Volume & Surface Area. Objectives: 7.2.02 Solve problems involving volume and surface area of...
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Transcript of Volume & Surface Area. Objectives: 7.2.02 Solve problems involving volume and surface area of...
Volume & Surface AreaObjectives:7.2.02 Solve problems involving volume and surface area of cylinders, prisms, and composite shapes.
Essential Question: How can I use what I know about area to calculate volume and surface area of cubes, prisms, and cylinders?
Cube: a 3D shape with six square or rectangular sides, a block.
Rectangular Prism: a polyhedron that has two parallel and congruent bases that are rectangles; a 3D solid with six rectangular faces.
Triangular Prism: a polyhedron that has two parallel, congruent bases that are triangles; a prism whose faces are triangles.
Cylinder: a 3D figure that has two parallel congruent bases.
Volume: the measure of space occupied by a solid region.
Surface Area: the sum of the areas of all the surfaces (faces) of a three dimensional figure.
Volume & Surface Area
Triangular Pyramid: a polyhedron with a three-sided polygon for a base and triangles for its sides; a pyramid with a triangular base.
Square Pyramid: a polyhedron with a four-sided polygon for a base and triangles for its sides; a pyramid with a square base.
Sphere: a perfectly rounded 3D object such as a ball.
Cone: a 3D figure with one circular base.
Volume & Surface Area
What is a 3D Figure:What do they look like…
Volume & Surface Area
In previous years you have studied 2D shapes like squares, rectangles, parallelograms, triangles, circles, and trapezoids.
But now it is time to add a third dimension…
Some 3D Figures:What do they look like…
Volume & Surface Area
Cube Rectangular Prism
Triangular Prism
Cylinder
Some Additional 3D Figures:What do they look like…
Volume & Surface Area
Triangular Pyramid
Square Pyramid
Sphere Cone
3D Characteristics:What makes what a what…
Volume & Surface Area
A 3D shape with two parallel congruent polygon bases
3D Characteristics:What makes what a what…
Volume & Surface Area
A cylinder falls under its own category because its bases are not considered polygons
3D Characteristics:What makes what a what…
Volume & Surface Area
Others include pyramids and cones because they contain only one base. Their name is derived based on the shape of the base
Important Volume Formulas:Volume & Surface Area
CubeRectangular
PrismTriangular
Prism Cylinder
V = s3 V = lwh V = ½bhw V = πr2h V = bhw
Example 1: CubeFind the volume of the cube whose sides measure 3 inches.
Volume & Surface Area
Volume = s3
V = 3in x 3in x 3inV = 27in3
Example 1: Rectangular PrismFind the volume of the rectangular prism whose length is 5in, width is 9in, and height is 4in.
Volume & Surface Area
Volume = lwh V = 5in x 9in x 4inV = 180in3
Example 1: Triangular PrismFind the volume of the triangular prism whose length is 6cm, width is 4cm, and height is 3cm.
Volume & Surface Area
Volume = ½bhw V = ½(6cm)(3cm)(4cm)V = 36cm3
Example 1: CylinderFind the volume of the cylinder whose height is 3ft and radius is 4ft.
Volume & Surface Area
Volume = πr2h V = (3.14)(4ft)2(3ft)V = 150.72ft3
Example 1: Rectangular PrismFind the volume of the rectangular prism.
Volume & Surface Area
V = lwh V = 4in x 6in x 5inV = 120in3
Example 1: Rectangular PrismFind the volume of the rectangular prism.
Volume & Surface Area
V = lwh V = 5in x 7in x 11inV = 385in3
Example 1: Triangular PrismFind the volume of the triangular prism.
Volume & Surface Area
Volume = ½bhw V = ½(15cm)(9cm)(4cm)V = 270cm3
Example 1: CylinderFind the volume of the cylinder.
Volume & Surface Area
Volume = πr2h V = (3.14)(3cm)2(12cm)V ≈ 339.3cm3
Find the volume of the block. A wooden block has a single hole drilled entirely through it. What is the volume of the block? Round to the nearest hundredth.
Volume & Surface Area
The block is a rectangular prism with a cylindrical hole. To find the volume of the block, subtract the volume of the cylinder from the volume of the prism.
Find the volume of the block. A wooden block has a single hole drilled entirely through it. What is the volume of the block? Round to the nearest hundredth.
Volume & Surface Area
The volume of the box is about 72 – 9.42 = 62.58 cubic centimeters.
Find the volume of the cube. A moving company has boxes of various sizes for packing. The smallest box available has the dimensions shown below. Find the volume of a larger box that is 3 times as large.
Volume & Surface Area
Answer
12 in.
12 in.
12 in.
Find the volume of the cylinder. A jumbo-size can of tomato soup is about 3 times the size of a standard-sized can of soup. The standard can has the dimensions shown. Find the surface area and volume of the jumbo-size can.
Volume & Surface Area
Answer
Formula Summary:Volume & Surface Area
CubeRectangular
PrismTriangular
Prism Cylinder
V = s3 V = lwh V = ½bhw V = πr2h V = bhw
So What’s The Difference:Now that we have studied volume it is time to move on to surface area…another important concept dealing with 3D shapes:
When you think about the words
SURFACE AREAWhat comes to mind?
Volume & Surface Area
Surface Area & Nets:When thinking about surface area we need to be able to break down the 3D solid by its faces…for instance:
If took the above cube and cut along the edges we could open the solid and see that there a total of 6 squares (we call these faces) – the figure on the right is called a net
Volume & Surface Area
Important Surface Area Formulas:Volume & Surface Area
CubeRectangular
Prism
SA = 6s2 SA = 2lw + 2lh + 2hw
Important Surface Area Formulas:Volume & Surface Area
Triangular Prism Cylinder
SA = 2(½bh) + lw1 + lw2 + lw3
SA = 2πr2 + 2πrh
Example 1: CubeFind the surface area.
Volume & Surface Area
SA = 6s2
SA = 6(4in)2
SA = 6(16in2)SA = 96in2
Example 1: Rectangular PrismFind the surface area.
Volume & Surface Area
SA = 2lw + 2lh + 2wh SA = 2(15)(9) + 2(15)(7) + 2(9)(7)SA = 270mm2 + 210mm2 + 126mm2
SA = 606mm2
Example 1: Triangular PrismFind the surface area.
Volume & Surface Area
SA = 2(½bh) + lw1 + lw2 + lw3
SA = 2(½)(4.5)(3) + (6x3.75) + (6x3.75) + (6x4.5)
SA = 13.5in2 + 22.5in2 + 22.5in2 + 27in2
SA = 85.5in2
Example 1: CylinderFind the surface area.
Volume & Surface Area
SA = 2πr2 + 2πrh SA = 2(3.14)(3mm)2 + 2(3.14)(3mm)(8mm)
SA = 56.52mm2 + 150.72mm2
SA = 207.24mm2
Camping. A family wants to reinforce the fabric of its tent with a waterproofing treatment. Find the surface area, including the floor, of the tent below.
Volume & Surface Area
Remember, a triangular prism consists of two congruent triangular faces and three rectangular faces.
Camping. A family wants to reinforce the fabric of its tent with a waterproofing treatment. Find the surface area, including the floor, of the tent below.
Volume & Surface Area
BOTTOMLEFT SIDE
RIGHT SIDE
TWO BASES
SA = 29 + 36.54 + 36.54 + 29 = 131.08