GCSE Maths Grade 9 Geometry Question

Hello all. Does anyone know where the radius (of the large cone) is represented on the smaller cone? I am trying to figure out how to solve the question. I have labelled AO and OB as L (the slanted side length of the cone) on the sector.
https://imgur.com/a/cUoqwtO

Reply 1

Nitrotoluene

8

Original post by Sichwünschen

Hello all. Does anyone know where the radius (of the large cone) is represented on the smaller cone? I am trying to figure out how to solve the question. I have labelled AO and OB as L (the slanted side length of the cone) on the sector.
https://imgur.com/a/cUoqwtO

I hope I got your handwriting right.

So we have to determine the size of angle ADB in the sector DACB. We have the volume and height of the cone. To do this, you first find out the radius of the cone’s base. Next we determine the slant height of the cone. Then, letting these two values relate the sector to the cone we can determine angle ADB.
Well, here is how you solve it, IMHO...of course!:
The volume formula will tell you to find the radius of the base of the cone:

Vol = (1/3) x pi x r^2 x h

Enough for now; next, find the slant height using the Pythagorean theorem:

l^2 = r^2 + h^2

Now look how the sector and cone fit together: the radius of the sector is equal to the slant height of the cone, and the length of the arc of the sector is equal to the circumference of the cone's base.

Use the circle circumference formula to find the arc length of the sector:

Circumference = 2 x pi x r

To calculate angle ADB, take the arc length of the sector and divide it by the circumference of the circle, then multiply the result by 360°.

The angle ADB in sector DACB comes out to about 264 degrees.

Errors and omissions excepted.

Kind regards from Italy,
Sandro

Reply 2

spreeder

5

Original post by Sichwünschen

Hello all. Does anyone know where the radius (of the large cone) is represented on the smaller cone? I am trying to figure out how to solve the question. I have labelled AO and OB as L (the slanted side length of the cone) on the sector.
https://imgur.com/a/cUoqwtO

Perhaps you should look for the length scale factor?

Reply 3

Sichwünschen

OP

12

Original post by Nitrotoluene

I hope I got your handwriting right.
So we have to determine the size of angle ADB in the sector DACB. We have the volume and height of the cone. To do this, you first find out the radius of the cone’s base. Next we determine the slant height of the cone. Then, letting these two values relate the sector to the cone we can determine angle ADB.
Well, here is how you solve it, IMHO...of course!:
The volume formula will tell you to find the radius of the base of the cone:

Vol = (1/3) x pi x r^2 x h

Enough for now; next, find the slant height using the Pythagorean theorem:

l^2 = r^2 + h^2

Now look how the sector and cone fit together: the radius of the sector is equal to the slant height of the cone, and the length of the arc of the sector is equal to the circumference of the cone's base.

Use the circle circumference formula to find the arc length of the sector:

Circumference = 2 x pi x r

To calculate angle ADB, take the arc length of the sector and divide it by the circumference of the circle, then multiply the result by 360°.

The angle ADB in sector DACB comes out to about 264 degrees.

Errors and omissions excepted.

Kind regards from Italy,
Sandro

Thanks a lot this really helped!

GCSE Maths Grade 9 Geometry Question

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