Surface Area of a Square Pyramid Formula

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Surface Area of a Square Pyramid Calculator

Are you in search of a surface area of a square pyramid calculator to estimate all types of areas of the Great pyramid?! If yes, you’re in the right place. You can calculate the total surface area, base area, lateral surface area, and face area of any square pyramid using our tool.

We also discuss how to find the surface area of a square pyramid using slant height and base length and how to calculate surface area using slant height and base perimeter. You can also find answers to some interesting numerical problems like the surface area of a pyramid of Giza and the amount of groundsheet required for any tent.

How do I calculate the surface area of a square pyramid?

To find the surface area of a square pyramid:

  1. Determine how many faces are there on a square pyramid: there are 4 triangular faces. Sum up their areas.
  2. Find the area of a square base.
  3. Add up 4 triangular faces area to 1 square base area to find the surface area of a square pyramid.

What is the formula for the surface area of a square pyramid?

The formula to calculate the surface area of a square pyramid is:

where:

  • SA is the surface area of the square pyramid;
  • a is the length of the base edge; and
  • h is the height of the square pyramid.

How do I calculate the base area of a square pyramid?

The base of a square pyramid is a square. Hence, the base area of the square pyramid of base edge a is a2.

How to find the lateral area of a square pyramid

First of all, let’s explain what the lateral area of a square pyramid is. The lateral surface area or lateral area of a square pyramid is the total area of its 4 triangular faces. We calculate the lateral area as:

How do I find the face area of a square pyramid?

The face area of a square pyramid is the area of one of its four triangular faces. We calculate the area of a triangular face, the face area (FA), using the formula:

How to find the surface area of a square pyramid using slant height

To find the surface area using the slant height, we use the formula:

  • SA = a2 + 2×a×l

🔎 Proof: The surface area of a square pyramid is the sum of the areas of its square base and four triangular faces:

  • SA = BA + (4 × FA)

The area of a triangle is half of the product of its base length (a) and height (l):

  • FA = a×l/2

Therefore, the area of four triangular faces or the lateral surface area of the square pyramid is:

  • 4 × FA = 2 × a × l

Thus, the lateral surface area (LSA) of the square pyramid of slant height l is

  • LSA = 2 × a × l

and the total surface area is

  • SA = a2 + 2 × a × l

How to use the surface area of a square pyramid calculator

❓ The Pyramid of Giza has a height of 480 feet. If the length of each side of the base is approximately 756 feet, what is its total surface area?

Okay, let’s see how to use this total surface area of a square pyramid calculator to solve the problem given above:

  1. Identify the measurements:

    • Height (h) = 480 ft
    • Base edge (a) = 756 ft
  2. Switch the units from cm to ft and cm2 to ft2 (or the desired units for the area) using the drop-down list near each variable in the square-based pyramid surface area calculator.

  3. Enter the values for height (480 ft) and base edge (756 ft).

  4. Ta-da! You got the results in no time! Our tool uses the surface area square pyramid formula to find:

    • Slant height (l) = 611 ft
    • Total surface area (SA) = 1,495,322 ft2
    • Base area (BA) = 571,536 ft2
    • Lateral surface area (LSA) = 923,786 ft2
    • Face area (FA) = 230,947 ft2

FAQ

How many faces are on a square pyramid?

A square pyramid has 5 faces: 4 equal triangles (side faces) and 1 square (base face).

How do I calculate the surface area of a square pyramid using its base perimeter?

The surface area of a square pyramid is half of the product of its base perimeter and its slant height:

  • SA = P × l / 2

where P is the base perimeter and l is the slant height.

How many square meters of groundsheet is required for a tent of base 5m and height 1.8m?

You need (5 m)2 = 25 m2 of groundsheet for the tent of base length 5 m irrespective of its height. The amount of groundsheet in square meters is the base area of the tent. The base area of a tent is BA = a2, where a is the base length.

Surface Area of a Square Pyramid

In this section, we will learn about the surface area of a square pyramid. A pyramid is a 3-D object whose all side faces are congruent triangles and whereas its base can be any polygon. One side of each of these triangles coincides with one side of the base polygon. A square pyramid is a pyramid whose base is a square. The pyramids are named according to the shape of their bases. Just like other three-dimensional shapes, a square pyramid also has two types of areas.

  • Total Surface Area (TSA)
  • Lateral Surface Area (LSA)

Let us learn about the surface area of a square pyramid along with the formula and a few solved examples here. You can find a few practice questions in the end.

What is the Surface Area 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.

