Geometry: Surface Area - Prisms, Cylinders, Cones, Pyramids and Spheres
This is a free lesson from our course in Geometry
 
   
This lesson explains the concepts of Surface area and combines important properties of Cube, Prism, Pyramids, Cylinders, Cones, and Spheres, to apply the relevant formulas for determining the surface area of the 3D solids. As a first step, you’ll need to review earlier learning about the solids and use the formulas for surface area. Moving forward you’ll be able to calculate lateral area and surface area of these shapes. Further on will solve several problems involving lateral and total surface areas of prisms. Once mastered the surface area of cubes, prisms etc. can move to other 3D figures including pyramids, cones, and spheres etc. The presentation covering such content will be done by the instructor in own handwriting, watching video and with the help of several examples with solution. This will help you apply important geometry properties and relationships to solve problems from day to day life situations and also doing geometry home work.(More text below video...)
<h2> Geometry - Surface Area - Prisms, Cylinders, Cones, Pyramids and Spheres</h2> <p> The prism surface area formula is: 2(lw + lh + wh), where l is length, w is width and h is height of the prism.A Cuboid where its length, breadth and height are equal i.e. where all side lengths equal and all angles measure 90 degrees, is known as ‘Cube’. </p> <p> The prism surface area formula is: 2(lw + lh + wh), where l is length, w is width and h is height of the prism.A Cuboid where its length, breadth and height are equal i.e. where all side lengths equal and all angles measure 90 degrees, is known as ‘Cube’. </p>
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Volume - Prisms, Cylinders, Cones, Pyramids and Spheres
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(Continued from above) Cube: A Cuboid; where its length, breadth and height are equal i.e. where all side lengths equal and all angles measure 90, is known as ‘Cube’.
Commonly in geometry, a cuboid is known by a solid figure bounded by six faces, forming a convex polyhedron (A polyhedron in geometry is a 3D solid consisting of a group of polygons, generally joining at their edges). Further the requirement is that these six faces each be a quadrilateral, and that the graph formed by the vertices and edges of the polyhedron should be similar to the graph of a cube. Also a ‘cuboid’ may be a shape of the type in which each of the faces is a rectangle, and where each pair of adjacent faces meets in a right angle. The later type is also known as a right cuboid, rectangular box, rectangular hexahedron, right rectangular prism, or rectangular parallelepiped. E.g. a chalk box, geometrical box, match box and a book etc. are the examples of cuboids.
Cuboid
Fig: 3, above shows a 3D figure called cuboid. Notice that it has six rectangular plane surfaces called faces.
Two adjacent faces meet along a line segment called an edge. It has 12 edges in all.
It has 8 corners called the vertices (A, B, C, D, E, F, G, and H). At every vertex, there are three edges meeting- one of these edges represents length, second the breadth and third as the height.
The main diagonal of the cuboid (ds) is defined by joining the line-segment of vertex D to the vertex F.
Note: If the length, breadth and height of a cuboid are equal it is a cube.
Walk through a few examples now:
Example 1: Given: AB= l, AE = b and AD = h, find out the surface area.
Step 1: the total surface area of cuboid, can be worked out by adding areas of six faces i.e. = sum of the areas of six faces ABCD, EFGH, ADGE, BCHF, ABFE and DCHG
Step 2: = (lh + lh + bh + bh + lb + lb) = 2(lb + bh + hl)
Step 3: total surface area of a cuboid = 2(lb + bh + hl)
Main diagonal of a cuboid= (l2 + b2 + h2)
Now if it is considered as cube : A cube is a special case of a cuboid when all the edges are equal i.e. l = b = h
If each one is a, l = b = h = a
Therefore surface area of the cube = 6a2 and main diagonal of cube is = 3 * a
Main Diagonal of a cube = 3 * a, as the final answer.
Example 2: Given- if the surface area of a cube is 96 sq cm, find length of each side.
Step 1: the surface area of a cube equals to = 6a2, where a is the side of cube
Step 2: thus 96 = 6a2 i.e. a2 = 16
Step 3: a = 4 cm.
Length of each side is 4 cm, as the final answer.
Example 3 Given- Karen grows cabbages in her garden square in shape. Each cabbage takes 1 square feet of area in the garden. This year, she has increased her output by 211 cabbages as compared to last year. The shape of the area used for growing the cabbages has remained a square in both these years. How many cabbages did she produce this year?
Step 1: the shape of the area used for growing cabbages is a square in both the years. Say the side of the square area used for growing cabbages this year be X ft, and last year it was Y feet.
Step 2: the ground area this year is X2 and last year it was Y2 sq ft.
Step 3: known that increase in cabbages is 211. Therefore, increase in area over previous year is 211 sq ft (given that each cabbage takes 1 square feet of area in the garden).
Step 4: thus (X2 - Y2) = 211 OR (X + Y) (X - Y) = 211
Step 5: As 211 is a prime number, therefore, its two factors can be represented by (106 + 105) * (106 - 105).
Step 6: (X + Y) (X - Y) = (106 + 105) (106 - 105). It can thus be deduced that X = 106 and Y = 105.
Step 7: Number of cabbages produced this year equals to X2 = (106)2 = 11236, as the final answer.
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