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Volume And Surface Areas

Published in: Mathematics
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  • Debarun S

    • Kolkata
    • 10 Years of Experience
    • Qualification: B.Tech
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this presentation describes volume and surface areas of different objects like cone, cylinder, cube etc

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    Surface Area and Volume
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    Day 1 - Surface Area of Prisms Surface Area = The total area of the surface of a three-dimensional object (Or think of it as the amount of paper you'll need to wrap the shape.) Prism = A solid object that has two identical ends and all flat sides. We will start with 2 prisms — a rectaneular prism and a trianeular prism.
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    Rectangular Prism Triangular Prism
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    Surface Area (SA) of a Rectangular Prism Surface Area of a Prism Like dice, there re six sides (or 3 pairs of sides) Top Bottom Front Back Right
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    Prism net - unfolded
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    Add the area of all 6 sides to find the Surface Area. 10 - 6 - height 5 - width length
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    SA 21w + 21h + 2wh 6 - height 5 - width length 10 - 21w + 21h + 2wh SA = SA— 2 (10 x 5) + 2 (10 x 6) + 2 (5 x 6) = 2 (50) + 2(60) + 2(30) = 100 + 120 + 60 = 280 units squared
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    Practice 12 ft 10 ft 22 ft SA = 21w + 21h + 2wh = 2(22 x 10) + 2(22x 12) + 2(10 x 12) = 2(220) + 2(264) + 2(120) = 440 + 528 + 240 = 1208 ft squared
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    Surface Area of a Triangular Prism •2 bases (triangular) •3 sides (rectangular)
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    Unfolded net of a triangular prism
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    2(area of triangle) + Area of rectangles Area Triangles = 1/2 (b x h) -1/2 (12 x 15) 20 ft 25 ft 20 ft 15f 12 ft Area Rect. 1 Area Rect. 2 = 1/2 (180) = 90 = 12 x 25 = 300 = 25 x 20 = 500 SA = 90 + 90 + 300 + 500 + 500 SA = 1480 ft squared
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    Practice Triangles 8 cm Rectangle 10 cm Rectangle Add them all up 1 2 - 1/2 (56) — 28 cm -10 x8 = 80 cm = 90 cm SA = 28 + 28 + 80 + 90 + 90 SA = 316 cm squared
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    Surface Area of a Cylinder
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    Review •Surface area is like the amount of paper you'll need to wrap the shape. •You have to "take apart" the shape and figure the area of the parts. •Then add them together for the Surface Area (SA)
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    Parts of a cylinder A cylinder has 2 main parts. A rectangle and A circle — well, 2 circles really. Put together they make a cylinder.
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    The Soup Can Think of the Cylinder as a soup can. You have the top and bottom lid circles) and you have the label (a rectangle — wrapped around the can). The lids and the label are related. The circumference of the lid is the same as the length of the label. CONDENSED TOMATO SOUP
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    Area of the Circles Formula for Area of Circle — 3.14 x 32 — 3.14 x 9 - 28.26 But there are 2 of them so 28.26 x 2 = 56.52 units squared
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    The Rectangle This has 2 steps, To find the area we need base and height, Height is given (6) but the base is not as easy, Notice that the base is the same as the distance around the circle (or the Circumference),
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    Find Circumference Formula is — 3.14 x 6 (radius doubled) — 18.84 Now use that as your base. — 18.84 x 6 (the height given) — 113.04 units squared
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    Add them together Now add the area of the circles and the area of the rectangle together. 56.52 + 113.04 - — 169.56 units squared The total Surface Area!
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    Formula SA = (T d x h) + 2 (Tf2) Label Lids (2) Area of Rectangle Area of Circles
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    Practice Be sure you know the difference between a radius and a diameter! SA = (rd x h) + 2 (Tr2) — (3.14 X 22 X 14) + 2 (3.14 X 112) - (367.12) + 2 (3.14 x 121) - (367.12) + 2 (379.94) - (367.12) + (759.88) - 1127 cm2 22 cm 14 cm
