SA B 1 p where is the slant height of the pyramid. V 1 3 Bh. 3D Solids Pyramids and Cones. Surface Area and Volume of a Pyramid

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1 Accelerated AAG 3D Solids Pyramids and Cones Name & Date Surface Area and Volume of a Pyramid The surface area of a regular pyramid is given by the formula SA B 1 p where is the slant height of the pyramid. 2 The volume of a pyramid is given by the formula V 1 3 Bh. 1) Use the square pyramid to the right to answer the following questions. A) Name the base. B) Name a lateral face. C) Name the apex. D) Name the height. E) Name the slant height. F) If the perimeter of the base is 48 inches and the height is 8 inches, find the slant height. G) Calculate the surface area of the pyramid. F) Calculate the volume.

2 Surface Area and Volume of a Cone The formulas for surface area and volume of a cone are the same as the formulas for a pyramid, except the base is a circle. SA B 1 p where is the slant height of the cone. 2 h Ä V 1 3 Bh r 2) Use the right cone to the right to answer the following. A) Find the slant height of the cone. B) Find the total surface area of the cone. C) Find the volume of the cone.

3 Accelerated AAG 3D Solids Pyramids and Cones Name & Date 3) Find the surface area and volume of the cone. Surface Area Volume 4) Find the surface area and volume of the square pyramid. Surface Area Volume

4 5) Find the surface area and volume of the rectangular pyramid. Surface Area Volume 6) Find the exact surface area and volume of the regular hexagonal pyramid with a base edge of 16 feet and a slant height of 24 feet. Surface Area Volume

5 Accelerated AAG 3D Solids Pyramids and Cones Name & Date 7) Find the surface area and volume of the regular pentagonal pyramid with a base edge of 10 feet and an altitude of 16 feet. Surface Area Volume 8) A right cone has a slant height of 25in and a lateral area of 175 in 2. A) Find the radius of the base. B) Calculate the total surface area. C) Calculate the volume of the cone.

6 9) Find the exact surface area and volume of the regular tetrahedron with an edge of length 24 inches. A tetrahedron is a solid with 4 equilateral triangles as faces. Surface Area Volume 10) A square pyramid has a volume of 900in 3 and a height of 45in. What is the exact length of a base edge?

7 Accelerated AAG 3D Solids Pyramids and Cones Name & Date 11) A cylinder has a lateral area of 40 m 2 and a height of 8m. A) What is the area of the base? B) What is the volume of the cylinder? 12) The surface area of a pentagonal pyramid is ft 2. If the perimeter of the base is 50 ft, what is the volume of the pyramid? Round to the nearest tenth. 13) A regular octahedron has 8 faces that are equilateral triangles. Find the volume of a regular octahedron with a side length of 10 cm.

8 14) The total surface area of a cone with radius 10 ft is 1130 ft 2. A) What is the lateral surface area of the cone to the nearest tenth? B) What is the slant height of the cone to the nearest tenth? 15) If the area of the square in the figure below is 81 and the perimeter of each of the 4 triangles is 30, what is the perimeter of the figure outlined by the solid line? 16) In the square pyramid below, AB = 10ft and m GFD 75. Find the surface area and volume of the pyramid. G A F B C D E Surface Area Volume

9 Accelerated AAG 3D Solids Pyramids and Cones Name & Date 17) Consider a regular tetrahedron with an edge of length e. Your goal will be to find a formula for the total surface area and volume in terms of e. A) Find the area of the base in terms of e. B) Find the total surface area in terms of e. C) Find the length of the segment connecting one base vertex and the center of the base. Use your length to find the height of the tetrahedron in terms of e.

10 D) Find the length of the apothem in terms of e. Recall, the apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides. Use your apothem to find the height of the tetrahedron in terms of e. You will also need the slant height you found in part (A). Verify that this height agrees with your answer in part (C). E) Find the total volume in terms of e.

11 Accelerated AAG 3D Solids Pyramids and Cones Name & Date 18) Find the surface are and volume of the Regular Tetrahedron (right, triangular pyramid) below that is made up of congruent, equilateral triangles with side lengths of 18 inches on all sides. 19) The figure below right is created by attaching two Right Regular Hexagonal Pyramids together at their bases. The slant height of each lateral face 12 inches and the length of each edge of the Regular Hexagon is 8 inches. A) Find the exact area of the Regular Hexagonal base. B) Find the exact height of each of the pyramids. C) Using your answers above, what is the exact volume of the polyhedron (entire solid)?

12 20) Pictured below right is the Net of a Right cone. The radius of the Sector is 24 inches and it will become the slant height of the cone when it is folded back together. In addition, the Length of the Arc on the Lateral Surface is 16 inches and it will wrap around the circumference of the circular base when it is folded back together. A) Find the surface area of the cone. B) Find the volume of the cone. C) Find the measure of the central angle of the sector.

13 Accelerated AAG 3D Solids Pyramids and Cones Name & Date 21) Draw the net of the cone below and then label all possible measurements on the diagram. Remember that the lateral surface of the cone opens up to become the sector and the slant height becomes the radius of the sector. Base Area Lateral Area Surface Area Volume Measure of Ratio of r to s

14 22) In the space below we will generalize the relationships between a cone and its net.

15 Accelerated AAG 3D Solids Pyramids and Cones Name & Date 23) A right cone has a height of 8 inches and a slant height of 10 inches. A) Draw and label a 3 model of the cone. B) Find the radius and the central angle of the lateral surface (theta). C) Draw close representation of the net of the cone. D) Find the Exact Surface Area and Volume of the cone. 24) A right cone has a slant height of 16 feet and a lateral area of 96 feet 2. A) Find the radius and the central angle of the lateral surface (theta). B) Draw and label a 3 model of the cone. C) Draw close representation of the net of the cone. D) Find the Exact Surface Area and Volume of the cone.

16 25) A right cone has a base diameter of 8 cm and a central angle of the lateral surface of 72 o. A) Draw and label a 3 model of the cone. B) Find the radius, the height, and the slant height. C) Draw close representation of the net of the cone. D) Find the Exact Surface Area and Volume of the cone. 26) A right cone has a height of 16 meters and a volume of 768 meters 3. A) Find the radius, the slant height, and the central angle of the lateral surface (theta). B) Draw and label a 3 model of the cone. C) Draw close representation of the net of the cone. D) Find the Exact Surface Area and Volume of the cone.

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