Question

# Find the lateral and total surface area of the following pyramids.(a) Square-based pyramid with base 6cm and slant height 14cm;(b) Triangular-based pyramid with base 12cm and slant height 20cm.

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Solution

## (a) The perimeter of the base is P=4s, since it is a square, therefore, P=4×6=24 cmThe general formula for the lateral surface area of a regular pyramid is LSA=12Pl where P represents the perimeter of the base and l is the slant height.Since the perimeter of the pyramid is P=24 cm and the slant height is l=14 cm, therefore, the lateral surface area is:LSA=12Pl=12×24×14=168 cm2Now, the area of the base B=s2 with s=6 cm is:B=s2=62=36cm2The general formula for the total surface area of a regular pyramid is TSA=12Pl+B where P represents the perimeter of the base, l is the slant height and B is the area of the base.Since LSA=12Pl=168cm2 and area of the base is B=36 cm2, therefore, the total surface area is:TSA=12Pl+B=168+36=204cm2Hence, lateral surface area of the pyramid is 168cm2 and total surface area is 204cm2.(b) The perimeter of the base is P=4s, since it is a triangle, therefore, P=3×12=36 cmThe general formula for the lateral surface area of a regular pyramid is LSA=12Pl where P represents the perimeter of the base and l is the slant height.Since the perimeter of the pyramid is P=36 cm and the slant height is l=20 cm, therefore, the lateral surface area is:LSA=12Pl=12×36×20=360 cm2Now, the area of the base B=√34s2 with s=12 cm is: B=√34s2=√34×12×12=36√3cm2The general formula for the total surface area of a regular pyramid is TSA=12Pl+B where P represents the perimeter of the base, l is the slant height and B is the area of the base.Since LSA=12Pl=360cm2 and area of the base is B=36√3 cm2, therefore, the total surface area is:TSA=12Pl+B=360+36√3=36(10+√3)cm2Hence, lateral surface area of the pyramid is 360cm2 and total surface area is 36(10+√3)cm2.

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