Pyramid is a three-dimensional geometric structure with a polygonal base and all its triangular faces meet at a common point called the apex or vertex. The real-life examples of pyramids are the pyramids of Egypt; rooftops; camping tents, etc. Depending upon the shape of the polygonal base, pyramids are classified into different types, such as triangular pyramids, square pyramids, rectangular pyramids, etc. The side faces, or lateral surfaces of a pyramid, are triangles that meet at a common point called the apex. The height, or altitude, of a pyramid, is the perpendicular distance between the apex and the center of the base, whereas a slant height is a perpendicular distance between the apex and the base of a lateral surface.
Surface Area of PyramidSurface Area of a Pyramid
The surface area of a pyramid is the sum of the areas of the lateral surfaces, or side faces, and the base of a pyramid. The surface area of a pyramid has basically two types of surface areas: the lateral surface area and the total surface area, and they are measured in terms of m2, cm2, in2, ft2, etc.
Lateral surface area of a pyramid (LSA) = Sum of areas of the lateral surfaces (triangles) of the pyramid
Total surface area of a pyramid (TSA) = Lateral surface area of the pyramid + Area of the base
To determine the lateral surface of a pyramid, the areas of the side faces (triangles) are calculated and the results are added, whereas the total surface area of a pyramid is calculated by adding the area of the pyramid's base and the lateral surface area.
Read More about Pyramid.
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 scalar triangle. It has three lateral (triangular) faces and a triangular base.
We know that,
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 (s)2
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 side length of the base, and
- l is slant height of the pyramid.
Read More,
Sample Problems on Surface Area of a Pyramid
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: A box is in the shape of a triangular pyramid, and every face of the box is in the shape of an equilateral triangle. Now, determine the total surface area of a triangular pyramid if the side length of the base is 24 cm.
Solution:
Given data,
The side of the base (a) = 24 inches, and
We know that,
The total surface area of a pyramid (TSA) = Area of the base + Area of lateral faces
We have,
Area of an equilateral triangle = √3/4 (a)2
Hence, the area of each triangular face = √3/4 (a)2
= √3/4 (24)2
= 249.415 sq. cm
Now, TSA = 249.415 + 3 × (249.415) = 997.66 sq. cm
Hence, the total surface area of the given pyramid is 997.66 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 the pyramid is 15 inches.
Solution:
Given,
The side of the pentagonal base (a) = 10 inches, and
Slant height, l = 15 inches
The perimeter of the square base (P) = 5a = 5(10) = 50 cm
The lateral surface area of a pentagonal pyramid is,
Lateral surface area (LSA) = (½) Pl
= (½ ) × (50) × 15
= 375 sq. cm
Hence, the lateral surface area of the given pyramid is 375 sq. cm.
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 on Surface Area of a Pyramid
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.
Similar Reads
Surface Area of a Triangular Pyramid Formula A triangular pyramid is a three-dimensional shape with four triangular faces. To calculate the surface area of a triangular pyramid, we need to find the area of each triangular face and summing them up. This process can be straightforward if you know the dimensions of the triangles involved. In simp
5 min read
Surface Area of a Cone Formula A geometric object with a smooth transition from a flat base to the apex or vertex is called a cone. It consists of segments, split lines, or lines that link a commonality, the pinnacle, to most of the spots on a base that are not in the same plane as the apex.Table of ContentSurface Area of ConeFor
8 min read
Surface Area of a Cube Formula Cube Formulas as the name suggest are all the formulas of a cube to calculate its volume, surface areas and diagonal. A cube is a three-dimensional figure that has equal length, breadth, and height. It is a special case of cuboid and is widely observed in our daily life. The most common example of t
6 min read
Surface Area of a Square Pyramid Pyramid is a three-dimensional geometric structure with a polygonal base and triangular faces equal to the number of sides in the base. The triangular faces or lateral surfaces of a pyramid meet at a single point known as the apex or the vertex. Surface Area of a Square Pyramid In a pyramid, the bas
8 min read
Volume of a Pyramid Formula Pyramid is a three-dimensional shape whose base is a polygon, and all its triangular faces join at a common point called the apex. The pyramids of Egypt are real-life examples of pyramids. Volume of a pyramid is the space occupied by that pyramid and is calculated by the formula, V = 1/3×(Area of Ba
9 min read