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Surface Area of a Pyramid

The surface area of a pyramid includes the area of the base and the area of all triangular faces that make up the sides. Surface Area of a Pyramid is expressed in square units. Given below is a square pyramid which has a square of side 's' as the base of the pyramid. The height of the Pyramid is 'h' and it's slant height is 'l'.

Surface Area of a Pyramid

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Surface Area of a Pyramid Formula

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The formula for total surface area of a pyramid is given as follows:

Surface Area of a Regular Pyramid, A = B + $\frac{1}{2}$ PL square units

where, B is the base area

P is the base perimeter

L is the slant height.

When the side of the base and the slant height of the pyramid are given, the formula to find the surface area of the square pyramid is given below:

Surface area of pyramid = Area of base + 2 * S * L square units

Where, Area of the base = S * S square.units

S is the side of the base

L is the slant height of the pyramid.

Lateral Surface Area of a Pyramid

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The lateral area of a pyramid is the sum of the area of all the lateral faces. Since each lateral face is a triangle, the lateral area is equal to the product of $\frac{1}{2}$ of its base edge and slant height of pyramid.

The lateral area of a pyramid is the sum of the area of all the lateral faces.

L.S.A. =
$\frac{1}{2}$ * Perimeter of base * Slant height of pyramid

Surface Area of a Triangular Pyramid

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A triangular pyramid is a pyramid having a triangular base. A triangular pyramid is a three dimension figure with all triangular sides (In this case, we are considering base triangle as a equilateral triangle).

We know that, Surface area of any pyramid = $\frac{1}{2}$ * Perimeter of the base * Slant height of pyramid + Area of the base

Area of Triangular Pyramid

Formula for Surface Area of a Triangular Pyramid:


Since given pyramid is triangular pyramid,

Perimeter of the base = 3a

Area of the base = $\frac{1}{2}$ * a * h

Surface Area of Triangular Pyramid = $\frac{1}{2}$ * 3a * l + ($\frac{1}{2}$ * a * h)

= $\frac{3}{2}$ al + $\frac{1}{2}$ah

Total Surface Area of a Triangular Pyramid = $\frac{3}{2}$ al + $\frac{1}{2}$ah

where, a - Side of the triangular base
h - Height of the triangular base and
l - Slant height of pyramid

Surface Area of a Square Pyramid

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A square pyramid is a pyramid with a square base.

Area  of Square Pyramid

Lateral Surface Area of a Square Pyramid:


The lateral area of a square pyramid is the sum of the area of all the lateral faces.

L.S.A. = $\frac{1}{2}$ * Perimeter of base * Slant height of pyramid

Let perimeter of base with side length "a"
is 4a and slant height is s, therefore the lateral area of the pyramid is 2as.

Lateral area of a square pyramid (L.S.A.) = $\frac{1}{2}$ (4a)s = 2as

where, a - Base side length
and s - Slant height of square pyramid

Surface Area of a Square Pyramid Formula:


Base area with side length 'a', (B) = a$^2$

Surface area of square pyramid = L.S.A. + B

= 2as + a$^2$

Total surface area of square pyramid = 2as + a$^2$

where, a - Base side length
and s - Slant height of square pyramid

Surface Area of a Right Square Pyramid:

If a line segment drawn from the apex to the base is perpendicular, then the square pyramid is called a right square pyramid.

Surface area of a right pyramid = L.S.A. + Base area

where, L.S.A. = $\frac{1}{2}$ * Perimeter of base * Slant height of pyramid

and base area = (Side length of base)$^2$

Surface Area of a Rectangular Pyramid

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A rectangular pyramid is a pyramid having a rectangular base and triangular faces that meet at a common point.

We know that, Surface area of any pyramid = $\frac{1}{2}$ * Perimeter of the base * Slant height of pyramid + Area of the base

With rectangular pyramid, we cannot use the standard formula of surface area, because there is different slant height with different base.

Surface Area of a Rectangular Pyramid Figure

The surface area is the area of the rectangular base plus the areas of the four triangular faces.

