Cone side

Calculate the volume and area of the cone whose height is 10 cm and the axial section of the cone has an angle of 30 degrees between height and the cone side.

Correct answer:

V =  349.0659 cm³
S =  314.1593 cm²

Step-by-step explanation:

$$\begin{gathered} h=10\text{ cm } \\ A=30^{\circ }\to \text{ rad}=30^{\circ }\cdot {\displaystyle \frac{\pi }{180}}=30^{\circ }\cdot {\displaystyle \frac{3.1415926}{180}}=0.5236=\pi \mathrm{/}6 \\ \tan A=r:h \\ r=h\cdot \tan (A)=10\cdot \tan (0.5236)\doteq 5.7735\text{ cm } \\ {S}_1=\pi \cdot r^2=3.1416\cdot 5.7735^2\doteq 104.7198{\text{cm}}^2 \\ V={\displaystyle \frac{1}{3}}\cdot S_1\cdot h={\displaystyle \frac{1}{3}}\cdot 104.7198\cdot 10=349.0659{\text{cm}}^3 \end{gathered}$$

$$\begin{gathered} s=\sqrt{h^2+r^2}=\sqrt{10^2+5.7735^2}\doteq 11.547\text{ cm } \\ {S}_2=\pi \cdot r\cdot s=3.1416\cdot 5.7735\cdot 11.547\doteq 209.4395{\text{cm}}^2 \\ S=S_1+S_2=104.7198+209.4395=314.1593{\text{cm}}^2 \end{gathered}$$

Try another example

Related math problems and questions:

• Axial section of the cone
  The axial section of the cone is an isosceles triangle in which the ratio of cone diameter to cone side is 2: 3. Calculate its volume if you know its area is 314 cm square.
• Axial section
  Calculate the volume and surface of a cone whose axial section is an equilateral triangle with side length a = 18cm.
• Angle of deviation
  The surface of the rotating cone is 30 cm² (with circle base), its surface area is 20 cm². Calculate the deviation of the side of this cone from the plane of the base.
• Axial cut
  The cone surface is 388.84 cm², the axial cut is an equilateral triangle. Find the cone volume.
• Angle of cone
  The cone has a base diameter of 1.5 m. The angle at the main apex of the axial section is 86°. Calculate the volume of the cone.
• Cone
  The rotating cone volume is 9.42 cm³, with a height 10 cm. What angle is between the side of the cone and its base?
• Pyramid 8
  Calculate the volume and the surface area of a regular quadrangular pyramid with the base side 9 cm and side wall with the base has an angle 75°.
• Axial section
  The axial section of the cone is an equilateral triangle with area 168 cm². Calculate the volume of the cone.
• Four sided prism
  Calculate the volume and surface area of a regular quadrangular prism whose height is 28.6cm and the body diagonal forms a 50-degree angle with the base plane.
• Lateral surface area
  The ratio of the area of the base of the rotary cone to its lateral surface area is 3: 5. Calculate the surface and volume of the cone, if its height v = 4 cm.
• Cone
  Calculate volume and surface area of ​​the cone with a diameter of the base d=15 cm and side of the cone with the base has angle 52°.
• Lampshade
  The cone-shaped lampshade has a diameter of 30 cm and a height of 10 cm. How many cm² of material will we need when we 10% is waste?
• Truncated cone
  Find the volume and surface area of the truncated cone if r1 = 12 cm, r2 = 5 cm and side s = 10 cm.
• Rotary cone
  The volume of the rotation of the cone is 472 cm³ and angle between the side of the cone and base angle is 70°. Calculate lateral surface area of this cone.
• Tetrahedral pyramid
  Determine the surface of a regular tetrahedral pyramid when its volume is V = 120 and the angle of the sidewall with the base plane is α = 42° 30'.
• Cone container
  Rotary cone-shaped container has a volume 1000 cubic cm and a height 12 cm. Calculate how much metal we need for making this package.
• Frustrum - volume, area
  Calculate the surface and volume of the truncated cone, the radius of the smaller figure is 4 cm, the height of the cone is 4 cm and the side of the truncated cone is 5 cm.