Volume of Pyramids and Cones

Volume of Pyramids and Cones “If I had influence with the good fairy… I should ask that her gift to each child in the world be a sense of wonder so indestructible that it would last throughout life.” Rachel Carson

Exit Activity 8 n 3 4 5 6 10 n+2 5 6 7 8 24 3n 9 12 15 18 16 2n 6 8 10 12 (½*a*s*8 )H (½*P*a )H (½ * b * h)H (l*w)H (½*a*s*5 )H (½*a*s*6 )H

Objectives • Discover formulas for the volumes of pyramids and cones. • Practice three-dimensional visual thinking skills.

Pyramids and Cones What do all these shapes have in common?

Investigation The Volume Formulas for Pyramids and Cones p 522 Pyramid-Cone Volume Conjecture If B is the area of the base of a pyramid or a cone and H is the height of the solid, then the formula for the volume is V = ___________. BH

Example The shape to the left is a regular pentagonal pyramid. Find the volume of the shape. Height = 8cm 6 cm

Solution We must find the length of the apothem B = ½ * a *p B = ½ * 3 (3)*(6)(5) B = ½ * 3 (3)*30 B = 15* 3 (3) B = 45 (3) 6cm V = * B * H 3cm 3cm V = * 45 (3) * 8 Apothem = 3(3) V = 120(3)cm3

Example • What is the height of the cone to the left if the total volume is 324 in3. Volume = BH Volume = r2H 324 = (92)H 324 = 27 * H 27 27 12 = H H 9 in

Volume What makes the volume of a cone and pyramid different from the volume of prisms and cylinders? How can we remember this difference?

Volume of Pyramids and Cones