Chapter 6: Perimeter, Area, and Volume

Regular Math

Section 6.1: Perimeter & Area of Rectangles & Parallelograms

Perimeter OUTSIDE

– the distance around the of a figure

Area

– the number of square units

INSIDE

a figure

Finding the Perimeter of Rectangles and Parallelograms Find the perimeter of each figure.

P = S + S + S + S P = 26 + 20 + 26 + 20 P = 92 feet

Try this one on your own…

Find the perimeter of each figure.

P = S + S + S + S P = 17.5x + 11x + 17.5x + 11x P = 57X units

Using a Graph to Find Area

Graph each figure with the given vertices. Then find the area of each figure. (-3, -1), (-3, 4), (1, 4), (1, -1) A = bH b = base ; H = height A = 4 X 5 A = 20 units squared

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Graph each figure with the given vertices. Then find the area of the figure.

(-4, 0), (2, 0), (4, 3), (-2, 3) A = bH A = 6 x 3 A = 18 units squared

Finding Area and Perimeter of a Composite Figure Step One: Fill in the missing sides.

Find the perimeter and area of the figure. Step Two: Solve for Perimeter Step Three: Break the figure into rectangles.

Step Four: Solve for Area of each rectangle.

Step Five: Add the areas of each individual rectangles.

Section 6.2: Perimeter and Area of Triangles and Trapezoids Find the perimeter of each figure.

P = S + S + S P = 22 + 22 + 27 P = 71 feet

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Find the perimeter of each figure.

P = S + S + S P = 2.5x + 5y + 2x + 2x + 4y P = 6.5x + 9y

Find the area of triangles and trapezoids.

A = ½ x h x (b1 + b2) Graph and find the area of each figure with the given vertices.

(-1,-3), (0,2), (3,2), (3, -3) A = ½ x 5 x (4 +3) A = ½ x 5 x (7) A = 2.5 x 7 A =17.5 units squared

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A = ½ x h x (B1 + B2) A = ½ x 3 x (3 + 5) A = ½ x 3 x (8) A = 1.5 x 8 A = 12 units squared Graph and find the area of each figure with the given vertices. (-3,-2), (-3,1), (0,1), (2, -2)

Section 6.3: The Pythagorean Theorem

Example 1: Finding the length of the hypotenuse.

Find the length of the hypotenuse.

Graph the triangle with coordinates (6,1), (0,9), and (0,1).

Find the length of the hypotenuse.

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Find the length of the hypotenuse.

Graph the triangle with the following coordinates (1,-2), (1,7), and (13,-2).

A = 9 B = 12 Find the length of the hypotenuse. C = 15 C = 6.40

Example 2: Finding the length of a Leg in a Right Triangle Solve for the unknown side in the right triangle.

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Solve for the unknown side in the right triangle.

b = 24

Example 3: Using the Pythagorean Theorem to Find Area Use the Pythagorean Theorem to find the height of the triangle. Then, use the height to find the area of the triangle.

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Use the Pythagorean Theorem to find the height of the triangle.

h = square root of 20 or 4.47

Then, use the height to find the area of the triangle.

A = 17.89 units squared

Section 6.4: Circles

Finding the circumference of a Circle.

Find the circumference of each circle, both in terms of pi and to the nearest tenth. Use 3.14 for pi.

Circle with radius 5 cm Circle with diameter 1.5 in

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Find the circumference of each circle, both in terms of pi and to the nearest tenth. Use 3.14 for pi.

Circle with radius 4 m C = 8pi m or 25.1 m Circle with diameter 3.3 ft C = 3.3pi or 10.4 ft

Finding the Area of a Circle.

Find the area of each circle, both in terms of pi and to the nearest tenth. Use 3.14 for pi.

Circle with radius 5 cm Circle with diameter 1.5 in

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Find the area of each circle, both in terms of pi and to the nearest tenth. Use 3.14 for pi.

Circle with radius 4 in A = 16pi inches squared or 50.2 inches squared Circle with diameter 3.3 m A = 2.7225pi meters squared or 8.5 meters squared

Finding Area and Circumference on a Coordinate Plane.

