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**Unit 2: Engineering Design Process**

Foundations of Technology Unit 2: Engineering Design Process Lesson 5: Prototyping and Modeling 3 Calculating Area and Volume

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The Big Idea Big Idea: At various intervals of the Engineering Design Process, conceptual, mathematical and physical models are used to evaluate the design solution.

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**Area and Volume Definitions:**

Area – the amount of space inside a two- dimensional shape. Surface area – the total area of all surfaces on a three-dimensional object. Volume – the amount of space a three- dimensional object occupies.

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**Area Square: a = the length of all sides, as all sides are equal**

Equation for Area (A) = s X s = s² = A s s s s

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**Area Rectangle: W = the width of rectangle H = the height of rectangle**

Equation for Area (A) = W x H = A W H

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**Area Triangle: b = the base of the triangle**

h = the height of the triangle Equation for Area (A) = ½(b x h) = A h b

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**Area Circle: r = the radius of the circle**

Equation for Area (A) = π(r²) = A r

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**Practice Questions Calculate the area for the square and rectangle:**

W = 3 H = 2

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**Practice Questions Calculate the area for the square and rectangle**

W = 3 Area = a² = 4² A = 16 Area = W x H = 3 x 2 A = 6 H = 2

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**Practice Questions Calculate the area for the triangle and circle:**

b = 4

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**Practice Questions Calculate the area for the triangle and circle:**

Area = π(r²) = π(2.5²) = π(6.25) A = 19.6 Area = ½(b x h) = ½(4 x 3) = ½(12) A = 6 b = 4

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Surface Area Cube: a = the length of all sides, as all six sides are equal Equation Surface Area (SA) = 6(s²) = SA s s s

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**Surface Area Rectangular Prism: W = the width of the prism**

D = the depth of the prism H = the height of the prism Equation Surface Area (SA) = 2(W x H) + 2(D x H) + 2(W x D) = SA W D H

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**Surface Area Sphere: r = the radius of the sphere**

Equation Surface Area (SA) = 4π(r²) = SA r

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**Surface Area Cylinder: r = the radius of the cylinder**

h = the height of the cylinder Equation Surface Area (SA) = 2π(r²) + 2π(r x h) = SA h r

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Practice Question Calculate the surface area for the sphere and cylinder: H = 3 s = 2 D = 2 W = 4

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Practice Question Calculate the surface area for the cube and rectangular prism: H = 3 s = 2 D = 2 W = 4 Surface Area = 2(W x D) + 2(H x D) + 2(W x H) = 2(4 x 2) + 2(3 x 2) + 2(4 x 3) = 2(8) + 2(6) + 2(12) = SA = 52 Surface Area = 6s² = 6(2²) = 6(4) SA = 24

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Practice Question Calculate the surface area for the sphere and cylinder: r = 2 h = 6 r = 1

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Practice Question Calculate the surface area for the sphere and cylinder: Surface Area = 2π(r²) + 2π(r x h) = 2π(1²) + 2π(1 x 6) = 2π(1) + 2π(6) = SA = 44 r = 2 h = 6 Surface Area = 4π(r²) = 4π(2²) = 4π(4) SA = 50.24 r = 1

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**Volume of Cube and Prism**

Volume of a Cube V = s3 Volume of a Rectangular prism V = W x H x D s W D H

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**Volume of Sphere and Cylinder**

Volume of a Sphere V = 4/3∏(r³) Volume of a Cylinder (V) = ∏(r²)h r h r

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Practice Question Calculate the volume of the cube and rectangular prism: H = 3 b = 2 D = 2 W = 4

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Practice Question Calculate the volume of the cube and rectangular prism: H = 3 s = 2 D = 2 W = 4 Volume = W x D x H V = 4 x 2 x 3 V = 24 Volume = s3 V = (23) V = 8

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**Practice Question Calculate the volume of the sphere and cylinder:**

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**Practice Question Calculate the volume of the sphere and cylinder:**

Volume = ∏(r²)h V = (3.14) (1.52) (6) V = (3.14) (2.25) (6) V = 42.39 r = 2 h = 6 Volume = 4/3∏(r³) V = (1.33) (3.14) (23) V = 4.19 x 8 V = 33.49 r = 1.5

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Modeling and Prototypes Presentation 4.4.1 Explanation © 2011 International Technology and Engineering Educators Association, STEM Center for Teaching.

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