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Volume of a pyramid/Sphere
Thursday, 23 May 2019
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Volume of a pyramid 𝑉𝑜𝑙𝑢𝑚𝑒= 1 3 ×𝐵𝑎𝑠𝑒 𝐴𝑟𝑒𝑎×ℎ𝑒𝑖𝑔ℎ𝑡 Square based pyramid
Tetrahedron cone 𝑉𝑜𝑙𝑢𝑚𝑒= 1 3 ×𝐵𝑎𝑠𝑒 𝐴𝑟𝑒𝑎×ℎ𝑒𝑖𝑔ℎ𝑡
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Example Find the volume for each of the following pyramids (i)
7×5 =35 Base Area = 7cm 5cm 4cm 1 3 ×35×4 Volume = =46.67 cm3
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(ii) 3cm 6cm 3×3 =9 Base Area = 1 3 ×9×6 Volume = =18 cm3
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(iii) 8cm 5cm 𝜋× 2.5 2 =19.63 1 3 ×19.63×8 =52.36 Base Area = Volume =
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(iv) 6cm 5cm 𝜋× 5 2 =78.54 1 3 ×78.54×6 =157.08 Base Area = Volume =
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Example Which has the greatest volume? 3.2cm 7cm 3.2cm 7cm 𝜋× 3.2 2 =32.17 Base Area = 3.2×3.2 =10.24 Base Area = 1 3 ×32.17×7 1 3 ×10.24×7 Volume = Volume = =75.06 cm3 =23.89 cm3 Cone has the greatest volume
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Example A teal plug is machined into the shape of a truncated cone, with end radii of 1.5cm and 2.4cm. Find the volume of the plug 𝜋× 1.5 2 =7.07 Base Area = 2.4cm 1.5 3.5cm 4.5cm 1 3 ×7.07×4.5 Volume = =10.60 𝜋× 2.4 2 =18.10 Base Area = 1 3 ×18.10×8 Volume = Volume of plug = 37.65 cm3 =48.25
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Volume of a sphere 𝑉𝑜𝑙𝑢𝑚𝑒= 4 3 𝜋 𝑟 3 radius
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Find the volume of the following spheres (i)
Example Find the volume of the following spheres (i) 3.2cm 𝑉𝑜𝑙𝑢𝑚𝑒= 4 3 𝜋 𝑟 3 = 4 3 𝜋× 3.2 3 =137.26 cm3
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(ii) 𝑉𝑜𝑙𝑢𝑚𝑒= 4 3 𝜋 𝑟 3 = 4 3 𝜋× 23 3 = cm3 =0.051 m3 46cm
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Example The diagram below is of a bowl which consists of a hemi-sphere of diameter 20cm with a smaller hemi-sphere, of radius 9cm, cut out. Find the volume of the bowl.
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𝑉𝑜𝑙𝑢𝑚𝑒= 2 3 𝜋 𝑟 3 = 4 3 𝜋× 10 3 = 2 3 𝜋× 9 3 𝑉𝑜𝑙𝑢𝑚𝑒 =2094.40 =1526.81
𝑉𝑜𝑙𝑢𝑚𝑒= 2 3 𝜋 𝑟 3 = 4 3 𝜋× 10 3 = 2 3 𝜋× 9 3 𝑉𝑜𝑙𝑢𝑚𝑒 = = 𝑉𝑜𝑙𝑢𝑚𝑒= − =567.58 cm3
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