Formula of Surface Area of a Square Pyramid

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

Lateral surface area of a square pyramid

The lateral surface area of a pyramid is the area occupied by its lateral surfaces or side faces. The formula for calculating the lateral surface area of a square pyramid using slant height is given as follows,

Lateral surface area (LSA) = ½ × perimeter × slant height

We know that,

The perimeter of a square = 4s

So, LSA = ½ × 4s × l = 2sl

Lateral surface area of a square pyramid (LSA) = 2sl square units

Where,

“s” is the base length of a square pyramid,

“l” is the slant height or height of each side face.

The slant height of the pyramid (l) = √(s2/4 + h2)

The formula for calculating the lateral surface area of a square pyramid using the height is given as follows,

Lateral surface area of a square pyramid = 2s√(s2/4 + h2) square units

where,

“s” is the base length of a square pyramid,

“h” is the height of the square pyramid.

Total surface area of a square pyramid

The total surface area of a square pyramid is the sum of the areas of its lateral faces and its base area. The formula for calculating the total  surface area of a square pyramid is given as follows,

Total surface area of a pyramid (TSA) = Lateral surface area of the pyramid (LSA) + Base area

The lateral surface area of the square pyramid (LSA)= 2sl square units

Base area = s2 square units

So, TSA = 2sl + s2

Total surface area of a square pyramid (TSA) = 2sl + s2  square units

where,

 “s” is the base length of a square pyramid,

“l” is the slant height or height of each side face.

Slant height of the pyramid (l) = √(s2/4 + h2)

The formula for calculating the lateral surface area of a square pyramid using the height is given as follows,

Total surface area of a square pyramid (TSA) = 2s√(s2/4 + h2) + s2  square units

where,

“s” is the base length of a square pyramid,

“h” is the height of the square pyramid.

How to Calculate Surface Area 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.

Answer: The surface area of the given square pyramid = 1095.96 square units.

FAQs on Surface Area of a Square Pyramid

What Is the Surface Area of the Square Pyramid?

The surface area of a square pyramid is the sum of the areas of all its 4 triangular side faces with the base area of the square pyramid. If a, h, and l are the base length, the height of the pyramid, and slant height respectively, then the surface area of

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

To find the lateral area of a square pyramid, find the area of one side face (triangle) and multiply it by 4. If a and l are the base length and the slant height of a square pyramid, then the lateral area of the square pyramid = 4 (½ × a × l) = 2al.

If h is the height of the pyramid, then the lateral area = 2a

What Is the Area of One of the Triangular Faces of a Square Pyramid?

If a and l are the base length and the slant height of a square pyramid, then the area of one of the 4 triangular side faces is, ½ × a × l.

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

The lateral area of a square pyramid is the sum of the areas of the side faces only, whereas the surface area is the lateral area + area of the base. The lateral area of a square pyramid = 2al (or) 2a

To get the total surface area, we need to add the area of the base (which is a2) to each of these formulas. The total surface area of a

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?

We know, slant height of a square pyramid is given in terms of

where,

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

What Is the Base Area of a Square Pyramid?

The base of a square pyramid is square-shaped. Thus, the base area of square pyramid can be calculated using the formula, Base Area of Pyramid = a2, where, a is the length of the base of square pyramid.

How Many Bases Does a Square Pyramid Have?

A square pyramid is a pyramid with a square-shaped base. A square pyramid thus has only one base.

Which Two Shapes Make up a Square Pyramid?

The base of a square pyramid is a square and its side faces are triangles. So the two shapes that make up a square pyramid are square and triangle.

Solved Examples based on Square Pyramid Formulas

Example 1: Determine the total surface area of a square pyramid if the base’s side length is 15 cm and the pyramid’s slant height is 21 cm.

Solution:

Given,

The side of the square base (s) = 15 cm

Slant height, (l) = 21 cm

The perimeter of the square base (P) = 4s = 4(15) = 60 cm

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

LSA = (½ ) × (60) × 21 = 630 sq. cm

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

= s2 + LSA

= (15)2 + 630

= 225 + 630 = 855 sq. cm

Therefore, the total surface area of the given pyramid is 855 sq. cm.

Example 2: Determine the lateral surface area of a square pyramid if the side length of the base is 18 inches and the pyramid’s slant height is 22 inches.

Solution:

Given,

The side of the square base (s) = 18 inches

Slant height, (l) = 22 inches

The perimeter of the square base (P) = 4s = 4(18) = 72 inches

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

LSA = (½ ) × (72) × 22 = 792 sq. in.

Therefore, the lateral surface area of the given pyramid is 792 sq. in.

Example 3: What is the slant height of the square pyramid if its lateral surface area is 200 sq. in. and the side length of the base is 10 inches?

Solution:

Given data,

Length of the side of the base of a square pyramid (s) = 10 inches

The lateral surface area of a square pyramid = 200 sq. in

Slant height (l) = ?

We know that,

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

The perimeter of the square base (P) = 4s = 4(10) = 40 inches

⇒ 200 = ½ × 40 × l

⇒ l = 10 in

Hence, the slant height of the square pyramid is 10 inches.

