Prism

Prism

Prism

A solid object in which two opposite faces are congruent and parallel and its cross section part is congruent to its bases is called a prism. For example: cube, cuboid, cylinder etc are prisms. In the figure given below is a cuboid (prism).
Cuboid (Prism)
The name of prism is determined with respect to its base.

Cuboid:

A rectangular prism is called a cuboid. A cuboid is also known as a rectangular parallelopiped.
Rectangular prism (Cuboid)
                Cross sectional area = Area of base
                                                  = l × b
                Total surface area = Area of all 6 faces of the cuboid
                                              = lb+lb+bh+bh+lh+lh
                                              = 2lb + 2bh + 2lh
                                              = 2(lb + bh + lh)
                Volume = Area of base × height
                             = l × b × h

Cube:

A cuboid in which all three dimensions (length, breadth and height) are equal is called a cube.
Cube (Prism)
                    Cross-sectional area = Area of base = l2
                      Total surface area = 6l2
                      Volume = l3

Cylinder:

A circular prism is called a cylinder.
Circular prism (Cylinder)
                    Cross-sectional area = Area of base = Area of circle = πr2
                      Total surface area = Curved surface area + area of two bases (circles) = 2πrh + 2πr2
                      Volume = Area of base × height = πr2h

Triangular Prism:

A prism having its base a triangular shaped is called a triangular prism.
Triangular prism
                    Cross-sectional area = Area of base triangle = A (formula according to the type of triangle)
                      Lateral surface area = Perimeter of Δ × height = Ph
Total surface area =  Lateral surface area + Area of 2 bases = Ph + 2A
                      Volume = Area of base triangle × height = Ah

Workout Examples

Example 1: Find the area of cross-section, total surface area and volume of the given prism.
Example 1: Prism
Solution: The given prism is a cuboid of,
                     Length (l) = 8cm
                     Breadth (b) = 6cm
                     Height (h) = 12cm

                  Area of cross section = l × b
                                                          = 8cm × 6cm
                                                          = 48cm2

                    Total surface area = 2(lb+bh+lh)
                                                    = 2(8×6 + 6×12 + 8×12)
                                                    = 2(48 + 72 + 96)
                                                    = 2 × 216
                                                    = 432cm2

                    Volume = l × b × h
                                  = 8cm × 6cm × 12cm
                                  = 576cm3


Example 2: Find the cross-section area, total surface area and the volume of the given prism.
Example 2: Prism
Solution: The given prism is a cube of,
                    Length (l) = 8cm

                  Area of cross section = l2
                                                          = (8cm)2
                                                          = 64cm2

                    Total surface area = 6l2
                                                    = 6 × (8cm)2
                                                    = 6 × 64cm2
                                                    = 384cm2

                    Volume = l3
                                  = (8cm)3
                                  = 512cm3


Example 3: Find the area of cross-section, total surface area and volume of the given prism.
Example 3: Prism
Solution: The given prism is a cylinder of,
                    Radius (r) = 7cm
                    Height (h) = 12cm

                  Area of cross section = πr2
                                                          = 22/7 × 72
                                                          = 154cm2

                    Total surface area = 2πrh + 2πr2
                                                    = 2 × 22/7 × 7 × 12 + 2 × 22/7 × 72
                                                    = 528 + 308cm2
                                                    = 836cm2

                    Volume = πr2h
                                  = 22/7 × 72 × 12
                                  = 1848cm3


Example 4: Find the area of cross-section, total surface area and volume of the given prism.
Example 4: Prism
Solution: The given figure is a triangular prism with right angled triangle base of,
                    base (b) = 4cm, perpendicular (p) = 3cm
hypotenuse

                     Perimeter = 4cm + 3cm + 5cm = 12cm
                       Height of prism (h) = 10cm

                  Area of cross section (A) = Area of right angled triangle
                                                                = ½ × b × h
                                                                = ½ × 4 × 3
                                = 6cm2

                    Total surface area = Ph + 2A
                                                    = 12cm × 10cm + 2 × 6cm2
                                                    = 120cm2 + 12cm2
                                                    = 132 cm2

                    Volume = A × h
                                  = 6cm2 × 10cm
                                  = 60cm3


Example 5: Find the area of cross-section, total surface area and volume of the given prism.
Example 5: Prism
Solution: Base is divided into two parts A1 and A2 as given below in the figure,
Solution 5: Prism
                  Area of cross section (A) = A1 + A2
                                                                = 5 × 3 + 4 × 2
                                                                = 15 + 8
                                                                = 23cm2

                    Total surface area = Perimeter of cross-section × h + 2 × cross-section area
                                                    = (5 + 3 + 3 + 4 + 2 + 7) × 8 + 2 × 23
                                                    = 24 × 8 + 46
                                                    = 192 + 46
                                                    = 238cm2

                    Volume = A × h
                                  = 23cm2 × 8cm
                                  = 184cm3


You can comment your questions or problems regarding the surface area and volume of prisms here.

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