Surface Area of a Pyramid Formula

The surface area of a pyramid is the total area occupied by all its faces, including the polygonal base and the triangular lateral faces that converge at a single point called the apex.

The diagram above shows the key parts of a pyramid—its apex, base, altitude (height), and slant height.

• The surface area of a pyramid is the total area covered by its base and all its triangular faces.
• The lateral surface area represents the area of all the triangular faces, and adding the base area gives the total surface area of the pyramid.

For a regular pyramid, it can be calculated using the following formulas:

• Lateral Surface Area (LSA) = (1/2) × P × s
• Total Surface Area (TSA) = (1/2) × P × s + B

Where:
P = perimeter of the base
s = slant height
B = base area

Proof

To derive the surface area formulas, consider a square pyramid with base side a and slant height l.

Base area (B) = a²
Perimeter of base (P) = 4a

Area of one triangular face = (1/2) × a × l

Since there are 4 triangular faces,
Total lateral surface area = 4 × (1/2 × a × l)
= (1/2) × (4a) × l
= (1/2) × P × l

So,
Lateral Surface Area (LSA) = (1/2) × P × l

Now, adding the base area,
Total Surface Area (TSA) = LSA + B
= (1/2) × P × l + B

This proves the formulas for the surface area of a pyramid.

Surface Area with Altitude

When the altitude (h) is given, the slant height (l) can be found using the Pythagorean theorem:

l² = h² + (a/2)²

Then, the surface area is calculated using:

TSA = (1/2) × P × l + B

Surface Area of a Triangular Pyramid

A triangular pyramid is a pyramid having a triangular base, where the triangular base can be equilateral, isosceles, or a scalene triangle. It has three lateral (triangular) faces and a triangular base. 

The total surface area of a pyramid (TSA) = Lateral surface area of the pyramid + Area of the base

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

So, TSA = ½ × perimeter × slant height + ½ × base × height

Total surface area (TSA) of a triangular pyramid = ½ ×  P ×  l + ½ bh

Where,

• P is perimeter of base,
• l is slant height of pyramid,
• b is base of base triangle, and
• h is height of pyramid.

Surface Area of a Square Pyramid

A square pyramid is a pyramid having a square base. It has four lateral (triangular) faces and a square base.

We know that,

The total surface area of a pyramid (TSA) = Lateral surface area of the pyramid + Area of the base

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

LSA = 4 × [½ × a × l] = 2al

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

So, TSA = 2al + a²

Total surface area of a square pyramid (TSA) = 2al + a²

Where,

• a is side of square base, and
• l is slant height of pyramid.

Surface Area of a Rectangular Pyramid

A rectangular pyramid is a pyramid having a rectangular base. It has four lateral (triangular) faces and a rectangular base.

We know that,

The total surface area of a pyramid (TSA) = Lateral surface area of the pyramid + Area of the base

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

The slant height of width face of the pyramid = √[h² + (w/2)²]

Lateral surface area of a rectangular pyramid = 2 × {½ × l ×√[h² + (l/2)²]} + 2 × {½ × w ×√[h² + (w/2)²]

So,

Total surface area of the rectangular pyramid = 2 × {½ × l ×√[h² + (l/2)²]} + 2 × {½ × w ×√[h² + (w/2)²] + l × w  

Where,

• l is length of rectangle base,
• w is breadth of rectangle base, and
• h is height of pyramid.

Surface Area of a Pentagonal Pyramid

A pentagonal pyramid is a pyramid having a pentagonal base. It has five lateral (triangular) faces and a pentagonal base.

We know that,

The total surface area of a pyramid (TSA) = Lateral surface area of the pyramid + Area of the base

Apothem length of the base = a

Side length of the base = s

Slant height of the pyramid = l

Area of the pentagonal base = 5⁄2 (a × s)

Now,

LSA = 5 × [½ × base × height] = 5/2 × s × l

Lateral surface area of the pentagonal pyramid = 5⁄2 (s × l)

Total surface area of the pentagonal pyramid = 5⁄2 (s × l) + 5⁄2 (a × s)

Where,

• l is  slant height of the pyramid,
• a apothem length of the base, and
• s side length of the base.

Surface Area of a Hexagonal Pyramid

A hexagonal pyramid is a pyramid having a hexagonal base. It has six lateral (triangular) faces and a hexagonal base.