Example 4: Calculate the side length of the base of the square pyramid if its lateral surface area is 480 sq. cm and the slant height is 24 cm.

Solution:

Given data,

The lateral surface area of a square pyramid = 480 sq. cm

Slant height (l) = 24 cm

Let the length of the side of the base of a square pyramid be “s”.

We know that,

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

The perimeter of the square base (P) = 4s

⇒ 480 = ½ × 4s × 24

⇒ s = 10 cm

Hence, the side length of the base of the square pyramid is 10 cm.

Example 5: Determine the total surface area of a square pyramid if the base’s side length is 14 cm and the pyramid’s height is 24 cm.

Solution:

Given,

The side of the square base (s) = 14 cm

The height of the square pyramid (h) = 24 cm

The slant height of the pyramid (l) = √[(s/2)2 + h2]

l = √[(14/2)2 + 242)] = √(49+576)

= √625 = 25 cm

The perimeter of the square base (P) = 4s = 4(14) = 56 cm

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

LSA = (½ ) × (56) × 25 = 700‬ sq. cm

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

= s2 + LSA

= (14)2 + 700‬

= 196 + 700‬ = 896 sq. cm

Therefore, the total surface area of the given pyramid is 896 sq. cm.

Example 6: Determine the surface area of a square pyramid if the base’s side length is 10 cm and the pyramid’s slant height is 15 cm.

Solution:

Given,

The side of the square base (s) = 10 cm

Slant height (l) = 15 cm

We know that,

The total surface area of a square pyramid (TSA) = 2sl + s2  square units

= 2 × 10 × 15 + (10)2

= 300 + 100 = 400 sq. cm

Therefore, the surface area of the given pyramid is 400 sq. cm.

FAQs on Square Pyramid Formula

Question 1: What is the area of the base of a Square Pyramid?

Answer:

Base of a square pyramid is shaped as square. Thus, the area of the base of square pyramid can be calculated using the formula, for area of square.

Area of base Square Pyramid = a2,

where,

a is the length of the base of the square pyramid.

Question 2: How many bases does a Square Pyramid have?

Answer:

A square pyramid is a pyramid with only one base in shape of a square. So, a square pyramid has only one base.

Question 3: Which two shapes make up a Square Pyramid?

Answer:

The base of a square pyramid is a square and all its side faces are triangles. So there are two shapes that make up a square pyramid which are square and triangle.

Question 4: What is the area of one of the triangular faces of a Square Pyramid?

Answer:

Suppose, a is the length of base and l is the slant height of a square pyramid, then the area of any one of the four triangular side faces is, 

Area = ½ × a × l.

Here is how to derive the surface area of a square pyramid.

Surface area of the square pyramid = area of the base + area of 4 triangles.

The area of the square base is s2

The area of one triangle is (s × l)/2

Since there are 4 triangles, the area is 4 × (s × l)/2 = 2 × s × l

Therefore, the surface area, call it SA is SA = s2  +  2 × s × l

A couple of examples showing how to find the surface area of a square pyramid.

Example #1:

Find the surface area of a square pyramid with a base length of 5 cm, and a slant height of 10 cm.

SA = s2  +   2 × s × l

SA = 52   +   2 × 5 × 10

SA = 25   +   10 × 10

SA = 25  +   100

SA = 125 cm2
Example #2:

Find the surface area with a base length of 3 cm, and a slant height of 2 cm.

SA = s2  +  2 × s × l

SA = 32  +   2 × 3 × 2

SA = 9  +   6 × 2

SA = 9  +   12

SA = 21 cm2
Example #3:

Find the surface area with a base length of 1/2 cm, and a slant height of 1/4 cm.

SA = s2  +   2 × s × l

SA = (1/2)2  +   2 × 1/2 × 1/4

SA = (1/2)×(1/2)  +   2 × 1/2 × 1/4

SA = 1/4  +  1 × 1/4

SA = 1/4 + 1/4

SA = 2/4

SA = 1/2 cm2

Question: What is the surface area of the square pyramid which has the base length of 8 cm and side 5 cm?

Solution:

Given,
Base length = 8 cm
Side length = 5 cm

Using the formula:

General Formula for Surface Area of Pyramid

The general formula for the surface area of the pyramid is as follows:

The lateral surface area of the regular pyramid formula is given by:

Lateral Surface Area of Regular Pyramid = (½) Pl Square units

Similarly, the total surface area of the regular pyramid formula is given by:

Total Surface Area of Regular Pyramid = (½)Pl + B Square units

Where “l” is the slant height of a pyramid

“P” is the base perimeter of a pyramid

“B” represents the base area of a pyramid

Surface Area of Pyramid Formulas

Generally, if we are asked to find the surface area of the pyramid without any specifications, it represents the total surface area. Now, let us discuss the surface area of pyramid formulas with different bases.