Surface Area of a Rectangular Pyramid Formula:


Surface area of Rectangular pyramid (SA) = lb + l $s_1$ + b $s_2$

where,
l - Length of rectangle
b - Width of rectangle
$s_1$ - Slant length of the triangles with base 'l'.
$s_2$ - Slant length of the triangles with base 'b'

Surface Area of a Pentagonal Pyramid

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A pentagonal pyramid is a pyramid with a pentagonal base and five triangular lateral faces.

Pentagonal Pyramid Surface Area

Surface Area of Pentagon Pyramid = L.S.A. + Area of base

where, L.S.A = $\frac{1}{2}$ Ph. Here, P is perimeter of base and h is the slant height

Perimeter of regular pentagon = 5b

Now, L.S.A. of pyramid = $\frac{1}{2}$ (5b)h

where, b = Side length of pentagon and h = Slant height of pyramid

Again, Area of base = $\frac{5}{2}$ ab (from figure)

where, b = Side length of pentagon and a = Apothem height of pentagon

So, surface area = $\frac{1}{2}$ (5b)h + $\frac{5}{2}$ ab

Total Surface Area of Pentagonal Pyramid = $\frac{5}{2}$ (bh + ab) square units

where, b - Side length of pentagon
a - Apothem height of pentagon
h - Slant height of pyramid

Surface Area of a Hexagonal Pyramid

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A hexagonal pyramid is a pyramid with a hexagonal base and six triangular lateral faces.

Hexagonal Pyramid Surface Area

Surface Area of Hexagonal Pyramid = $\frac{1}{2}$ Ph + Area of each face

where, P is perimeter of base and h is the slant height

Since base is hexagonal with base length 'b',

Perimeter of base (P) = 6b

and Base area = 3ab

Area of hexagonal pyramid = $\frac{1}{2}$ (6b)h + 3ab = 3bh + 3ab

Surface Area of Hexagonal Pyramid Formula = 3bh + 3ab

where, a - Apothem length of the hexagon
b - Side length of the hexagon and
h - Slant height of the hexagonal pyramid

How to Find the Surface Area of a Pyramid?

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Given below are some of the examples to find the surface area of a pyramid.

Solved Examples

Question 1: Find the surface area of the square pyramid whose base side is 4 cm and the slant height of the pyramid is 6 cm.
Solution:

The side of the square is 4 cm

The slant height of the pyramid is 6 cm

Surface area of a square pyramid = Area of base + 2SL square units
where, S is the base side length and L is the slant height of pyramid

Area of the base = (4 $\times$ 4) = 16

=> Area of the base is 16 cm2

Surface area of a square pyramid = 16 + (2 $\times$ 4 $\times$ 6)

= 16 + 48

= 64 

The surface area of the square pyramid is 64 cm2



Question 2: Find the surface area of the square pyramid whose base side is 5 m and the slant height of the pyramid is 8 m.
Solution:

The side of the square is 5 m

The slant height of the pyramid is 8 m

Surface area of a square pyramid = (Area of base + 2 $\times$ base side length $\times$ slant height of pyramid) square units

Area of the base = (5 $\times$ 5) = 25

=> Area of the base of pyramid = 25 cm2

Surface area of a square pyramid = 25 + (2 $\times$ 5 $\times$ 8)

= 25 + (2 $\times$ 40)

= 25 + (80)

= 105

The surface area of the square pyramid is 105 m2



Surface Area of a Pyramid Practice Problems

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Given below are some of the practice problems on surface area of a pyramid.

Practice Problems

Question 1: Find the surface area of the square pyramid whose base side is 6m and the slant height of the pyramid is 8m.
Question 2: Find the surface area of the square pyramid whose base side is 6m and the slant height of the pyramid is 10m.
More topics in Surface Area of a Pyramid
Lateral Surface Area of a Pyramid Surface Area of a Triangular Pyramid
Surface Area of a Square Pyramid Surface Area of a Rectangular Pyramid
Surface Area of a Hexagonal Pyramid
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