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Mathematics – Sphere and Prism
- 1. MATHEMATICS – SPHEREAND PRISM MUHAMMAD NAZMI YEE JYH LIN ERIK ONG ISYRAQ NASIR CHANG MAYCHEN RIFAI NUSAIR
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- 3. WHAT IS ASPHERE? A sphere is a geometrical figure that is perfectly round, 3- dimensional and circular - like a ball.
- 4. WHAT IS ASPHERE? Geometrically, a sphere is defined as the set of all points equidistant from a single point in space. It is a shape of biggest volume with the smallest surface area.
- 5. SPHEROID? Watermelon andEarth is not exactly a sphere as it is not perfectly round. This are known as spheroid.
- 6. PROPERTY OF ASPHERE It is perfectly symmetrical All points on the surface are the same distance from the center. It has no edges or vertices (corners)
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- 8. VOLUME OF SPHERE Volume= 4 3 𝜋𝑟3 General Formula for Volume of sphere R is radius By rearranging the above formula, you can find the radius: Radius= 3 3𝑣 4𝜋
- 9. EXAMPLE Find thevolume of a sphere of radius 9.6 m, rounding your answer to two decimal places. V = 4 3 𝜋𝑟3 4 3 × 𝜋 × 9.63 (replace r with 9.6) 4 3 × 𝜋 × 884.736 = 3705.97 𝑚3 9.6 m
- 10. SURFACE AREA OFSPHERE Surface Area = 4𝜋𝑟2 By rearranging the above formula, you can find the radius: Radius= 𝑎 4𝜋
- 11. EXAMPLE Find thesurface area of a sphere of diameter 28 cm. Radius = ½ Diameter Surface Area = 4𝜋𝑟2 4 × 𝜋 × 142 (28 divide by 2 and replace r with it) = 2464 𝑐𝑚2 28 m
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- 14. VOLUME OF HEMISPHERE It is exactly half of the sphere so: 4 3 𝜋𝑟3 ÷ 2 Volume = 2 3 𝜋𝑟3
- 15. EXAMPLE Find thevolume of a hemisphere, whose radius is 10 cm. V = 2 3 𝜋𝑟3 2 3 × 𝜋 × 103 (replace r with 10) 2 3 × 𝜋 × 1000 = 2093.3 𝑚3 10 cm
- 16. SURFACE AREA OFHEMISPHERE The surface area of hemisphere is equals to half of surface area of sphere plus the area of the base (circle). 2𝜋𝑟2 + 𝜋𝑟2 Therefore Surface Area =3𝜋𝑟2 this is only if the question asked about total surface area.
- 17. EXAMPLE Find thetotal surface area of a hemisphere, whose radius is 8 cm. Surface Area = 3𝜋𝑟2 3 × 𝜋 × 82 (replace r with 10) 2 3 × 𝜋 × 64 = 602.88 𝑐𝑚2 10 cm
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- 19. WHAT IS APRISM? A prism is a geometrical solid object with two identical ends and flat sides.
- 20. CROSS SECTION Across section is the shape made by cutting straight across an object. A prism must have the same cross section all along its length.
- 21. NO CURVES! Aprism is a polyhedron which means all faces must be flat.
- 22. PARALLEL SIDES Theside faces of a prism are parallelograms. When to ends are not parallel it is not a prism.
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- 24. VOLUME OF PRISM Volume = Base Area × Length Base Area is calculated normally depends on the shape.
- 25. EXAMPLE What isthe volume of a prism where the base area is 25 m2 and which is 12 m long: Volume = Base Area × Length 𝑉𝑜𝑙𝑢𝑚𝑒 = 25 𝑥 12 = 300 𝑚3
- 26. SURFACE AREA OFPRISM Surface Area = (2 x Base Area) + (Base Perimeter x Length)
- 27. EXAMPLE What isthe surface area of a prism where the base area is 25 m2, the base perimeter is 24 m, and the length is 12 m: Surface Area = (2 x Base Area) + (Base Perimeter x Length) 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = (2 × 25) + (24 × 12) 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 50 𝑚2 + 288 𝑚2 = 338 𝑚2
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- 29. Find the totalsurface area of a large round bowl (hollow hemisphere) with outer radius of 12 cm and thickness of 1 cm.
- 30. ANSWER Outer Hemisphere: S.A =2𝜋𝑟2 S.A = 2 × 𝜋 × 122 S.A = 2 × 𝜋 × 144 S.A = 288𝜋 Inner Hemisphere: S.A = 2𝜋𝑟2 S.A = 2 × 𝜋 × 112 S.A = 2 × 𝜋 × 121 S.A = 242𝜋 The ‘Ring’: S.A = 𝐵𝑖𝑔𝑔𝑒𝑟 𝑐𝑖𝑟𝑐𝑙𝑒 − 𝑆𝑚𝑎𝑙𝑙𝑒𝑟 𝑐𝑖𝑟𝑐𝑙𝑒 S.A = (𝜋 × 122) − (𝜋 × 112) S.A = 144𝜋 − 121𝜋 S.A = 23𝜋 Add All: 242𝜋 + 288𝜋 + 23𝜋 = 553𝜋 ≈ 𝟏𝟕𝟑𝟕. 𝟑 𝒄𝒎 𝟐
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