Volume & Surface Area

Volume &

Surface Area

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.


What is a 3D Figure:What do they look like…


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…

What is a 3D Figure:What do they look like…


Some 3D Figures:What do they look like…


Cube Rectangular Prism

Triangular Prism

Cylinder

Some Additional 3D Figures:What do they look like…


Triangular Pyramid

Square Pyramid

Sphere Cone

Cubes:What do they look like…


Prisms:What do they look like…


Cylinders:What do they look like…


3D Characteristics:What makes what a what…


A 3D shape with two parallel congruent polygon bases



A cylinder falls under its own category because its bases are not considered polygons



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


Triangular 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 = 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 = 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 = 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 = πr2h V = (3.14)(4ft)2(3ft)V = 150.72ft3

Example 1: Rectangular PrismFind the volume of the rectangular prism.


V = lwh V = 4in x 6in x 5inV = 120in3

Example 1: Rectangular PrismFind the volume of the rectangular prism.


V = lwh V = 5in x 7in x 11inV = 385in3

Example 1: Triangular PrismFind the volume of the triangular prism.


Volume = ½bhw V = ½(15cm)(9cm)(4cm)V = 270cm3

Example 1: CylinderFind the volume of the cylinder.


Volume = πr2h V = (3.14)(3cm)2(12cm)V ≈ 339.3cm3

Independent Practice: Volume


Find the volume of each 3d shape.1. 2.

3. 4.

Independent Practice: Volume


Answers.1. 27 m3 2. 1260 cm3

3. 27 m3 4. 336 mm3

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.


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.


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.


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.


Answer

Formula Summary:Volume & Surface Area



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?


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


Surface Area & Nets:Each solid has its own set of faces or NET:


Important Surface Area Formulas:Volume & Surface Area


SA = 6s2 SA = 2lw + 2lh + 2hw

Important Surface Area Formulas:Volume & Surface Area


SA = 2(½bh) + lw1 + lw2 + lw3

SA = 2πr2 + 2πrh

Example 1: CubeFind the surface area.


SA = 6s2

SA = 6(4in)2

SA = 6(16in2)SA = 96in2

Example 1: Rectangular PrismFind the 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.


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.


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.


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.


BOTTOMLEFT SIDE

RIGHT SIDE

TWO BASES

SA = 29 + 36.54 + 36.54 + 29 = 131.08

HOMEWORK


Volume & Surface Area

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