✅ ​Regular Square Pyramid Formula ⭐️⭐️⭐️⭐️⭐️

Pyramid Formula

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.

3] Volume of the Pyramid

Volume of a 3 -dimensional solid is the amount of space occupies by the solid.

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

b	base length
s	slant height
h	height

2] Triangular Pyramid

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

a	apothem length
b	base length
s	slant height
h	height

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

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 W₅. 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)l²
• V = (√2/6)l³

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 = edge²

Surface Area of a Square Pyramid

Where a is the base edge length,

h is the height of square pyramid.

The general formula for the total surface area of a regular pyramid is given by:

T.S.A.= ½ Pl + B

where,

P is the perimeter of the base

l the slant height and

B the area of the base

Lateral Edge Length

Where h is the height

And l is the base length

Height of square pyramid

Where h is the height

And l is the base length

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:

1. 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.
2. 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.

n 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 10² or 100  cm².

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  cm³.

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  cms²

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 = s²

Base area = 16² = 256  cms²

Therefore, the total surface area of the pyramid,

T.S.A.=½ × 64 × 17 + 256  = 544 + 256  = 800  cms²

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 Shape

h = height
s = slant height
a = side length
P = perimeter of base
e = lateral edge length
r = a/2
V = volume
L = lateral surface area
B = base surface area
A = total surface area
m = h/r = rise/run = side face slope
θ = tan^-1(h/r) × 180/π = side face angle

Calculator Use

This online calculator will calculate the various properties of a square pyramid given 2 known variables. The square pyramid is a special case of a pyramid where the base is square. It is a regular pyramid since it has a square base which is a regular polygon. This is also a right square pyramid where “right” refers to the fact that the apex lies directly above the centroid of the base. In other words the point at the top of the pyramid is directly above the center point of the square base.

Units: Note that units are shown for convenience but do not affect the calculations. The units are in place to give an indication of the order of the results such as ft, ft² or ft³. For example, if you are starting with mm and you know r and h in mm, your calculations will result with s in mm, V in mm³, L in mm², B in mm² and A in mm².

NAN: means not a number. This will show as a result if you are using

values that just do not make sense as reasonable values for a pyramid.

Below are the standard formulas for a pyramid. Calculations are based on algebraic manipulation of these standard formulas.

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

Volume of a Square Pyramid

• V = (1/3)a²h

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.

Slant Height of a square pyramid

• By the pythagorean theorem we know that
• s² = r² + h²
• since r = a/2
• s² = (1/4)a² + h², and
• s = √(h² + (1/4)a²)
• 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√(h² + (1/4)a²)
• Squaring the 2 to get it back inside the radical,
• L = a√(a² + 4h²)

Base Surface Area of a square pyramid

• B = a²

Total Surface Area of a square pyramid

• A = L + B = a² + a√(a² + 4h²))
• A = a(a + √(a² + 4h²))

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

Square Pyramid Calculations:

Other formulas for calculations are derived from the formulas above.

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 m³, cm³, in³, 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 × b² × 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

The word “surface” means ” the exterior or outside part of an object or body”. So, the total surface area of a square pyramid is the sum of the areas of its lateral faces and its base. We know that a square pyramid has:

• a base which is a square.
• 4 side faces, each of which is a triangle.

All these triangles are isosceles and congruent, each of which has a side that coincides with a side of the base (square).

So, the surface area of a square pyramid is the sum of the areas of four of its triangular side faces and the base area which is square.

Let us consider a square pyramid whose base’s length (square’s side length) is ‘a’ and the height of each side face (triangle) is ‘l’ (this is also known as the slant height). i.e., the base and height of each of the 4 triangular faces are ‘a’ and ‘l’ respectively. So the base area of the pyramid which is a square is a × a = a² and the area of each such triangular face is 1/2 × a × l. So the sum of areas of all 4 triangular faces is 4 ( ½ al) = 2 al. Let us now understand the formulas to calculate the lateral and total surface area of a square pyramid using height and slant height.