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    SA More Practice! = (rd x h) + 2 (Tr2) = (3.14 x 11 x 7) +2 ( 3.14 x 5.52) = (241.78) + 2 (3.14 x 30.25) = (241.78) + 2 (3.14 x 94.99) 71 = (241.78) + 2 (298.27) = (241.78) + (596.54) = 838.32 cm2 11 cm
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    End of Day 2
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    Day 3 Surface Area of a Pyramid
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    Pyramid Nets TOTAL SURFACE A pyramid has 2 AREA OF A PYRAMID shapes: One (1) square Four (4) triangles Step One Step Two Step Three
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    slant height Since you know how to find the areas of those shapes and add them,
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    slant height you can use a formula... Where lis the Slant Height and p is the perimeter and B is the area of the Base
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    Perimeter Slant height 8 ; = 1/2 (8 X 26) + (7 x 6) = 1/2 (208) + (42) 104 + 42 2 = 146 units - 26 8 5 6 7 *area of the base*
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    Practice 10 = 1/2 (18 x 24) + (6 x 6) = 1/2 (432) + (36) = 216 + 36 = 252 units2 What is the extra information in the diagram? 6 Slant height = 18 Perimeter = 6x4 = 24
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    End Day 3
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    Day 4 Volume of Prisms and Cylinders
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    Volume The number of cubic units needed o to fill the shape. Find the volume of this prism by counting how many cubes tall, long, and wide the prism is and then multiplying. There are 24 cubes in the prism, so the volume is 24 cubic units. 2 — height 3 — width 4 — length
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    Formula for Prisms VOLUME OF A PRISM The volume V of a prism is the area of its base B times its height h. Note — the capital letter stands for the AREA of the BASE not the linear measurement.
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    4 ft- widt 8 ft - length Try It 3 ft - height Find area of the base = (32) x 3 Multiply it by the height = 96 ft3
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    Practice v 12 cm 10 cm 22 cm = (22 x 10) x 12 = (220) = 2640 CT113
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    Cylinders VOLUME OF A CYLINDER The volume V of a cylinder is the area of its base, nr2, times its height h. Notice that is the formula for area of a circle.
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    5m 4.2 m Try It The radius of the cylinder is 5 m, and the height is 4.2 m V = 3 .14 • 52 • 4.2 Substitute the values you know.
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    Practice 13 cm - radius 7 cm - height Start with the formula 3.14 X 132 X 7 Substitute what you know 3.14 X 169 X 7 Solve using order of Ops. 3714.62 cm3
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    Lesson Quiz Find the volume of each solid to the nearest tenth. Use 3.14 for T. 4,069.4 rn3 3. triangular prism: 312 ft3 11.4 cm 8.4 cm 9 cm 861.8 crn3 base area = 24 ft2, height =
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    End of Day 4
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    Day 5 Volume of Pyramids
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    Remember that Volume of a Prism is B x h where b is the area of the base. You can see that Volume of a pyramid will be less than that of a prism, How much less? Any guesses?
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    If you said 2/3 less, you win! Volume of a Pyramid: V = (1/3) Area of the Base x height V = (1/3) Bh Volume of a Pyramid = 1/3 x Volume of a Prism
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    Find the volume of the square pyramid with base edge length 9 cm and height 14 cm. The base is a square with a side length of 9 cm, and the height is 14 Ctn. V = 1/3 Bh 1/3 (9 x 1/3 1/3 (1134) cm3 = 378 1 cm 9 cm 9 cm
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    v Practice 1/3 Bh 1/3 (5 x 5) (10) 1/3 (25) (10) 1/3 250 3 — 83,33 units 10
  • 47
    Quiz Find the volume of each figure, a rectangular pyramid with length 25 cm, 1. width 17 cm, and height 21 cm 2975 cm3 2, a triangular pyramid with base edge length 12 in. a base altitude of 9 in. and height 10 in. 360 in3
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    End of Day 5

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