Graph the circle with center (-1,1) that passes through (-1,3). Find the area and circumference, both in terms of pi and to the nearest tenth. Use 3.14 for pi. Step One: Graph Circle Step Two: Find the radius Step Three: Use the Area and Circumference Formula

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Graph the circle with center (-2,1) that passes through (1,-1). Find the area and circumference, both in terms of pi and to the nearest tenth. Use 3.14 for pi.

A = 9pi units squared and 28.3 units squared C = 6pi units and 18.8 units

A bicycle odometer recorded 147 revolutions of a wheel with diameter 4/3 ft. How far did the bicycle travel? Use 22/7 for pi.

The distance traveled is the circumference of the wheel times the number of revolutions.

C = pi(d) = (22/7) (4/3) = 88/21 Circumference x Revolutions 88/21 x 147 = 616 feet

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A Ferris wheel has a diameter of 56 feet and makes 15 revolutions per ride. How far would someone travel during a ride? Use 22/7 for pi.

C = 22/7(56) = 176 feet Distance = 176 (15) = 2640 feet

Section 6.5: Drawing Three Dimensional Figures

Example 1: Drawing a Rectangular Box

 Use isometric dot paper to sketch a rectangular box that is 4 units long, 2 units wide, and 3 units high.

    Step 1: Lightly draw the edges of the bottom face. It will look like a parallelogram.

 2 units by 4 units Step 2: Lightly draw the vertical line segments from the vertices of the base.  3 units high Step 3: Lightly draw the top face by connecting the vertical lines to form a parallelogram.

 2 units by 4 units Step 4: Darken the lines.

 Use solid lines for the edges that are visible and dashed lines for the edges that are hidden.

Example 2: Sketching a One-Point Perspective Drawing Step 1: Draw a rectangle.

This will be the front face.

Label the vertices A through D.

Step 2: Mark a vanishing point “V” somewhere above your rectangle, and draw a dashed line from each vertex to “V”.

Step 3: Choose a point “G” on line BV. Lightly draw a smaller rectangle that has G as one of its vertices.

Step 4: Connect the vertices of the two rectangles along the dashed lines.

Step 5: Darken the visible edges, and draw dashed segments for the hidden edges. Erase the vanishing point and all the lines connecting it to the vertices.

Example 3: Sketching a Two-Point Perspective Drawing Step 1: Draw a vertical segment and label it AD. Draw a horizontal line above segment AD. Label vanishing points V and W on the line. Draw dashed segments AV, AW, DV, and DW.

Step 2: Label point C on segment DV and point E on segment DW. Draw vertical segments through C and E. Draw segment EV and CW.

Step 3: Darken the visible edges. Erase horizon lines and dashed segments.

Section 6.6: Volume of Prisms and Cylinders

Example 1: Finding the Volume of Prisms and Cylinders Find the volume of each figure to the nearest tenth.

Step One: Figure out what formula to use.

Step Two: Plug the numbers into the formula.

Step Three: Solve

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Find the volume of each figure to the nearest tenth.

Example 2: Exploring the Effects of Changing Dimensions A juice can has a radius of 1.5 inches and a height of 5 inches. Explain whether doubling the height of the can would have the same effect on the volume as doubling the radius.

Original Double Radius Double Height

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A juice can has a radius of 2 inches and a height of 5 inches. Explain whether tripling the height would have the same effect on the volume as tripling the radius.

Example 1: Finding the Volume of Prisms and Cylinders Find the volume of each figure to the nearest tenth.

A rectangular prism with base 1 meter by 3 meters and height of 6 meters

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Find the volume of each figure to the nearest tenth.

A rectangular prism with base 2 cm by 5 cm and a height of 3cm

Example 2: Exploring the Effects of Changing Dimensions A juice box measures 3 inches by 2 inches by 4 inches. Explain whether doubling the length, width, or height of the box would double the amount of juice the box holds.

Original Length Width Height

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A juice box measures 3 inches by 2 inches by 4 inches. Explain whether tripling the length, width, or height would triple the amount of juice the box holds.

Original Length Width Height

Example 3: Construction Application Kansai International Airport is a man-made island that is a rectangular prism measuring 60 ft deep, 4000 ft wide, and 2.5 miles long. What is the volume of rock, gravel, and concrete that was needed to build the island?

Try this one on your own… A section of an airport runway is a rectangular prism measuring 2 feet thick, 100 feet wide, and 1.5 miles long. What is the volume of material that was needed to build the runway?