Total Surface Area of Square Pyramid Using Slant Height

The total surface area of a square pyramid is the total area covered by the four triangular faces and a square base. The total surface area of a square pyramid using slant height can be given by the formula,
Surface area of a square pyramid = a² + 2al
where,

• a = base length of square pyramid
• l = slant height or height of each side face

Total Surface Area of a Square Pyramid Using Height

Let us assume that the height of the pyramid (altitude) be ‘h’. Then by applying Pythagoras theorem (you can refer to the below figure),

Lateral Surface Area of a Square Pyramid

The lateral surface area of a square pyramid is the area covered by the four triangular faces. The lateral surface area of a square pyramid using slant height can be given by the formula,
Lateral surface area of a square pyramid = 2 al
or,

where,

• a = base length of square pyramid
• l = slant height or height of each side face
• h = height of square pyramid

The surface area of a square pyramid can be calculated by representing the 3D figure into a 2D net. After expanding the 3D figure into a 2D net we will get one square and four triangles.

The following steps are used to calculate the surface area of a square pyramid :

• To find the area of the square base: a², ‘a’ is the base length.
• To find the area of the four triangular faces: The area of the four triangular side faces can be given as: 2al, ‘l’ is the slant height. If slant height is not given, we can calculate it using height, ‘h’ and base

• Add all the areas together for the total surface area of a square pyramid, while the area of 4 triangular faces gives the lateral area of the square pyramid.
• Thus, the surface area of a square pyramid is a² + 2al and lateral surface area as 2al in squared units.

Now, that we have seen the formula and method to calculate the surface area of a square pyramid, let us have a look at a few solved examples to understand it better.

Examples on Surface Area of a Square Pyramid

Example 2: The height of a square pyramid is 25 units and the base area of a square pyramid is 256 square units. Find its surface area.

Solution

Let the side of the base (square) be ‘a’ units.

Then it is given that a² = 256 ⇒ a = 16 units.

The height of the given square pyramid is h = 25 units.

How Do You Find the Lateral Area of a Square Pyramid?

How Do You Find the Lateral Area and Surface Area of a Square Pyramid?

where,

• a = Length of the base (square)
• l = Slant height
• h = Height of the pyramid

How To Calculate Surface Area of a Square Pyramid Without Slant Height?

where,

• a = Length of the base (square)
• l = Slant height
• h = Height of the pyramid

Practise Problems based on Regular Square Pyramid Formula

Problem 1: Calculate a square pyramid’s total surface area if the base’s side length is 20 inches and the pyramid’s slant height is 25 inches.

Solution:

Given,

The side of the square base (a) = 20 inches, and

 

Slant height, l = 25 inches

The perimeter of the square base (P) = 4a = 4(20) = 80 inches

The lateral surface area of a regular square pyramid = (½) Pl

LSA = (½ ) × (80) × 25 = 1000‬ sq. in

Now, the total surface area = Area of the base + LSA

= a² + LSA

= (20)² + 1000‬ 

= 400 + 1000 = 1400 sq. in

Hence, the total surface area of the given pyramid is 1400 sq. in.

Problem 2: Calculate the slant height of the regular square pyramid if its lateral surface area is 192 sq. cm and the side length of the base is 8 cm.

Solution:

Given data,

Length of the side of the base (a) = 8 cm

The lateral surface area of a regular square pyramid = 192 sq. cm

Slant height (l) = ?

We know that,

The lateral surface area of a regular square pyramid = (½) Pl

The perimeter of the square base (P) = 4a = 4(8) = 32 cm

⇒ 192 = ½ × 32 × l

⇒ l = 12 cm

Hence, the slant height of the square pyramid is 12 cm.

Problem 3: What is the volume of a regular square pyramid if the sides of a base are 10 cm each and the height of the pyramid is 15 cm?