We know that,

The total surface area of a pyramid (TSA) = Lateral surface area of the pyramid + Area of the base

Side length of the base = s

Slant height of the pyramid = l

Area of the hexagonal base = 3√3/2 * s²

Now, 

LSA = The sum of areas of the lateral surfaces (triangles) of the pyramid

⇒ LSA = 6 × [½ × base × height] = 3(s × l)

Lateral surface area of the hexagonal pyramid = 3(s × l)

Total surface area of the hexagonal pyramid = 3(s × l) + 3√3/2 (s)²

Where,

• s is the side length of the base, and
• l is the slant height of the pyramid.

Related Articles

• Surface Area of a Square Pyramid
• Total Surface Area of Rectangular Pyramid Formula
• Area of a Pentagonal Pyramid
• Surface Area of a Triangular Pyramid
• Surface Area Formulas for Different Geometrical Figures

Sample Problems

Problem 1: Determine the surface area of a square pyramid if the side length of the base is 16 inches and the slant height of the pyramid is 18 inches.

Solution:

Given,

The side of the square base (a) = 16 inches, and

Slant height, l = 18 inches

The perimeter of the square base (P) = 4a = 4(16) = 64 inches

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

LSA = (½ ) × (64) × 18 = 576 sq. in

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

= a² + LSA

= (16)² + 576 = 832 sq. in

Hence, the surface area of the given pyramid is 832 sq. in.

Problem 2: Find the total surface area of a triangular pyramid in which all faces are equilateral triangles of side 24 cm.

Solution:

Given:
Side of triangle (a) = 24 cm
Area of equilateral triangle = (√3/4) × a²

Calculation:
Area of one triangle = (√3/4) × (24)²
= (√3/4) × 576
= 144√3
≈ 144 × 1.732
≈ 249.41 sq. cm

Total surface area of pyramid = 4 × area of one triangle
= 4 × 249.41
= 997.64 sq. cm

Problem 3: Determine the lateral surface area of a pentagonal pyramid if the side length of the base is 10 cm and the slant height of pyramid is 38.1 cm.

Solution:

Given,
Side of the pentagonal base, a = 10 cm
Slant height, l = 38.1 cm

Perimeter of the base,
P = 5a = 5 × 10 = 50 cm

Lateral Surface Area of a pentagonal pyramid,
LSA = (1/2) × P × l
= (1/2) × 50 × 38.1
= 25 × 38.1
= 952.5 cm²

Problem 4: Determine the surface area of a hexagonal pyramid if the side length of the base is 12 inches and the slant height of the pyramid is 14 inches.

Solution:

Given,

The side of the hexagonal base (a) = 12 inches, and

Slant height, l = 14 inches

The perimeter of the square base (P) = 6a = 6(12) = 72 inches

The surface area of a pentagonal pyramid = Area of the base + Area of lateral faces

Lateral surface area (LSA) = (½) Pl

= (½ ) × (72) × 14 = 504 sq. in

Area of the hexagonal base = 3√3/2 (a)² = 3√3/2 (12) = 374.123 sq. in

The surface area of a pyramid = Area of the base + Area of lateral faces

= 374.123 sq. in + 504 sq. in = 878.123 sq. in

Hence, the surface area of the given pyramid is 878.123 sq. in.

Problem 5: Determine the side length of a square pyramid if its lateral surface area is 600 square inches and the slant height of the pyramid is 20 inches.

Solution:

Given,

The  lateral surface area = 600 square inches

Slant height, l = 20 inches

Let the side length of a square base be "a".

Now, the perimeter of the base (P)= 4a

We know that

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

⇒ 600 = (½ ) × (4a) × 20

⇒ 80a = 1200 ⇒ a = 15 inches

Hence, the side length of the given pyramid is 15 in.

Practice Problems

Problem 1: A square pyramid has a base side length of 6 cm and a slant height of 10 cm. Calculate the surface area of the pyramid.

Problem 2: The base of a triangular pyramid has an area of 24 square cm, and each triangular face has a base of 8 cm with a height of 9 cm. Find the surface area of the pyramid.

Problem 3: A regular hexagonal pyramid has a base side length of 4 cm and a slant height of 7 cm. Determine the surface area of the pyramid.

Problem 4: A pentagonal pyramid has a base perimeter of 25 cm and a slant height of 12 cm. What is the surface area of the pyramid?

Problem 5: Calculate the surface area of a square pyramid with a base side length of 5 cm and a height of 12 cm.