Triangular Pyramid

In the triangular pyramid, the base of the pyramid is a triangle.

Surface Area of Triangular Pyramid = 1⁄2(a × b) + 3⁄2(b × s) square units

Where, b = side length, a = height, s = slant height.

Square Pyramid

In the Square pyramid, the base of the pyramid is a square.

Surface Area of Square Pyramid = 2bs + b2 Square units

Where, b = side length, s = slant height.

Pentagonal Pyramid

In the pentagonal pyramid, the base of the pyramid is a pentagon.

Surface Area of Pentagonal Pyramid = 5⁄2(a × b) + 5⁄2(b × s) Square units

Where, b = side length, s = slant height, a = apothem length.

Hexagonal Pyramid

In the hexagonal pyramid, the base of the pyramid is a hexagon.

Surface Area of Hexagonal Pyramid =3(a × b) + 3(b × s) Square units

Where, b = side length , s = slant height, a= apothem length.

Surface Area of Pyramid Examples

Go through the solved examples on the surface area of the pyramid:

Example 1:

Determine the surface area of the square pyramid, given that side length = 4 cm and slant height = 8 cm.

Solution:

Given: Side length, b = 4 cm

Slant height, s = 8 cm

We know that the surface area of square pyramid = 2bs + b2 Square units

Now, substitute the known values in the formula

S.A = 2(4)(8) + (4)2

S.A = 64+16 = 80

Hence, the surface area of the square pyramid is 80 cm2.

Example 2:

Compute the base area of the pentagonal pyramid, given that side length = 6.4 m and apothem = 16 m.

Solution:

Given: Side length, b = 6.4 m

Apothem = 16 m

We know that the base of a pentagonal pyramid is a pentagon. Hence, base area = 5⁄2(a × b) square units.

Base area = (5/2)(16×6.4)

Base area = 5(8×6.4) = 5(51.2) = 256

Thus, the base area of a pentagonal pyramid is 256 m2.

Practice Questions

Solve the following problems:

  1. Determine the surface area of the triangular pyramid, given that side length = 2 cm, height = 4 cm and slant height = 5 cm.
  2. Compute the base area of the square pyramid whose side length is equal to 7 cm.
  3. Find the surface area of the square pyramid given that side length = 5 cm and slant height = 7 cm.

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Frequently Asked Questions on Surface Area of Pyramid

What is the surface area of a pyramid?

The surface area of the pyramid is the total area occupied by the surface of the pyramid.

What is the general formula for the lateral surface area of a regular pyramid?

If “P” is the base perimeter and “l” is the slant height, then the lateral surface area of the regular pyramid formula is given by (½) Pl Square units.

What is the general formula for the total surface area of a regular pyramid?

If “P” is the base perimeter, “l” is the slant height, and “B” is the base area, then the total surface area of the regular pyramid formula is given by (½)Pl + B Square units.

Surface Area Of A Pyramid

A pyramid is a solid with a polygonal base and several triangular lateral faces. The lateral faces meet at a common vertex. The number of lateral faces depends on the number of sides of the base. The height of the pyramid is the perpendicular distance from the base to the vertex.

A regular pyramid has a base that is a regular polygon and a vertex that is above the center of the polygon. A pyramid is named after the shape of its base. A rectangular pyramid has a rectangle base. A triangular pyramid has a triangle base.

We can find the surface area of any pyramid by adding up the areas of its lateral faces and its base.

Surface area of any pyramid = area of base + area of each of the lateral faces

If the pyramid is a regular pyramid, we can use the formula for the surface area of a regular pyramid.
Surface area of regular pyramid = area of base + 1/2 ps
where p is the perimeter of the base and s is the slant height.

If the pyramid is a square pyramid, we can use the formula for the surface area of a square pyramid.

Surface area of square pyramid = b2 + 2bs
where b is the length of the base and s is the slant height.

Example:
Calculate the surface area of the following pyramid.

Solution:
Sketch a net of the above pyramid to visualize the surfaces.

Since the given pyramid is a square pyramid, we can use any of the above formulas.

Using the formula for the surface area of any pyramid:

Area of base = 6 × 6 = 36 cm2

Area of the four triangles = 1/2 × 6 × 12 × 4 = 144 cm2

Total surface area = 36 + 144 = 180 cm2

Using the formula for a regular pyramid

Surface area of regular pyramid = area of base + 1/2 ps

= 6 × 6 + 1/2 × 6 × 4 × 12 = 180 cm2

Using the formula for a square pyramid

Surface area of square pyramid = b2 + 2bs

= 6 × 6 + 2 × 6 × 12 = 180 cm2

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