Illustrative Math Alignment: Grade 6 Unit 1

In this program we look at applications of area and volume. We do this in the context of three real-world applications. In the first, we look at the sinking of the Titanic in the context of volume and density. In the second application we look at the glass pyramid at the Louvre Museum and calculate its surface area. In the third application we look at the Citibank Tower in New York City to study the ratio of surface area to volume to learn about heat loss in tall buildings.

The sinking of the Titanic provides an opportunity to explore volume, density, and buoyancy. Students construct a mathematical model of the Titanic to determine why it sank and what could have been done to prevent it from sinking.

The glass-paneled pyramid at the Louvre Museum in Paris is a tessellation of rhombus-shaped glass panels. Students create a model of the pyramid to calculate the number of panels used to cover the surface area of the pyramid.

The Citibank Tower in New York City presents some unique design challenges. In addition it has to cope with a problem that all tall structure have to deal with: heat loss. By managing the ratio of surface area to volume, a skyscraper can effective manage heat loss.

Topic

3D Geometry

Definition

Surface area is the total area that the surface of a three-dimensional object occupies.

Description

In the realm of three-dimensional geometry, surface area is a fundamental concept that quantifies the extent of a 3D shape's exterior surface. This measure is crucial for various applications, including engineering, architecture, and everyday tasks. For example, when painting a room, the surface area of the walls, ceiling, and floor must be calculated to determine the amount of paint required.

The formula for the Surface Area of a Cube.

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The formula for the Surface Area of a Cylinder.

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The formula for the Surface Area of a Rectangular Prism.

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The formula for the Surface Area of a Sphere.

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The formula for the Surface Area of a Square Pyramid.

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The formula for the Surface Area of a Triangular Prism.

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The formula for the Surface Area of a Triangular Pyramid.

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Topic

Geometric Models

This set of tutorials provides an overview of the 24 worked-out examples that show how to calculate the surface area of different three-dimensional figures.

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Topic

Geometry Concepts

Description

This image presents an irregular polygon superimposed on a square grid. The grid serves as a crucial tool for estimating the area of the polygon by providing a systematic method of counting squares.

To estimate the area, students should count the number of full squares entirely contained within the irregular polygon. For partial squares intersected by the polygon's edges, they need to approximate what fraction of each square is covered. By summing these full and partial square counts, students can arrive at a reasonable estimate of the polygon's area.

Topic

Geometry Concepts

Description

This image showcases another closed curve area superimposed on a square grid. The grid acts as a vital tool for estimating the area enclosed by the curve through a square-counting method.

To estimate the area, students should begin by counting the number of whole squares that are completely enclosed within the closed curve. For squares that are partially covered by the curve, students need to approximate what fraction of each square is included. By combining these full and partial square counts, students can arrive at a reasonable estimate of the area within the closed curve.

Topic

Geometry Concepts

Description

This image showcases another irregular polygon overlaid on a square grid. The grid functions as an essential tool for estimating the area of the polygon through a square-counting method.

To estimate the area, students should first count the number of complete squares that fall entirely within the boundaries of the irregular polygon. For squares that are only partially covered by the polygon, students need to assess what fraction of each square is included. By adding these full and partial square counts, students can develop a reasonable estimate of the polygon's total area.

Topic

Geometry Concepts

Description

This image depicts an irregular polygon placed on a square grid. The grid acts as a vital tool for estimating the area of the polygon using a square-counting approach.

To estimate the area, students should begin by counting the number of whole squares that are completely enclosed within the irregular polygon. For squares that are partially covered by the polygon's edges, students need to approximate what portion of each square is included. By combining these full and partial square counts, students can arrive at a reasonable estimate of the polygon's total area.

Topic

Geometry Concepts

Description

This image features an irregular polygon superimposed on a square grid. The grid serves as a crucial tool for estimating the area of the polygon through a square-counting method.

To estimate the area, students should first count the number of complete squares that fall entirely within the boundaries of the irregular polygon. For squares that are only partially covered by the polygon, students need to assess what fraction of each square is included. By summing these full and partial square counts, students can develop a reasonable estimate of the polygon's total area.

Topic

Geometry Concepts

Description

This image showcases a closed curve area outlined on a square grid. The grid functions as an essential tool for estimating the area enclosed by the curve through a square-counting approach.

To estimate the area, students should first count the number of complete squares that fall entirely within the closed curve. For squares that are only partially enclosed by the curve, students need to assess what fraction of each square is included. By adding these full and partial square counts, students can develop a reasonable estimate of the total area within the closed curve.

Topic

Geometry Concepts

Description

This image presents a closed curve area superimposed on a square grid. The grid serves as a crucial tool for estimating the area enclosed by the curve using a square-counting method.

To estimate the area, students should begin by counting the number of whole squares that are completely enclosed within the closed curve. For squares that are partially covered by the curve, students need to approximate what portion of each square is included. By combining these full and partial square counts, students can arrive at a reasonable estimate of the total area within the closed curve.

