Surface Area of Pyramid

In this section we will discuss about surface area of pyramid.

Pyramid :

Regular pyramid Triangular Quadrilateral Pentagonal Hexagonal
Shape of the base Triangle Square or rhombus Regular pentagon Regular hexagon
Top Vertex Vertex Vertex Vertex
Shape of the lateral faces Equilateral triangle Equilateral triangle Equilateral triangle Equilateral triangle
Number of lateral faces
3
4
5
6
Total number of faces
4
5
6
7

Right pyramid : If a line drawn through the vertex of pyramid and perpendicular to the base meets the base in its center, it is called a right pyramid and length of the perpendicular is called the height of the pyramid.

If the vertex of a pyramid is P and its base is ΔABC, the pyramid is denoted by
(P – ABC ). The perpendicular from P will meet the base at the centroid G of ΔABC. Hence the height of the pyramid is PG. PB is a slant height.

Lateral surface area = 1/2 x perimeter x slant height = 1/2 pl
Total surface area = area of the base + Lateral area = Area of the base + 1/2 pl

Some solved examples on surface area of pyramid

1) A pyramid with a triangular base with each side of length 10 cm. The slant height of the pyramid is 12 cm. Find the lateral surface area and the total surface area of this pyramid.
Solution :

In pyramid (P – ABC), AB= BC = CA = 10 cm and slant height PD = 12 cm
Lateral surface area = ½ pl = ½ x 30 x 12 = 180 cm
2
Area of ΔABC = (√3 x side
2 ) / 4
⇒ = (√3 x 10
2 ) / 4
⇒ = (√3 x 100) / 4
∴ Area of ΔABC = 25 √3 = 25 x 1.73 = 43.25 cm
2
Total surface area = area of base + Lateral area
⇒ = 43.25 + 180
∴ Total surface area = 223.25 cm
2
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2) If the length of each edge of a regular pyramid (P – ABC ) is 30 units, find its slant height.
Solution :

In ΔPDC, ∠D is a right angle,
So by Pythagorean theorem
c
2 = a 2 + b 2
30
2 = a 2 + 15 2
900 = a
2 + 225
∴ a
2 = 900 – 225 = 675
∴ a = PD = √675 = 25.98 units.

Surface Area :

Surface Area of Cube
Surface Area of Rectangular Prism(Cuboid)
Surface Area of Cylinder
Surface Area of Cone
Surface Area of Sphere and Hemisphere
Surface Area of Prism
Surface Area of Pyramid

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