Glencoe Geometry Interactive Chalkboard Copyright by The Mc

Glencoe Geometry Interactive Chalkboard Copyright © by The Mc. Graw-Hill Companies, Inc. Developed by FSCreations, Inc. , Cincinnati, Ohio 45202 Send all inquiries to: GLENCOE DIVISION Glencoe/Mc. Graw-Hill 8787 Orion Place Columbus, Ohio 43240

Lesson 12 -1 Three-Dimensional Figures Lesson 12 -2 Nets and Surface Area Lesson 12 -3 Surface Areas of Prisms Lesson 12 -4 Surface Areas of Cylinders Lesson 12 -5 Surface Areas of Pyramids Lesson 12 -6 Surface Areas of Cones Lesson 12 -7 Surface Area of Spheres

Example 1 Use Orthogonal Drawings Example 2 Identify Solids Example 3 Slicing Three-Dimensional Figures

Draw the back view of a figure given its orthogonal drawing. Use blocks to make a model. Then use your model to draw the back view.

The top view indicates one row of different heights one column in the front right. and The front view indicates that there are four standing columns. The first column to the left is 2 blocks high, the second column is 3 blocks high, the third column is 2 blocks high, and the fourth column to the far right is 1 block high. The dark segments indicate breaks in the surface. The right view indicates that the front right column is only 1 block high. The dark segments indicate breaks in the surface. Check the left side of your model. The back left column is 2 blocks high. The dark segments indicate breaks in the surface.

Check to see that all views correspond to the model. Now that your model is accurate, turn it around to the back and draw what you see. The blocks are flush, so no heavy segments are needed. Answer:

Draw the corner view of the figure. Turn your model so you are looking at the corners of the blocks. The lowest columns should be in the front so the differences in height between the columns are visible. Connect the dots on isometric dot paper to represent the edges of the solid. Shade the tops of each column.

Answer:

a. Draw the back view of the figure given its orthogonal drawing. Answer:

b. Draw the corner view of the figure. Answer:

Identify the solid. Name the bases, faces, edges, and vertices. The bases are rectangles, and the four remaining faces are parallelograms.

Answer: This solid is a rectangular prism. Bases: Faces: Edges: Vertices: A, B, C, D, E, F, G, H

Identify the solid. Name the bases, faces, edges, and vertices. The bases are circular and congruent. Answer: This solid is a cylinder. Bases:

Identify the solid. Name the bases, faces, edges, and vertices. The base is a triangle, and the remaining three faces meet at a point. Answer: This solid is a triangular pyramid. Base: Faces: Edges: Vertices: A, B, C, D

a. Identify the solid. Name the bases, faces, edges, and vertices. Answer: triangular prism Bases: Faces: Edges: Vertices: A, B, C, E, F, G

b. Identify the solid. Name the bases, faces, edges, and vertices. Answer: cylinder Bases:

c. Identify the solid. Name the bases, faces, edges, and vertices. Answer: pentagonal pyramid Bases: ABCDE Faces: Edges: Vertices: A, B, C, D, E, F

BAKERY A customer ordered a two-layer sheet cake. Describe the possible cross sections of the cake.

Answer: If the cake is cut horizontally, the cross section will be a rectangle. If the cake is cut vertically, the cross section will also be a rectangle.

A solid cone is going to be sliced so that the resulting flat portion can be dipped in paint and used to make prints of different shapes. How should the cone be sliced to make prints of a circle, triangle, and an oval? Answer: If the cone were to be cut parallel to the base, the cross -section would be a circle.

Answer: If the cone were to be cut perpendicular to the base, the slice would be a triangle. If the cone were to be cut on an angle to the base, the slice would be an oval.

Example 1 Draw a Solid Example 2 Nets for a Solid Example 3 Nets and Surface Area

Sketch a rectangular prism 4 units long, 3 units wide, and 2 units high using isometric dot paper. Step 1 Draw the corner of the solid: 2 units down, 4 units to the left, and 3 units to the right.

Sketch a rectangular prism 4 units long, 3 units wide, and 2 units high using isometric dot paper. Step 2 Draw a parallelogram for the top of the solid.

Sketch a rectangular prism 4 units long, 3 units wide, and 2 units high using isometric dot paper. Step 3 Draw segments 2 units down from each vertex for the vertical edges.

Sketch a rectangular prism 4 units long, 3 units wide, and 2 units high using isometric dot paper. Answer: Step 4 Connect the corresponding vertices. Use dashed lines for the hidden edges. Shade the top of the solid.

Sketch a rectangular prism 1 unit high, 5 units long, and 2 units wide. Answer:

MULTIPLE- CHOICE TEST ITEM Which net could be folded into a triangular prism if folds are made only along dotted lines? Read the Test Item You are given four nets, only one of which can be folded into a triangular prism.

