Wize High School Grade 9 Math Textbook > 3D Geometry - Measurements

Surface Area of a Right Cone
A right cone is a solid 3D object with a circular base, and with a vertex that is directly above the center of the base. 

Surface Area
The surface area of a right cone can be broken down into the following:
Area of the circular base - πr2\pi r^2πr2
Area of the curved surface -           π r L          

The ratio of the area of the curved surface to the circumference of the curved surface must be equal to the area of the full circle to the circumference of the full circle:
Area of the curved surfaceCircumference of the curved surface=Area of the full circleCircumference of the full circleArea of the curved surface2πr=πL22πLArea of the curved surface2πr=πL22πLArea of the curved surface2πr=L2Area of the curved surface2πr×2πr=L2×2πrArea of the curved surface=πrL\begin{array}{rcl}
\dfrac{\text{Area of the curved surface}}{\text{Circumference of the curved surface}}&=&\dfrac{\text{Area of the full circle}}{\text{Circumference of the full circle}}\\[1.5em]
\dfrac{\text{Area of the curved surface}}{2\pi r}&=&\dfrac{\pi L^2}{2\pi L}\\[1.5em]
\dfrac{\text{Area of the curved surface}}{2\pi r}&=&\dfrac{\cancel{\pi} L^{\cancel{2}}}{2\cancel{\pi} \cancel{L}}\\[1.5em]
\dfrac{\text{Area of the curved surface}}{2\pi r}&=&\dfrac{L}{2}\\[1.5em]
\dfrac{\text{Area of the curved surface}}{2\pi r}\colorTwo{\times2\pi r}&=&\dfrac{L}{2}\colorTwo{\times2\pi r}\\[1.5em]
\text{Area of the curved surface}&=&\pi r L
\end{array}Circumference of the curved surfaceArea of the curved surface​2πrArea of the curved surface​2πrArea of the curved surface​2πrArea of the curved surface​2πrArea of the curved surface​×2πrArea of the curved surface​======​Circumference of the full circleArea of the full circle​2πLπL2​2π​L​π​L2​​2L​2L​×2πrπrL​

So, the surface area of a right cone is S.A.cone=πr2+πrL\large\boxed{\text{S.A.}_\text{cone}=\pi r^2+\pi r L}S.A.cone​=πr2+πrL​

Practice: Surface Area of a Right Cone
What is the surface area of a right cone with a height of 10" and a circular base that has a radius of 3"?

Enter the surface area in

inch^2

rounded to 2 decimal places, do not include units.

I don't know

0:00 / 0:00

Volume of a Right Cone
The volume of a right cone is 13\dfrac{1}{3}31​ the volume of the cylinder with the same circular base and the same height.


Vcone=13πr2h\large\boxed{V_\text{cone}=\dfrac{1}{3}\pi r^2h}Vcone​=31​πr2h​

Practice: Volume of a Right Cone
What is the volume of a right cone with a height of 10" and a circular base that has a radius of 3"?

Enter the volume of the cone in

inch^3

rounded to 2 decimal places, do not include units.

I don't know

Practice: Right Cones
A right cone has a circular base with area 16π m216\pi~m^216π m2 and a slant height of 5 m5~m5 m.

a) Determine the radius of circular base.

b) Determine the height of this cone.

c) Determine the surface area of this cone.

d) Determine the volume of this cone.

a) Enter the radius in

m

, do not include units.

b) Enter the height in

m

, do not include units.

c) Enter the EXACT surface area in

m^2

, do not include units, do not round your answer

d) Enter the EXACT volume in

m^3

, do not include units, do not round your answer.

I don't know

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Practice: Right Cones
A right cone and a cylinder have the same volume and the same size circular base. How do their heights compare?

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