Topic

Geometry Concepts

Description

This image features a closed curve area outlined on a square grid. The grid acts as a vital tool for estimating the area enclosed by the curve through a square-counting approach.

To estimate the area, students should first count the number of complete squares that fall entirely within the closed curve. For squares that are only partially enclosed by the curve, students need to assess what fraction of each square is included. By summing these full and partial square counts, students can develop a reasonable estimate of the total area within the closed curve.

Topic

Geometry Concepts

Description

This image depicts a radiating figure superimposed on a square grid. The grid serves as an essential tool for estimating the area of the figure using a square-counting method.

To estimate the area, students should begin by counting the number of whole squares that are completely enclosed within the radiating figure. For squares that are partially covered by the figure's edges, students need to approximate what portion of each square is included. By combining these full and partial square counts, students can arrive at a reasonable estimate of the figure's total area.

Topic

Geometry Concepts

Description

This image presents a closed curve area outlined on a square grid. The grid functions as a crucial tool for estimating the area enclosed by the curve using a square-counting approach.

To estimate the area, students should first count the number of complete squares that fall entirely within the closed curve. For squares that are only partially enclosed by the curve, students need to assess what fraction of each square is included. By summing these full and partial square counts, students can develop a reasonable estimate of the total area within the closed curve.

Topic

Surface Area

Description

This example demonstrates the calculation of surface area for a rectangular prism. The prism has dimensions of 25, 15, and 12 units. The surface area is determined by summing the areas of all faces, which is represented by the formula SA = 2ab + 2bc + 2ac. For this specific prism, the calculation yields a surface area of 1710 square units.

Topic

Surface Area

Description

This example presents a general formula for calculating the surface area of a cylinder with radius y and height x. The surface area consists of two circular ends and the rectangular region forming the curved surface. The formula used is SA = 2πr2 + 2πrh, which simplifies to SA = 2πy(x + y), where y is the radius and x is the height of the cylinder.

Topic

Surface Area

Description

This example demonstrates the calculation of surface area for a hollow cylinder with an outer radius of 10 units, an inner radius of 9 units, and a height of 15 units. The surface area is determined by subtracting the area of the hollow region from the full cylinder's surface area. The formula used is SA = 2πr12 - 2πr22 + 2πr1h, where r1 is the outer radius and r2 is the inner radius. For this specific hollow cylinder, the calculation yields a surface area of 338π square units.

Topic

Surface Area

Description

This example presents a general formula for calculating the surface area of a hollow cylinder with an outer radius y, an inner radius y - 1, and a height x. The surface area is determined by subtracting the hollow region's area from the full cylinder's surface area. The formula used is SA = 2πr12 - 2πr22 + 2πr1h, which simplifies to SA = 2π(2y + xy - 1), where r1 is the outer radius (y) and r2 is the inner radius (y - 1).

Topic

Surface Area

Description

This example demonstrates the calculation of surface area for a rectangular-based pyramid with dimensions 11, 9, and 15 units. The pyramid is depicted with its net unfolded, showing two triangular faces and a rectangular base. The surface area is calculated using the formula SA = b1 * h + b2 * h + l * w, where b1 and b2 are the bases of the triangular faces, h is the height of the triangular faces, and l and w are the length and width of the rectangular base. For this specific pyramid, the calculation yields a surface area of 399 square units.

Topic

Surface Area

Description

This example presents a general formula for calculating the surface area of a rectangular-based pyramid with dimensions labeled as x, y, and z. The pyramid is depicted with its net unfolded into two triangular faces and one rectangular base. The surface area is calculated using the formula SA = b1 * h + b2 * h + l * w, which simplifies to SA = xz + yz + xy, where x and y are the dimensions of the rectangular base, and z is the height of the triangular faces.

Topic

Surface Area

Topic

Surface Area

Description

This example presents a general formula for calculating the surface area of a truncated rectangular-based pyramid with dimensions labeled as x - 3, y - 3, z, x, and y. The virtual pyramid is shown with its net unfolded into four parallelogram regions and two rectangles. The surface area is calculated using the formula SA = 2yz + 2xz + xy + (x - 3)(y - 3), which simplifies to SA = (yz + xz + xy) - (x + y - 9).

Topic

Surface Area

Description

This example demonstrates the calculation of surface area for a sphere with a radius of 3 units. The surface area is calculated using the formula SA = 4πr2, where r is the radius of the sphere. For this specific sphere, the calculation yields a surface area of 36π square units.

Topic

Surface Area

Description

This example presents a general formula for calculating the surface area of a sphere with an unknown radius labeled as "x". The surface area is calculated using the formula SA = 4πr2, which in this case becomes SA = 4πx2, where x is the radius of the sphere.

Topic

Surface Area

Description

This example demonstrates the calculation of surface area for a cube with a side length of 7 units. The surface area is calculated using the formula SA = 6s2, where s is the length of one side of the cube. For this specific cube, the calculation yields a surface area of 294 square units.

