Regular Square Pyramid Formula

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A regular square pyramid is a three-dimensional geometrical figure that has square bases with four triangular sides. All four triangular faces meet at a point called the vertex. This figure has five vertices and eight edges.

A regular square pyramid is a type of polyhedron that is classified according to the shape of its polygonal base.
Triangular pyramids, Rectangular pyramids, Square pyramids and Pentagonal pyramids are common shapes of polygonal bases.
Since it has five faces, it is also known as pentahedron.
Volume and Surface area are two major formulas for this three dimensional figure.
The Great Pyramid of Giza is the most common example of a Regular Square Pyramid.

Volume of a square pyramid: 1/3.a².h

Surface area of the square pyramid: a(a+√a²+4h²)

Table of Content

Square Pyramid
Types of Regular Square Pyramid
Regular Square Pyramid Formula
Things to Remember
Sample Questions

Key Terms: Regular Square Pyramid Formula, Square Pyramid, Polyhedron, Square, Pentahedron, Volume of Square Pyramid, Surface Area of Square Pyramid, Polygon, Triangle, Equilateral Square Pyramid, Right Square Pyramid

Square Pyramid

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Square Pyramid is a three-dimensional polyhedron geometrical figure formed by connecting a polygonal base and a point called the apex. The apex and the edge of each base form a triangle called a lateral side.

A pyramid has n side base, n+1 vertices, n+1 faces, and 2n edges.
It is a pyramid that has a square base and four triangular lateral faces.
The figure is differentiated on the basis of the lengths of their edges and the position of the apex.
The Louvre, a museum in Paris, is another architectural example of the Square Pyramid.

Regular Square Pyramid Formula consists of the following components which are as follows:

An apex which is the top vertex or point of the pyramid
Next is the base of the pyramid in a square shape
Lastly, there are four triangular faces in the figure.

Pyramids are named according to their bases.

If the base is a triangle, it is a triangular pyramid.
If the base square, it is a square pyramid.
If the base is a pentagon, it is a pentagonal pyramid.

Example of Square Pyramid

Example: Assume that the height (h) and the length of the base edge (a) are 8 units and 3 units, respectively. Then, the volume of the square pyramid is:

Volume = 1/3 x 3² x 8 = 1/3 x 3 x 3 x 8 = 24 cubic units

Pyramid

Pyramid

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Types of Regular Square Pyramid

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There are three different types of regular square pyramids:

Equilateral Square Pyramid

If the edges of all the triangular faces are of equal length, then the sides of the figure form an equilateral triangle, and the pyramid formed is called an equilateral square pyramid.

Height (h) = ( 1/ √2 )

Area (a) = (1+ √3)²

Volume(V) = (√2/6)³

Equilateral Square Pyramid

Right Square Pyramid

If the top of the square pyramid is right above the center of the base forming a perpendicular with the base also all the lateral edges are of the same length and the sides other than the base are congruent isosceles triangles it will call a right square pyramid.

Volume (V) = \(V = \frac{1}{3}.b^{2}.h\)

Example of Right Square Pyramid

Example: Assume that the height (h) and the length of the base edge (a) are 8 units and 15 units, respectively. Then, the volume of the square pyramid is:

Volume = 1/3 x 15² x 8 = 1/3 x 15 x 15 x 8 = 600 cubic unit

Right Square Pyramid

Oblique Square Pyramid

If the top of the square pyramid is not aligned right above the center of the base of a pyramid, then it is called an oblique square pyramid.

Oblique Square Pyramid

Regular Square Pyramid Formula

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Regular Square Pyramid Formula consists of formulas for volume, height, base area, and surface area. It also helps determine the total surface area (TSA) and lateral surface area (LSA) of the square pyramid.

Volume of a Square Pyramid

The volume of square pyramid is given by the formula:

V= [(1/3)a²h]

where, a is the length of the base and h is the perpendicular height.

Example of Volume of a Square Pyramid

Example: Assume that the height (h) and the length of the base edge (a) are 12 units and 3 units, respectively. Then, the volume of the square pyramid is:

Volume = 1/3 x 3² x 8 = 1/3 x 3 x 3 x 12 = 36 cubic unit

Surface Area of Square Pyramid

The surface area of a square pyramid is given by the formula:

Total Surface area of the square pyramid: a(a+√a²+4h²)

Total Surface area of the square pyramid = 2al + a² square units

where, a is the length of the base, l is the slant height and h is the perpendicular height.

Example of Surface Area of Square Pyramid

Example: Assume that the height h and the length of the base edge a are 2 units and 4 units, respectively.

Then, the surface area of the square pyramid is:

Surface area = (4)² + 2 x 4 √[(4²/4) + 2²]

= 16 + 8 √[(4) + 4]

= 16 + 8√(8)

= 38.56 square units.

Lateral Surface Area of Square Pyramid

The lateral surface area of square pyramid are as follows:

Lateral surface area of the regular square pyramid (LSA)= 2al

Lateral surface area of the square pyramid (LSA)= 2a√(a²/4 + h²)

where, a is the length of the base, l is the slant height and h is the perpendicular height.