Solution:

Given data,

Length of the side of the base (a)= 10 cm

Height of the pyramid (h) = 15 cm.

The volume of a regular square pyramid (V) = 1/3 × Area of square base × Height

Area of square base = a² = (10)² = 100 sq. cm

V = 1/3 × (100) ×15 = 500 cu. cm

Hence, the volume of the given square pyramid is 500 cu. cm.

Problem 4: Calculate the lateral surface area of a regular square pyramid if the side length of the base is 7 cm and the pyramid’s slant height is 12 cm.

Solution:

Given,

The side of the square base (a) = 7 cm

Slant height, l = 12 cm

The perimeter of the square base (P) = 4a = 4(7) = 28 cm

The lateral surface area of a regular square pyramid = (½) Pl

LSA = (½ ) × (28) × 12 = 168‬ sq. cm

Hence, the lateral surface area of the given pyramid is 168 sq. cm.

Problem 5: Calculate the height of the regular square pyramid if its volume is 720 cu. in. and the side length of the base is 12 inches.

Solution:

Given,

The side of the square base (a) = 12 cm

Volume = 720 cu. in

Height (H) =?

We know that,

The volume of a regular square pyramid (V) = 1/3 × Area of square base × Height

Area of square base = a² = (12)² = 144 sq. in

⇒ 720 = 1/3 × 144 × H

⇒ 48H = 720

⇒ H = 720/48 = 15 inches

Hence, the height of the square pyramid is 15 inches.

Problem 6: Calculate the volume of a regular square pyramid if the base’s side length is 8 inches and the pyramid’s height is 14 inches.

Solution:

Given data,

Length of the side of the base of a square pyramid = 8 inches

Height of the pyramid = 14 inches.

The volume of a regular square pyramid (V) = (1/3)a²h cubic units

V = (1/3) × (8)² ×14 

= (1/3) × 64 × 14

= 298.67 cu. in

Hence, the volume of the given square pyramid is 298.67 cu. in.

Problem 7: Find the surface area of a regular square pyramid if the base’s side length is 15 units and the pyramid’s slant height is 22 units.

Solution:

Given,

The side of the square base (a) = 15 units, and

Slant height, l = 22 units.

We know that,

The total surface area of a regular square pyramid (TSA) = 2al + a² square units

= 2 × 15 × 22 + (15)²

= 660 + 225= 885 sq. units

Hence, the total surface area of the given pyramid is 885 sq. units.

Frequently Asked Questions on Square Pyramid

Question 1: What is a square pyramid?

Answer:

Square pyramid is a 3-D figure with a square base and four triangular faces joined at a vertex.

Question 2: Mention the few properties of a square pyramid?

Answer:

Few properties of square pyramid are:

• Square Pyramid has the base of a square.
• It has five vertices and four triangular faces.
• Square Pyramid has 8 edges.

Question 3: What is the formula for the volume of a square pyramid?

Answer:

Formula for the volume of a square pyramid is 

Volume = (⅓)×(Base area of a square)×(Height of the square pyramid) cubic units.

Surface Area of a Pyramid

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

The total surface area of a regular pyramid is the sum of the areas of its lateral faces and its base.

The general formula for the lateral surface area of a regular pyramid is L.S.A.=¹/₂pl where p represents the perimeter of the base and l the slant height.

Example 1:

Find the lateral surface area of a regular pyramid with a triangular base if each edge of the base measures 8

inches and the slant height is 5 inches.

The general formula for the total surface area of a regular pyramid is T.S.A.=¹/₂pl+B where p represents the perimeter of the base, l the slant height and B the area of the base.

Example 2:

Find the total surface area of a regular pyramid with a square base if each edge of the base measures 16

inches, the slant height of a side is 17 inches and the altitude is 15 inches.

There is no formula for a surface area of a non-regular pyramid since slant height is not defined. To find the area, find the area of each face and the area of the base and add them.

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