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SURFACE AREA AND VOLUME 2

This guide provides a comprehensive approach to calculating the volume and surface area of various three-dimensional shapes: pyramids, cones, and spheres. It includes step-by-step methods for determining the volume of a rectangular-based pyramid and a cone, along with their respective surface areas. Additionally, it covers the concepts of frustums, as well as the volume and surface area of hemispheres and spheres. Utilizing geometric principles like the Pythagorean theorem, we simplify complex calculations for educational purposes.

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SURFACE AREA AND VOLUME 2

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  1. SURFACE AREA AND VOLUME 2

  2. Volume of a pyramid h

  3. Calculate the volume of the rectangular-based pyramid. E 6 cm D C 4 cm A 5 cm B

  4. Surface area of a pyramid h Surface area = sum of the areas of all the faces of the pyramid

  5. Calculate the surface area of the rectangular-based pyramid. E First find the length of EX and EY. Use Pythagoras on triangle EOX. 6 cm D C Use Pythagoras on triangle EOY. O X 4 cm A B Y 5 cm

  6. E Area of rectangle ABCD = 4 × 5 NET OFPYRAMID = 20 cm2 Area of triangle BCE = ½ × 4 × 6.5 = 13 cm2 6.325 D C Area of triangle CDE = ½ × 5 × 6.325 5 cm = 15.81 cm2 6.5 6.5 4 cm E E A B Surface area = sum of areas of faces = 20 + 13 + 13 + 15.81 + 15.81 6.325 = 77.6 cm2 E

  7. Volume of a cone h r

  8. Calculate the volume of the cone. 7 cm 4 cm

  9. Surface area of a cone The surface of a cone is made from a flat circular base and a curved surface. The curved surface is made from a sector of a circle. CURVED SURFACE = + FLAT BASE Curved surface area of a cone = , where is the slant height Total surface area of a cone =

  10. Calculate a the curved surface area of the cone, b the total surface area of the cone. a First calculate the slant height using Pythagoras. 12 cm Curved surface area 5 cm b Total surface area

  11. The straight edges of the sector are joined together to make a cone. Calculate a the curved surface area of the cone, b the radius of the base of the cone, c the height of the cone. 280o 4 cm a Curved surface area = area of sector 4 cm c Using Pythagoras b Curved surface area 4 3.11

  12. When you make a cut parallel to the base of a cone and remove the top part, the part that is left is called a frustum. FRUSTUM Volume of frustum = volume of large cone – volume of smaller cone

  13. 3 Calculate the volume of the frustum. All lengths are in cm. 8 6 You must first find the height of the smaller cone using similar triangles. 3 Volume of large cone 8 6 Volume of small cone Volume of frustum

  14. Volume and surface area of a sphere Volume of a sphere Volume of a hemisphere Curved surface area of a hemisphere Surface area of a sphere Volume and surface area of a hemisphere A hemisphere is half a sphere.

  15. The sphere has radius 10 cm. Calculate a the volume of the sphere, b the surface area of the sphere. a Volume b Surface area

  16. The solid hemisphere has radius 6 cm. Calculate a the volume of the hemisphere, b the curved surface area of the hemisphere, c the total surface area of the hemisphere. 6 cm a Volume b Curvedsurface area c Totalsurface area = area of base circle + curved surface area

  17. The solid is made from a cylinder and a hemisphere. The cylinder has a height of 8 cm and a radius of 3 cm. Calculate the volume of the solid. Volume of cylinder Volume of hemisphere Total volume

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