# The volume of a right circular cone is 768πcm ³ and the ratio of its base radius to its height is3:4find the total surface area of the cone?

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### Priya Ahmed

Passionate KS1-GCSE Tutor!

To help answer this question we need two formulae. 🔹Surface Area of a cone: πrl+πr² (where l is the slanted height) 🔹Volume of a cone: 1/3*πr²h (where h is the perpendicular height) The question gives us the volume of the cone already. So we can equal this volume to our volume of cone equation: 1/3*πr²h = 768π If we divide by π on both sides, it will disappear, giving; 1/3*r²h = 768 Let’s remove the 1/3 by multiplying by 3 on both sides; r²h = 2304 Now we need an expression for radius and height. The questions says the ratio of radius to the height is 3:4. So let can let the radius = 3x And the height = 4x Let’s substitute this into our equation r²h = 2304, this gives us; (3x)² * (4x) = 2304 Using index laws; 9x²*4x = 2304 Multiplying out; 36x³ = 2304 Now we have an equation where we can solve for x. To do this we need to divide each side by 36, this gives; x³ = 64 Now we need to take the cube root of both sides, giving; x = 4 We now know what x is. We can use this to quantify our radius and height. Earlier we let radius = 3x, we worked out x to be 4. Therefore the radius is 3*4 = 12. r = 12 height = 4x, so 4*4 = 16 h = 16. Now with the radius and height, we can now work out the SLANTED height which is what we need to work out surface area. In a cone, the base radius, perpendicular height and slanted height form a right angled triangle, where the slanted height is the hypotenuse. We can now apply Pythagoras’ Theorem (a² + b² = c²) to work out slanted height. This gives; 16² + 12² = 400 The 100 represents c² (hypotenuse squared) so to work out c, we need to take the square root so √400 = 20 Therefore our slanted height is 20. l = 20 Referring back to out equation at the beginning Surface Area of a cone: πrl+πr² We can now work out the surface area by substituting in the radius and the slanted height. This gives; π*12*20+π*12² Multiply out; = 240π + 144π Simplify; = 384 π Therefore, answer: 384π cm² (Don’t forget units for area) 🙂

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