Solve the Test Item A triangular prism has two triangular bases and three rectangular faces. Analyze each answer choice carefully and locate the one that has each of these elements. This net has two triangles for the bases, but only one rectangle. This net has four nonoverlapping triangles and a square for the base. It folds into a rectangular pyramid.

This net has four squares and one triangle. This net has three adjacent rectangles and two triangles. The triangles are adjacent to each of the shorter sides of the middle rectangle. Each base and face of the prism is represented. This choice is correct. Answer: D

Which net could be folded into a triangular pyramid if folds are made only along the dotted lines? Answer: B

Draw a net for the triangular prism. We need to know the length of the legs of the isosceles triangular bases in order for us to know the width of two of the rectangular faces of the prism. Use the Pythagorean Theorem.

Pythagorean Theorem Simplify. Take the square root of each side. Use rectangular dot paper to draw a net. Let one unit on the dot paper represent 1 centimeter.

Answer:

Use the net to find the surface area of the prism. To find the surface area of the prism, add the areas of the three rectangles and the two triangles.

Write an equation to solve for the surface area. Answer: The surface area of the triangular prism is 60 square centimeters.

a. Draw a net for the rectangular pyramid. Answer:

b. Use the net to find the surface area of the prism. Answer: 84 cm 2

Example 1 Lateral Area of a Pentagonal Prism Example 2 Surface Area of a Triangular Prism Example 3 Use Surface Area to Solve a Problem

Find the lateral area of the regular hexagonal prism. The bases are regular hexagons. So the perimeter of one base is 6(5) or 30 centimeters. Lateral area of a prism P = 30, h = 12 Multiply. Answer: The lateral area is 360 square centimeters.

Find the lateral area of the regular octagonal prism. Answer: 216 cm 2

Find the surface area of the square prism.

Surface area of a prism L = Ph Substitution Simplify. Answer: The surface area is 360 square centimeters.

Find the surface area of the triangular prism. Answer: 416 units 2

SCULPTURE A solid block of marble will be used for a sculpture. If the block is 3 feet wide, 4 feet long, and feet high, find the surface area of the block. The block is shaped like a rectangular prism.

The perimeter of a base is 2(3) + 2(4) or 14 feet. The area of a base is 3(4) or 12 square feet. Formula for surface area Simplify. Answer: The surface area of the block of marble is 157 square feet.

A television is packaged in a box that is feet long, 2 feet wide, and 2 feet high. Find the surface area of the box. Answer: 28 ft 2

Example 1 Lateral Area of a Cylinder Example 2 Surface Area of a Cylinder Example 3 Find Missing Dimensions

A fruit juice can is cylindrical with aluminum sides and bases. The can is 12 centimeters tall, and the diameter of the can is 6. 3 centimeters. How many square centimeters of aluminum are used to make the sides of the can? The aluminum sides of the can represent the lateral area of the cylinder. If the diameter of the can is 6. 3 centimeters, then the radius is 3. 15 centimeters. The height is 12 centimeters. Use the formula to find the lateral area.

Lateral area of a cylinder Use a calculator. Answer: About 237. 5 square centimeters of aluminum are used to make the sides of the can.

A set of toy blocks are sold in a cylindrical shape container. A product label wraps around all sides of the container without any overlaps or gaps. How much paper is used to make the label the appropriate size if the diameter of the container is 12 inches and the height is 18 inches? Answer: about 678. 6 in 2

Find the surface area of the cylinder. The radius of the base and the height of the cylinder are given. Substitute these values in the formula to find the surface area.

Surface area of a cylinder Use a calculator. Answer: The surface area is approximately 2814. 9 square feet.

Find the surface area of the cylinder. Answer: about 1156. 1 ft 2

Find the radius of the base of a right cylinder if the surface area is square feet and the height is 10 feet. Use the formula for surface area to write and solve an equation for the radius. Surface area of a cylinder Substitution Simplify. Divide each side by

Subtract 264 from each side. Factor. Solve. Since a radius of a circle cannot have a negative value, – 22 is eliminated. Answer: The radius of the base is 12 feet.

Find the radius of the base of a right cylinder if the surface area is square inches and the height is 22 inches. Answer: 14 in.

Example 1 Use Lateral Area to Solve a Problem Example 2 Surface Area of a Square Pyramid Example 3 Surface Area of Pentagonal Pyramid

CANDLES A candle store offers a pyramidal candle that burns for 20 hours. The square base is 6 centimeters on a side and the slant height of the candle is 22 centimeters. Find the lateral area of the candle. We need to find the lateral area of the square pyramid. The sides of the base measure 6 centimeters, so the perimeter is

Lateral area of a regular pyramid Multiply. Answer: The lateral area of the candle is 264 square centimeters.

