Prism:
A prism is a polyhedron, meaning its cross section will be a polygon and all
sides will be flat. These are 3D figures that occupy space. The triangular prism
(Fig: 4) has- 5 Faces, 6 corner points i.e. vertices, and 9 edges.
You should be familiar with the common shapes, sizes and characteristics like:
• Lateral edges: are defined by the lines connecting the corresponding vertices, which form generally a sequence of parallel segments.
• Lateral faces: The parallelograms formed by the lateral edges.
• Bases: a prism has two bases, which are congruent polygons and lying in parallel planes.
(More text below video...)
• Altitude: A segment perpendicular to the planes of the bases with an endpoint in each plane.
• Right prism: A prism whose lateral edges are perpendicular to the bases. In a right prism, a lateral edge is an altitude also.
• Oblique prism: A prism whose lateral edges are not perpendicular to the base. Its lateral faces are parallelograms and lateral edge is not the height of the prism. Notice that so far in the explanation and examples, what have been discussed are those where the sides made right angles with the base. Now think of how does surface area may change in oblique prisms? Is there a surface area formula for oblique prisms which is independent of the slant angle? In general the slant and height will influence the length of the lateral faces, which will vary depending on the height and the slant angle of the oblique prism.
The formula for a right prism can be used for a oblique prism. Note: If you stack a set of 100 pieces of paper on top of each other you make a right rectangular prism. If you push the stack into an oblique prism the number of pieces of paper does not change. Which means the volume does not change. Both stacks have the same number of paper sheets, and the same amount of height. Also, because every piece of paper has the same size and shape, they have the same area. Any card in either stack represents a cross section of each prism.
Look at some of the illustrative examples:
Example 1: Given- if the height of a prism is 5 m and its bases are right triangles with sides
9 m, 12 m, and 15 m, determine the lateral area and the total surface area (Fig: 6, N.T.S.).
Step 1: you may recall, the base of a right triangular prism is a right triangle and
its perimeter is the sum of the lengths of its sides i.e. 9, 12, 15m.
Step 2: the perimeter of base thus equals to (9 + 12 + 15) = 36 m.
Step 3: lateral area of prism = P (perimeter)* h (altitude)
= 36 * 5 = 180 m2
Step 4: area of triangle ABC = 1/2 * (b) * h, where b
is base (9)
and h is altitude (12). Thus area works out to
1/2 * 9 * 12 = 54.
Step 5: area of two bases of prism = 2 * 54 = 108 m2
Step 6: total surface area of a prism = Lateral area + area of bases
= 180 + 108 = 288 m2
Example 2: the dimensions of a box are- length 3 feet, a width 2 feet and its surface area is 52 square feet. Calculate height of the box and check the solution.
Step 1: Given length (l), width (w) of the box and its surface
area and its height (h) is to be determined.
You know that a rectangular box has six sides (faces).
Step 2: surface area of a rectangular box =2(lw + wh + lh)
= 2(3 * 2 + 2 * h + 3 * h)
= 2(6 + 5h) = 12 + 10 h
= 52 sq ft (surface area given)
Step 3: simplification gives, 10 h = 52 -12 = 40
Step 4: h = 4
Height of the box is 4 ft, as the final answer.
Step 5: check- surface area = 2(lw + wh + lh)
= 2(3 * 2
+ 2 * 4 + 3 * 4)
= 2(6 + 8 +12)
= 52 sq ft., as given. Found correct.
Example 3:
Given- Find the surface area of a regular pentagonal prism (Fig: 7), if its base area is 62 cm2
Step 1: base area of the regular pentagonal prism = 62 cm2 (Given)
Step 2: as all the five rectangles will have the same dimensions i.e. 6 cm * 15 cm, area of rectangles equals to,
5 (6 * 15) = 5 * 90 = 450 cm2.
Step 3: surface area of the prism = area of 5 rectangles + 2(base area)
= 450 + (2 * 62)
= 450 + 124 = 574
= 574cm2
Surface area of the prism is 574 cm2, as the final answer.
Example 4: Given- there are 25 cans in a box, each can having radius of 3 cm and height of 12 cm. Before getting it to market, the cans are required to have papers labels on them. Find out the quantity of paper required.
Step 1: the radius and the height of one can (given i.e. 3 cm and 12 cm respectively).
Step 2: understand it carefully that labels are to cover the curved surface and area of one can equals to –
A = 2r
* h
= 2
* 3 * 12
= 226.08.
Area for 25 cans equals to = 5652cm2
So, about 5652 cm2 of paper is needed to make the labels for the 25 cans, as the final answer.
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