12.5 You Notes Volume of Pyamids and Cones Goal p Find volumes of pyamids and cones. THEOREM 12.9: VOLUME OF A PYRAMID The volume V of a pyamid is V 5, h whee B is the aea of the base and h is the height. B Example 1 Find the volume of a solid Descibe the solid. Find its volume. In an oblique pyamid the vetex is not diectly ove the cente of the base. By Cavaliei's Pinciple, the volume fomula fo a pyamid woks fo oblique pyamids. The solid is a hexagonal pyamid. Fist, find the aea of the base using the fomula fo the aea of a egula polygon, 1 } ap. The apothem a of the hexagon is 2 inches and the peimete P is 6 p 5 inches. So, the aea of the base B is 1 } 2 ( ) ( ) 5 in. 2. Then, find the volume. 10 in. 4 in. V 5 } 1 Bh Fomula fo volume of a pyamid 3 5 } 1 3 ( )( ) Substitute. 5 Simplify. < Use a calculato. The volume of the pyamid is about inches. cubic 4 in. 2 3 in. Copyight Holt McDougal. All ights eseved. Lesson 12.5 Geomety Notetaking Guide 341
12.5 You Notes Volume of Pyamids and Cones Goal p Find volumes of pyamids and cones. THEOREM 12.9: VOLUME OF A PYRAMID The volume V of a pyamid is V 5 } 1 3 Bh, whee B is the aea of the base and h is the height. h B Example 1 Find the volume of a solid Descibe the solid. Find its volume. In an oblique pyamid the vetex is not diectly ove the cente of the base. By Cavaliei's Pinciple, the volume fomula fo a pyamid woks fo oblique pyamids. The solid is a hexagonal pyamid. Fist, find the aea of the base using the fomula fo the aea of a egula polygon, 1 } ap. The apothem a of the hexagon is 2 2 Ï } 3 inches and the peimete P is 6 p 4 5 24 inches. So, the aea of the base B is 1 } 2 ( 2 Ï } 3 )( 24 ) 5 24 Ï } 3 in. 2. Then, find the volume. 10 in. 4 in. V 5 } 1 Bh Fomula fo volume of a pyamid 3 5 } 1 3 ( 24 Ï } 3 )( 10 ) Substitute. 5 80 Ï } 3 Simplify. < 138.56 Use a calculato. The volume of the pyamid is about 138.56 cubic inches. 4 in. 2 3 in. Copyight Holt McDougal. All ights eseved. Lesson 12.5 Geomety Notetaking Guide 341
THEOREM 12.10: VOLUME OF A CONE The volume V of a cone is V 5 5, h whee B is aea of the base, h is the height, and is the adius of the base. B Example 2 Find volumes of cones Find the volume of the cone. a. Right cone b. Oblique cone 16.8 mm 12 ft 9.5 mm 4.5 ft a. Use the fomula fo volume of a cone. 3 π2 h Fomula fo volume of a cone In an oblique cone the vetex is not diectly ove the cente of the base. By Cavaliei's Pinciple, the volume fomula fo a ight cone woks fo oblique cones. 5 1 } 3 π ( ) 2 ( ) Substitute. 5 π Simplify. < Use a calculato. The volume of the ight cone is about mm 3. b. Use the fomula fo volume of a cone. 3 π2 h Fomula fo volume of a cone 5 1 } 3 π ( ) 2 ( ) Substitute. 5 π Simplify. < Use a calculato. The volume of the oblique cone is about ft 3. 342 Lesson 12.5 Geomety Notetaking Guide Copyight Holt McDougal. All ights eseved.
THEOREM 12.10: VOLUME OF A CONE The volume V of a cone is V 5 } 1 3 Bh 5 } 1 3 π2 h, whee B is aea of the base, h is the height, and is the adius of the base. B h Example 2 Find volumes of cones Find the volume of the cone. a. Right cone b. Oblique cone 16.8 mm 12 ft 9.5 mm 4.5 ft a. Use the fomula fo volume of a cone. 3 π2 h Fomula fo volume of a cone In an oblique cone the vetex is not diectly ove the cente of the base. By Cavaliei's Pinciple, the volume fomula fo a ight cone woks fo oblique cones. 5 1 } 3 π ( 9.5 ) 2 ( 16.8 ) Substitute. 5 505.4 π Simplify. < 1587.8 Use a calculato. The volume of the ight cone is about 1587.8 mm 3. b. Use the fomula fo volume of a cone. 3 π2 h Fomula fo volume of a cone 5 1 } 3 π ( 4.5 ) 2 ( 12 ) Substitute. 5 81 π Simplify. < 254.5 Use a calculato. The volume of the oblique cone is about 254.5 ft 3. 342 Lesson 12.5 Geomety Notetaking Guide Copyight Holt McDougal. All ights eseved.
Example 3 Find the volume of the ight cone. Use tigonomety to find the volume of a cone 12 m 12 m To find the adius of the base, use tigonomety. 628 5 opp. } Wite atio. adj. 628 628 tan 628 5 Substitute. 5 < Solve fo. Use the fomula fo the volume of a cone. 3 ( π 2 )h < 1 } 3 π ( ) 2 ( ) < The volume of the cone is about cubic metes. Example 4 Find volume of a composite solid Find the volume of the solid shown. The cone and the cylinde ae ight. 3 cm 7 cm Volume of solid 5 Volume of 1 Volume of 5 π 2 h 1 1 } 3 π2 h 5 π( ) 2 ( ) 1 } 1 3 π ( ) 2 ( ) Substitute. 5 π 1 π Simplify. 5 Use a calculato. The volume of the solid is about centimetes. cubic 3 cm Copyight Holt McDougal. All ights eseved. Lesson 12.5 Geomety Notetaking Guide 343
Example 3 Find the volume of the ight cone. Use tigonomety to find the volume of a cone 12 m 12 m To find the adius of the base, use tigonomety. tan 628 5 opp. } Wite atio. adj. 628 628 tan 628 5 12 } Substitute. 5 12 tan 628 < 6.38 Solve fo. Use the fomula fo the volume of a cone. 3 ( π 2 )h < 1 } 3 π ( 6.38 ) 2 ( 12 ) < 511.5 The volume of the cone is about 511.5 cubic metes. Example 4 Find volume of a composite solid Find the volume of the solid shown. The cone and the cylinde ae ight. 3 cm 7 cm Volume of solid 5 Volume of cylinde 1 Volume of cone 5 π 2 h 1 1 } 3 π2 h 5 π( 3 ) 2 ( 7 ) 1 1 } 3 π ( 3 ) 2 ( 3 ) Substitute. 5 63 π 1 9 π Simplify. 5 226.19 Use a calculato. The volume of the solid is about 226.19 cubic centimetes. 3 cm Copyight Holt McDougal. All ights eseved. Lesson 12.5 Geomety Notetaking Guide 343
Checkpoint Find the volume of the solid. Round you answe to two decimal places. 1. Pyamid with egula base 15 cm 9 cm 2. Right cone 10 ft 6 ft 3. Right cone 14 yd 328 4. Composite solid Homewok 344 Lesson 12.5 Geomety Notetaking Guide Copyight Holt McDougal. All ights eseved.
Checkpoint Find the volume of the solid. Round you answe to two decimal places. 1. Pyamid with egula base 15 cm 9 cm 1052.22 cm 3 2. Right cone 10 ft 6 ft 301.59 ft 3 3. Right cone 14 yd 328 574.82 yd 3 4. Composite solid Homewok 97.92 m 3 344 Lesson 12.5 Geomety Notetaking Guide Copyight Holt McDougal. All ights eseved.