A Prism is a polyhedron with an n-sided polygonal base, another congruent parallel base (with the same rotational orientation), and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases. All cross-sections parallel to the base faces are congruent to the bases. (from Wikipedia)

A Pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. It is a conic solid with polygonal base  (from Wikipedia)

A Tetrahedron is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedron and the only one that has fewer than 5 faces.(from Wikipedia)

Height (h) = (√2)/(√3) * (Side)
Total Surface Area (TSA) = (3√3)/4 * (Side)2
Lateral Surface Area (LSA) = (4√3)/4 * (Side)2
Volume (V) = (√2)/12 * (Side)3

#### Questions for Practice:

1. A prism with triangular base with sides AB=5cm, BC=8cm, and CA=7cm and Altitude (H)= 9cm. Find the Lateral surface Area, Total surface area, and Volume?
Ans: Since, triangle ABC is scalene triangle.
Perimeter (P) = 5+8+7 = 20cm
The area of ABC = √10*(10-5)*(10-8)*(10-7) = √(10*5*2*3) = 10√3.

LSA = Perimeter * Altitude = 20 * 9 = 108 cm2

TSA = LSA + 2 x Area of ABC = 108+20√3 cm2

Volume = Area x altitude = 10√3 * 9 = 90√3 cm3

2. A pyramid with trapezium base with sides AB=9cm, BC=12cm, CD=15cm, and DA=18cm. The Volume of pyramid = 1458 cm3. Find the Height (H)?
Ans:

3. A rectangle based pyramid, length and width of the base is 18 cm and 10 cm respectively. Find the total surface area (TSA), if its height is 12 cm?
Ans:

4. A regular tetrahedron has surface area of 144√3. Find the side, volume and Total surface area (TSA)?
Ans: Lateral Surface area = 144√3 = √3 * side2
=> side2 = 144 => side = 12 cm.
=> TSA = 3√3/4 * side2 = 3√3/4 * 122 = 27√3 cm2.
=> Volume = √2/12 * side3 = √2/12 * 123 = 144V2

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