CAMPING A pyramidal shaped tent is put up by two campers. The square base is 7 feet on a side and the slant height of the tent is 7. 4 feet. Find the lateral area of the tent. Answer: 103. 6 ft 2

Find the surface area of the regular pyramid to the nearest tenth. To find the surface area, first find the slant height of the pyramid. The slant height is the hypotenuse of a right triangle with legs that are the altitude and a segment with a length that is one-half the side measure of the base.

Pythagorean Theorem Use a calculator. Now find the surface area of a regular pyramid. The perimeter of the base is and the area of the base is

Surface area of a regular pyramid Use a calculator. Answer: The surface area is 179. 4 square meters to the nearest tenth.

Find the surface area of the regular pyramid to the nearest tenth. Answer: 89. 8 m 2

Find the surface area of the regular pyramid. Round to the nearest tenth. The altitude, slant height, and apothem form a right triangle. Use the Pythagorean Theorem to find the apothem. Let x represent the length of the apothem.

Pythagorean Theorem Simplify. Now find the length of the sides of the base. The central angle of the hexagon measures Let a represent the measure of the angle formed by a radius and the apothem.

Use trigonometry to find the length of the sides. Multiply each side by 9. Multiply each side by 2. Use a calculator.

Next, find the perimeter and area of the base.

Finally, find the surface area. Surface area of a regular pyramid Simplify. Answer: The surface area is approximately 748. 2 square centimeters.

Find the surface area of the regular pyramid. Answer: about 298. 2 cm 2

Example 1 Lateral Area of a Cone Example 2 Surface Area of a Cone

ICE CREAM A sugar cone has an altitude of 8 inches and a diameter of inches. Find the lateral area of the sugar cone. Explore We are given the altitude and the diameter of the base. We need to find the slant height of the cone.

Plan The radius of the base, height, and slant height form a right triangle. Use the Pythagorean Theorem to solve for the slant height. Then use the formula for the lateral area of a right circular cone.

Solve Write an equation and solve for. Pythagorean Theorem Simplify. Take the square root of each side. Next, use the formula for the lateral area of a right circular cone. Lateral area of a cone Use a calculator.

Examine Use estimation to check the reasonableness of this result. The lateral area is approximately Compared to the estimate, the answer is reasonable. Answer: The lateral area is approximately 31. 8 square inches.

A hat for a child’s birthday party has a conical shape with an altitude of 9 inches and a diameter of 5 inches. Find the lateral area of the birthday hat. Answer: 73. 4 in 2

Find the surface area of the cone. Round to the nearest tenth. Surface area of a cone Use a calculator. Answer: The surface area is approximately 20. 2 square centimeters.

Find the surface area of the cone. Round to the nearest tenth. Answer: about 63. 6 cm 2

Example 1 Spheres and Circles Example 2 Surface Area Example 3 Surface Area

In the figure, O is the center of the sphere, and plane P intersects the sphere in circle R. If OR 6 centimeters and OS 14 centimeters, find RS.

The radius of circle R is segment and S is a point on circle R and on sphere O. Use the Pythagorean Theorem for right triangle ORS to solve for RS. Pythagorean Theorem Simplify. Subtract 36 from each side. Use a calculator. Answer: RS is approximately 12. 6 centimeters.

In the figure, O is the center of the sphere, and plane U intersects the sphere in circle L. If OL 3 inches and LM 8 inches, find OM. Answer: about 8. 5 in.

Find the surface area of the sphere, given a great circle with an area of approximately 907. 9 square centimeters. The surface area of a sphere is four times the area of the great circle. Surface area of a sphere Multiply. Answer: The surface area is approximately 3631. 6 square centimeters.

Find the surface area of a hemisphere with a radius of 3. 8 inches. A hemisphere is half of a sphere. To find the surface area, find half of the surface area of the sphere and add the area of the great circle. Surface area of a hemisphere Substitution Use a calculator. Answer: The surface area is approximately 136. 1 square inches.

a. Find the surface area of the sphere, given a great circle with an area of approximately 91. 6 square centimeters. Answer: about 366. 4 cm 2 b. Find the surface area of a hemisphere with a radius of 6. 4 inches. Answer: about 386. 0 in 2

Find the surface area of a ball with a circumference of 24 inches to determine how much leather is needed to make the ball. First, find the radius of the sphere. Circumference of a circle Use a calculator.

Next, find the surface area of the sphere. Surface area of a sphere Use a calculator. Answer: The surface area is approximately 183. 3 square inches.

Find the surface area of a ball with a circumference of 18 inches to determine how much leather is needed to make the ball. Answer: about 103. 1 in 2

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