Example of Lateral Surface Area of Square Pyramid

Example: Assume that the slant height h and the length of the base edge a are 12 units and 4 units, respectively.

Then, the lateral surface area of the square pyramid is:

Lateral Surface area = 2al

= 2 x 12 x 4

= 96 square units.

Base Area of a Square Pyramid

The base area of a square pyramid is as follows:

Base Area of a Square Pyramid = a²

where, a is the side of the base.

Example of Base Area of a Square Pyramid

Example: Assume that the base edge of a square pyramid is given as 8 units. Then, the base area of the square pyramid is: BA = 8 × 8 = 64 square units

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Things to Remember

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The regular Square Pyramid Formula consists of five vertices and eight edges.
The four sides of the three dimensional figure are a triangle.
The apex is the common point where all triangular faces meet.
Equilateral Square Pyramid, Right Square Pyramid and Oblique Square Pyramid are three types of square pyramids.
The regular Square Pyramid Formula is mainly used to find the total surface area and volume of the figure.

Sample Questions

Ques: Find the volume of a regular square pyramid of base length 6 cm and height 10cm? (2 marks)

Ans: As the base length ‘a’ = 6cm

And the Height is given ‘h’= 10cm

Then according the formula of a regular square pyramid is

V = (1/3)a²h

V = (1/3)x6×6×10

V = 120 cm³

Ques: Find the surface area of the regular square pyramid of base area 4cm and Slant height 5cm? (2 marks)

Ans: Slant height of the pyramid given = 5cm

Base area ‘a’ = 4cm

Then according to the Square Pyramid formula = a² + 2al

= 4×4+2×4×5

= 56 cm²

Ques: A close square pyramid-shaped aquarium has a base edge of 10cm and the slant height is 20 cm. What is the total surface area of the aquarium? (2 marks)
Ans: Given, the slant height of the square pyramid is 20cm

And the base edge is 10cm

Then according to the total surface area of the aquarium is a² + 2al

The Surface area of the aquarium = 10×10 + 2×10×20

Surface area = 500cm²

The total surface of the aquarium is 500cm²

Ques: A vessel in the form of an inverted regular square pyramid has to be filled with water. The altitude of the vessel is 11cm and the base edge is 9 cm. What is the volume of the water that can be filled in the vessel? (2 marks)

Ans: The height of the vessel is given as 11 cm

Base edge is given as 9cm

Then according to the formula of the volume of the square pyramid

V = (1/3)a²h

V= (1/3)×9×9×11

V= 297cm²

The water that can be filled in the vessel is 297cm³

Ques: The base area of the square pyramid is 12cm and its volume is given as 720cm². Calculate the height of the square pyramid? (2 marks)

Ans: Given,

Base area of the square pyramid = 12cm

And the volume is = 720cm²

Then according to the formula, the height will be

V = (1/3) a² h

720 = (1/3)×12×12×h

720× 3/12×12 = h

Height will be 15 cm

Ques: Jiya Constructed a square pyramid whose perimeter of the base is 20 cm and the height measure is 15 cm. Manika made a square pyramid with a base edge of 10cm and a height of 18cm. Find the volume of the pyramids and compare both these measurements? (4 marks)

Ans: Jiya’s square pyramid dimensions are,

The Base edge is 20cm (A1) and the Height is 15cm (H1).

Volume = (1/3)a² h

The Volume of Jiya’s Square pyramid = (1/3)× 20 × 20 × 15

Volume = 2000 cm³

Manika’s square pyramid dimensions are,

The Base edge is 10cm (A2) and the Height is 18cm (H2).

Volume = (1/3)a² h

Volume of Manika’s Square pyramid = (1/3)× 10 × 10 × 18

Volume = 600 cm³

Volume of Jiya’s Square pyramid = Volume of Manika’s Square pyramid

Now, Comparing the measurements,

Dimension of Jiya’s = Dimension of manika’s

A1 = 20 A2 = 10

H1 = 15 H2 = 18

A1 / A2 = 20/10

A1 = 2A2

Jiya’s square pyramid base edge is 2 times the base edge of Manika’s square pyramid

H1 / H2 = 15/8

H1 = 1.8H2

Jiya’s square pyramid height is 1.8 times the height of Manika’s square pyramid

Ques: The base edge of the regular square pyramid is given as 8cm and the vertical height is 3cm. Calculate its slant height and lateral edge of the square pyramid? (4 marks)

Ans: Given the base edge perimeter = 8cm

And height ‘h’= 3cm

Now,

Slant Height = Height of a lateral face

Pyramid

Lateral edge = Edges between the lateral face

In XOY

XY² + OY² = XY²

3² + 4² = XY²

XY = √16+ 9 = 5cm

Slant Height = 5cm

Lateral Edge =

In XOC,

XO² + OC² = XC²

5² + 4² = XC²

XC = √25+16

XC = √41 cm

Hence the Slant Height and Lateral Edge = 5 cm and √41 cm

Ques: Calculate the surface area of a square pyramid with base edges 10 cm and lateral edges 13 cm? (3 marks)