Topic

Surface Area

Description

This example introduces a general formula for calculating the surface area of a rectangular prism with dimensions x, y, and z. The surface area is computed using the formula SA = 2xy + 2xz + 2yz, which can be simplified to 2(xy + xz + yz). This generalized approach allows for easy application to prisms of various sizes.

Surface area calculations are fundamental in mathematics, with applications ranging from manufacturing to environmental science. This collection of examples aids in teaching the topic by presenting both specific and general cases, helping students understand how to apply formulas flexibly to different scenarios.

Topic

Surface Area

Description

This example presents a general formula for calculating the surface area of a cube with an unknown side length labeled as "x". The surface area is calculated using the formula SA = 6s2, which in this case becomes SA = 6x2, where x is the length of one side of the cube.

Topic

Surface Area

Description

This example demonstrates the calculation of surface area for a hollow cube with an outer side length of 9 units and an inner hollow square side length of 7 units. The surface area is calculated using the formula SA = 6 * s12 - 2 * s22, where s1 is the outer side length and s2 is the inner side length. For this specific hollow cube, the calculation yields a surface area of 388 square units.

Topic

Surface Area

Description

This example presents a general formula for calculating the surface area of a hollow cube with variable dimensions. The outer side length is labeled as "x" and the inner hollow side length as "x - 2". The surface area is calculated using the formula SA = 6 * s12 - 2 * s22, where s1 = x and s2 = x - 2. After expanding and simplifying, the result is SA = 4(x2 + 2x - 2) square units.

Topic

Surface Area

Description

This example demonstrates the calculation of surface area for a cone with a radius of 5 units and a slant height of 13 units. The surface area is calculated using the formula SA = π * r2 + π * r * s, where r is the radius and s is the slant height. For this specific cone, the calculation yields a surface area of 90π square units.

Topic

Surface Area

Description

This example presents a general formula for calculating the surface area of a cone with variable dimensions. The radius is labeled as "x" and the slant height as "y". The surface area is calculated using the formula SA = π * r2 + π * r * s, where r = x and s = y. This simplifies to SA = π * x(x + y) square units.

Topic

Surface Area

Description

This example demonstrates the calculation of surface area for a hollow rectangular prism. The outer dimensions are 60, 20, and 16 units, while the inner dimensions are 18 and 14 units. The surface area is determined by subtracting the area of the hollow region from the surface area of the full rectangular prism, resulting in a surface area of 4456 square units.

Topic

Surface Area

Description

This example presents a general formula for calculating the surface area of a hollow rectangular prism. The outer dimensions are represented by x, y, and z, while the hollow dimensions are represented by (y - 2) and (x - 2). The surface area is determined by subtracting the hollow region's surface area from that of the full rectangular prism, resulting in the formula SA = 2(xz + yz + 2x + 2y - 4).

Topic

Surface Area

Description

This example demonstrates the calculation of surface area for a triangular prism with an equilateral triangular base. The base is 12 units, the height is 6 * √(3), and the length of the prism is 25 units. The surface area is calculated using the formula SA = b * h + 3 * l * w, where b is the base, h is the height of the triangle, l is the length of the prism, and w is the width of the triangle. The final result is 72 * √(3) + 900 square units.

Topic

Surface Area

Description

This example presents a general formula for calculating the surface area of a triangular prism with an equilateral triangular base. The dimensions are represented by x for the base, y for the height of the triangle, and z for the length of the prism. The surface area is determined using the formula SA = b * h + 3 * l * w, which simplifies to SA = xy + 3zx = x(y + 3z).

Topic

Surface Area

Description

This example demonstrates the calculation of surface area for a hollow triangular prism with an equilateral triangular base. The outer triangle has a base of 10 units and a height of 5 * √(3), while the inner triangle has a base of 8 units and a height of 4 * √(3). The length of the prism is 35 units. The surface area is calculated by subtracting the area of the hollow region from the full prism's surface area, resulting in SA = b1 * h1 - b2 * h2 + 3 * l * w = (18 * √(3)) + 1050 square units.

Topic

Surface Area

Description

This example presents a general formula for calculating the surface area of a hollow triangular prism with an equilateral triangular base. The outer triangle has a base labeled as x, and the inner triangle has a base labeled as a. The heights are given as √(3)/2 for both triangles, and the length of the prism is z units. The surface area is determined by subtracting the area of the hollow region from the full prism's surface area, resulting in the formula SA = (√(3)/2) (x2 - a2) + 3zx.

Topic

Surface Area

Description

This example demonstrates the calculation of surface area for a cylinder with a radius of 10 units and a height of 8 units. The surface area consists of two circular ends and the rectangular region forming the curved surface. The formula used is SA = 2πr2 + 2πrh, where r is the radius and h is the height. For this specific cylinder, the calculation yields a surface area of 360π square units.

This Teacher's Guide provides an overview of the 24 worked-out examples that show how to calculate the surface area of different three-dimensional figures.

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11/14/11. In this issue we commemorate the 125th anniversary of the Statue of Liberty. We also look at the geometry and architecture of this monument.

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