface area of all its five faces.

## Square Pyramid Solved Examples

**Example 1: **

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

**Solution:**

The formula for the volume of a square pyramid is given by:

V = ⅓ B h

As the base of the pyramid is a square, the base area is 10^{2} or 100 cm^{2}.

So, put 100 for B and 18 for h in the formula.

V=13×100×18=600

Therefore, the volume of the given square pyramid is 600 cm^{3}.

**Example 2:**

Find the lateral surface area of a regular pyramid that has a triangular base and each edge of the base is equal to 8 cms and the slant height is 5 cms.

**Solution: **

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

Perimeter = 3 × 8 = 24 cms

L.S.A.=12 × 24 × 5 = 60 cms^{2}

**Example 3: **

Find the total surface area of a regular pyramid having a square base where the length of each edge of the base is equal to 16 cms, the slant height of a side is 17 cms and the altitude is 15 cms.

**Solution: **

The perimeter of the base is 4s because it is a square.

P = 4 × 16 = 64 cms

The area of the square base = s^{2}

Base area = 16^{2} = 256 cms^{2}

Therefore, the total surface area of the pyramid,

T.S.A.=½ × 64 × 17 + 256 = 544 + 256 = 800 cms^{2}

## Frequently Asked Questions on Square Pyramid

### What is a square pyramid?

A square pyramid is a three-dimensional figure with a square base and four triangular faces joined at a vertex.

### Mention the few properties of a square pyramid?

A square pyramid has a square base.

It has four triangular faces and 5 vertices.

It has 8 edges.

### What is the formula for the volume of a square pyramid?

The formula for square pyramid is (⅓)(Base area of a square)(Height of the square pyramid) cubic units.

### What is the total surface area of a square pyramid?

The total surface area of a square pyramid is (½)Pl + B

Where “P” is the perimeter of the square pyramid

“l” is the slant height

“B” is the base area of a square.

### What is the right square pyramid?

If the apex of the square pyramid is perpendicularly above the centre of the square, then it is called the right square pyramid.

## Square Pyramid Formulas derived in terms of side length = a and height = h:

### Volume of a Square Pyramid

**V = (1/3)a**^{2}h

### Slant Height of a square pyramid

- By the pythagorean theorem we know that
- s
^{2}= r^{2}+ h^{2} - since r = a/2
- s
^{2}= (1/4)a^{2}+ h^{2}, and **s = √(h**^{2}+ (1/4)a^{2})- This is also the height of a triangle side

### Lateral Surface Area of a square pyramid (× 4 isosceles triangles)

- For the isosceles triangle Area = (1/2)Base x Height. Our base is side length a and for this calculation our height for the triangle is slant height s. With 4 sides we need to multiply by 4.
- L = 4 x (1/2)as = 2as = 2a√(h
^{2}+ (1/4)a^{2}) - Squaring the 2 to get it back inside the radical,
**L = a√(a**^{2}+ 4h^{2})

### Base Surface Area of a square pyramid (square)

**B = a**^{2}

### Total Surface Area of a square pyramid

- A = L + B = a
^{2}+ a√(a^{2}+ 4h^{2})) **A = a(a + √(a**^{2}+ 4h^{2}))

### Slope of Pyramid Side Face

- To find the pyramid slope of the side face we want to calculate the slope of the line s = slant height
- We know that the slope of a line is m = rise/run
- For the line s the rise is h = height of the pyramid
- r = a/2 and this is the run as it forms a right angle where r meets h at the center of the base
- m = h/(a/2) – in terms of h and a
- m = h/r – in terms of h and r

### Angle of Pyramid Side Face

- The angle of the pyramid side face is the angle formed between the side face and the base
- Let’s name theta θ = Side Face Angle and alpha α = the right angle (90°) formed by h and r
- Using the Law of Sines we can say that s/sin(α) = h/sin(θ)
- Solving for the unknown θ we have
- θ = sin
^{-1}[ (h × sin(α)) / s ] - We have another formula for θ in terms of the tangent from trigonometric ratios
- Since tan(θ) = side opposite θ / side adjacent θ we can say
- tan(θ) = h/r
- Solving for the unknown θ
- θ = tan
^{-1}(h/r) - θ in both calculations is in radians. Convert radians to degrees by multiplying θ by 180/π

** Example: **

Find the volume of a regular square pyramid with base sides 10

cm and altitude 18cm.

** Solution **

Draw a figure.

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