Ans: Given,

In right OAB,

OA² + AB² = OB²

H² + 5² = 13²

H = √169 - 25

H = √144

H = 12cm

Area of square base ‘a²’ = 10×10 = 100cm²

Area of four triangles = 4 × ½ bh

= 2 × 10× 12

= 240cm²

TSA of square pyramid = 100 + 240

= 340 cm²

The surface area of the square pyramid is 340cm²

Ques: The lateral sides of a square pyramid are given as equilateral triangles whose base perimeter is 30 cm. Calculate its surface area? (4 marks)

Ans: Given base perimeter = 30 cm

Now,

The surface area of the square pyramid = area of the base square + area of four equilateral triangles

Area of base square

Side = 30 cm

Area = (30)² = 900

Area of four equilateral triangles,

Side = 30cm

Area = √3/4 a²

= √3/4 (30)²

Area of 1 equilateral triangles = 225√3 cm²

Area of 4 equilateral triangles = 4(225√3)cm²

= 900√3 cm²

Surface area of the square pyramid = area of the base square+ area of 4 equilateral triangle

= 900cm² + 900√3 cm²

= 900(1+√3)cm²

Hence the surface area of the square pyramid is 900(1+√3)cm²

Ques: All the edges of the square pyramid are 20cm. What is its volume? (4 marks)

Ans: Given all the edges are 18cm

Now,

In ABC, ∠B = 90°

and AC = hypotenuse

AC² = AB² + BC² …………….(Pythagoras Theorem)

= 18² +18²

= √324+324

= √2×324

= 18√2 cm

AC = 18√2 cm

AE = ½ of AC

= 18√2/2

= 9√2 cm

AE² + OE² = OA² ……………..(Pythagoras Theorem)

(9√2)² + h² = 18²

h = √(18)² - [9√2]²

= √324 - 81×2

= √162

= 9√2

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

= (1/3) x (18)² × 9√2

= 972√2 cm³

Hence the Volume of the square pyramid is 972√2 cm³.

Ques: Dany is constructing a closed square pyramid-shaped aquarium in his backyard. The base edge of the square pyramid is 20 inches and the slant height is 10 inches. Help Dany determine the total surface area of the aquarium? (2 marks)

Ans: Given, the slant height l of the square pyramid is 10 inches, the base edge a of the square pyramid is 10 inches.

Thus, The total surface area of the aquarium is given by: a²+ 2al

= 20²+ 2 × 20 × 10

= 400 + 400 = 800 in²

Ques: Sophia has a vessel in the form of an inverted regular square pyramid that has to be filled with water. The altitude of the vessel is 20 inches and the base edge is 21 inches. What is the volume of water Sophia can fill in the vessel? (2 marks)

Ans: Given, the height h of the vessel is 20 inches and the base edge of the vessel is 21 inches.

Thus, the volume of the vessel is given by: [(1/3)a²h]

= [(1/3) × (21)²× (20)]

= 2940 inch³

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Regular Square Pyramid Formula

Square Pyramid

Example of Square Pyramid

Types of Regular Square Pyramid

Equilateral Square Pyramid

Right Square Pyramid

Example of Right Square Pyramid

Oblique Square Pyramid

Regular Square Pyramid Formula

Volume of a Square Pyramid

Example of Volume of a Square Pyramid

Surface Area of Square Pyramid

Example of Surface Area of Square Pyramid

Lateral Surface Area of Square Pyramid

Example of Lateral Surface Area of Square Pyramid

Base Area of a Square Pyramid

Example of Base Area of a Square Pyramid

Things to Remember

Sample Questions

CBSE X Related Questions

1.
The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

2.
An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

3.
Use graphical method to solve the system of linear equations : $x = -3$ and $5x - 2y = -5$.

4.
Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent.
Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

5.
Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

6.
A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.

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Regular Square Pyramid Formula

Square Pyramid

Example of Square Pyramid

Types of Regular Square Pyramid

Equilateral Square Pyramid

Right Square Pyramid

Example of Right Square Pyramid

Oblique Square Pyramid

Regular Square Pyramid Formula

Volume of a Square Pyramid

Example of Volume of a Square Pyramid

Surface Area of Square Pyramid

Example of Surface Area of Square Pyramid

Lateral Surface Area of Square Pyramid

Example of Lateral Surface Area of Square Pyramid

Base Area of a Square Pyramid

Example of Base Area of a Square Pyramid

Things to Remember

Sample Questions

CBSE X Related Questions

1.The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

2.An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

3.Use graphical method to solve the system of linear equations : $x = -3$ and $5x - 2y = -5$.

4.Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent. Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

5.Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

6.A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.

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SUBSCRIBE TO OUR NEWS LETTER

1.
The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

2.
An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

3.
Use graphical method to solve the system of linear equations : $x = -3$ and $5x - 2y = -5$.

4.
Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent.
Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

5.
Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

6.
A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.