Volume and Total Surface Area of RIGHT Prisms and CYLINDERS

Volume  The amount of space occupied by an object.  How many 1by1by1 unit cubes that will fit inside. Example: The VOLUME of this cube is all the space contained by the sides of the cube, measured in cube units (units 3 ).

Volume  Once we know the area of the base (lxw), this is then multiplied by the height to determine the VOLUME of the prism (how many cubes will fit inside. IMPORTANT! We know that: Volume of a prism = (Area of Base)x(Height of the prism) Not so important: Volume of this rectangular prism = (l x w) x H Because… the base area IS length times width l w h

Volume  Find the volume of this prism… Formula: V = B x H Where B is the area of the BASE H is the height of the PRISM 5 cm 4 cm 7 cm  BASE area: a 5 by 4 rectangle  B=20 square cm V = B x H V = 20 x 7 V = 140 cubic cm

Total Surface Area a totaling of the surface areas  Draw each surface 5 cm 4 cm 7 cm Find each area and add them up: 5x7 = 35 sq. cm 4x7 = 28 sq. cm 5x4 = 20 sq. cm total = Label the sides

Volume  Volume of a Triangular PRISM  (area of the BASE) x (Height of the prism)

Volume  The same principles apply to the triangular prism. To find the volume of the triangular prism, we must first find the area of the triangular base (shaded in yellow). b h

Volume  To find the area of the Base… Area (triangle) = b x h 2 This gives us the Area of the Base (B). b h

Volume  Now to find the volume… We must then multiply the area of the base (B) by the height (H) of the prism. This will give us the Volume of the Prism. B H

Volume  Volume of a Triangular Prism Volume (triangular prism) V = B x H B H

Volume V = B x H  Isosceles triangle based prism

Volume BASE area = (8 x 4) = 16 sq. cm 2 the Height of the prism is 12 cm V = B x h V =16 x 12 V =_____ cubic cm

Total Surface Area a totaling of the surface areas  Draw each surface Find each area and add them up: Label the sides 12 ?? ?? 4 4 Find the missing length using the right triangle inside the isosceles triangle and the Pythagorean Theorem…

Volume Volume of a Cylinder A cylinder is like a prism with a circle base, so we can use the SAME VOLUME formula V = B x H Where B is the area of the base (circle) And H is the height of the cylinder H H r r

VOLUME Formula for Area of Circle A=  r 2 =  x 3 2 =  x 9 = 9  square units H = 6 units VOLUME = B x H = 9  x 6 = 54  _____ cubic units

Parts of a cylinder for TSA 3 parts 1 rectangle and 2 circles

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.

Area of the Circles and rectangle Area of the Circles Area of the RECTANGLE TOTAL SURFACE AREA = circle + circle + rectangle _______ square units

Area of the Circles and rectangle Area of the Circles A=  r 2 =9  (from before) Circle with radius of 3

Area of the Circles and rectangle Area of the RECTANGLE A=(circumference)(Height of the cylinder ) =(  d)(H) = (  6)(6) = 36  Circle with radius of 3 CIRCUMFERENCE OF THE CIRCLE Diameter times pi 6 times  Height of cylinder 6 units

Area of the Circles and rectangle Area of the Circles A=  r 2 =9  (from before) Area of the RECTANGLE A=(circumference)(Height of the cylinder ) =(  d)(H) = (  6)(6) = 36  Circle with radius of 3 CIRCUMFERENCE OF THE CIRCLE 6 times  Height of cylinder 6 units TOTAL SURFACE AREA = circle + circle + rectangle _______ square units

For all this you need: Area Formulas Area Circle = π x r 2 r Area Rectangle (and parallelograms) = base x height h b b h Area Triangle = ½ x base x height h b Area Trapezoid = ½ x (a + b) x h a