Example 4: Finding the Volume of Composite Figures Find the volume of the milk carton.

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Find the volume of the barn.

Section 6.7: Volume of Pyramids and Cones

Example 1: Finding the Volume of Pyramids and Cones Find the volume of each figure.

Try this one on your own… Find the volume of each figure.

Example 2: Exploring the Effects of Changing Dimensions A cone has a radius 7 feet and height 14 feet. Explain whether tripling the height would have the same effect on the volume of the cone as tripling the radius.

Original Triple Height Triple Radius

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A cone has a radius 3 feet and height 4 feet. Explain whether doubling the height would have the same effect on the volume as doubling the radius.

Original Double Height Double Radius

Example 1: Finding the Volume of Pyramids and Cones Find the volume of each figure.

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Find the volume of each figure.

Example 3: Social Studies Application The Great Pyramid of Giza is a square pyramid. Its height is 481 feet, and its base has 756 feet sides. Find the volume of the pyramid.

Try these on your own… The pyramid of Kukulcan in Mexico is a square pyramid. Its height is 24 meters and its base has 55 meter sides. Find the volume of the pyramid.

Section 6.8: Surface Area of Prisms and Cylinders

Example 1: Finding Surface Area Find the surface area of each figure.

Try this one on your own…

Try this one on your own… Find the surface area of each figure.

Find the surface area of each figure.

Example 1: Finding Surface Area Finding the surface area of each figure.

Try this one on your own… Finding the surface area of each figure.

Example 2: Exploring the Effects of Changing Dimensions A cylinder has a diameter of 8 inches and a height of 3 inches. Explain whether doubling the height would have the same effect on the surface area as doubling the radius.

Original Double Height Double Radius

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Original Triple Radius Triple Height A cylinder has a diameter of 8 inches and a height of 3 inches. Explain whether tripling the height would have the same effect on the surface area as tripling the radius.

Example 3: Art Application

A web site advertises that it can turn your photo into an anamorphic image. To reflect the picture, you need to cover a cylinder that is 32mm in diameter and 100 mm tall with reflective material. How much reflective material do you need?

Try this one on your own… A cylindrical soup can has a radius of 7.6 cm and is 11.2 cm tall. What is the area of the label that covers the side of the can?

Section 6.9: Surface Area of Pyramids and Cones

Example 1: Finding Surface Area Find the surface area of each figure.

Try this one on your own… Find the surface area of each figure.

Try this one on your own… Find the surface area of each figure.

Find the surface area of each figure.

Example 1: Finding Surface Area Try this one on your own… Find the surface area of each figure.

Find the surface area of each figure.

Example 2: Exploring the Effects of Changing Dimensions A cone has a diameter 8 in. and slant height 5 in. Explain whether doubling the slant height would have the same effect on the surface area as doubling the radius.

Original Double Slant Height Double Radius

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Original Triple Radius Triple Slant Height A cone has diameter of 8 in. and slant height 3 in. Explain whether tripling the slant height would have the same effect on the surface area as tripling the radius.

Example 3: Life Science Application An ant lion pit is an inverted cone with the dimensions shown. What is the lateral surface area of the pit?

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The upper portion of an hourglass is approximately an inverted cone with the given dimensions. What is the lateral surface area of the upper portion of the hourglass?

Section 6.10: Spheres

Example 1: Finding the Volume of a Sphere Find the volume of a sphere with a radius of 6 ft, both in terms of pi and to the nearest tenth.

Try this one on your own… Find the volume of a sphere with radius 9 cm, both in terms of pi and to the nearest tenth.

Example 2: Finding Surface Area of a Sphere Find the surface area, both in terms of pi and to the nearest tenth.

Try this one on your own… Find the surface area, both in terms of pi and to the nearest tenth.

Example 3: Comparing Volumes and Surface Areas Compare the volume and surface area of a sphere with radius 21 cm with that of a rectangular prism measuring 28 x 33 x 42cm.

Sphere – Volume Sphere – Surface Area Prism – Volume Sphere – Surface Area

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Compare volumes and surface areas of a sphere with radius 42 cm and a rectangular prism measuring 44 cm by 84 cm by 84 cm.

Sphere – Sphere – Volume Surface Area Prism – Volume Prism – Surface Area