If you are aware of the volume V and the height of the cone H, we can Express the radius of the base R from the formula V=1/3∙πR2H. Get: R2=3V/πH, where R=√(3V/πH).
If you know the lateral surface area of the cone S and the length of its generatrix L, Express the radius R from the formula: S=πRL. You will get R=S/πL.
The following ways of finding the radius of the base of the cone based on the assertion that the cone formed by rotating a right triangle around one of the legs to the axle. So, if you are aware of the height of the cone H and the length of its generatrix L, that to find the radius R you can use the Pythagorean theorem: L2=R2+H2. Express this formula R, we get: R2=L2–H2, and R=√(L2–H2).
Use the rules of the relationships between the sides and angles in a right triangle. If you know the forming of the cone L and the angle α between the height of the cone and its generating line, find the base radius R, equal to one of the legs of a right triangle, using the formula: R=L∙sinα.
If you know the forming of the cone L and the angle β between the base radius of the cone and its generating line, find the base radius R by the formula: R=L∙cosβ. If you know the height of the cone H and the angle α between the generatrix and the radius of the base, find the base radius R by the formula: R=H∙tgα.
Example: forming of a cone L is 20 cm and the angle α between the generatrix and cone height equal to 15º. Find the base radius of the cone. Solution: In a right triangle with hypotenuse L and an acute angle α opposite the side angle R is calculated by formula R=L∙sinα. Substitute the corresponding values, we get: R=L∙sinα=20∙sin15º. Sin15º is from formulas of trigonometric functions of half argument and equal to 0.5√(2–√3). Hence the leg R=20∙0,5√(2–√3)=10√(2–√3)see, Respectively, the base radius of the cone R is equal to 10√(2–√3)cm.
A special case: in a right triangle the side opposite the 30º angle is half the hypotenuse. Thus, if we know the length of the generatrix of the cone and the angle between its generatrix and a height equal to 30º, then find the radius using the formula: R=1/2L.
Advice 2 : How to find the surface of the cone
Cone is a body which lies at the base circle. Outside the plane of this circle is a point called the vertex of the coneand the segments that connect the vertex of a cone of circular base is called a forming cone.
You will need
- Paper, pencil, calculator
The complete surface of the cone consists of the sum of the lateral surface of the cone and its base. To start the calculation of the surface of the cone , you can calculate the surface. Since the base of the cone is a circle, use the formula for the area of a circle: S=?R2, where S is the area of the base of the cone, ? is a constant equal to 3.14, and R2 is the radius of the base squared.
Then, calculate the lateral surface of the cone. This requires the radius of the base multiplied by the length of the generatrix and the resulting value multiplied by mentioned in the previous step ?. (S=Rl?, where S is the lateral surface area of cone, R is the base radius, l is the length of the guide, huh ? = 3.14).
To calculate the full surface of the cone, find the sum of the squares of the base and the lateral surface of the cone.
Described formula is not suitable for calculating the area of a truncated cone. If you want to find the area of a truncated cone, use this formula: S = πR2 + πr2 + π(R+r)l, where S is the full area of a truncated cone, π is a constant equal to 3.14, R2 is the radius of the larger base squared, r2 is the radius of the smaller base of the square, l - forms.
Advice 3 : How to calculate the volume of a cone
Cone can be defined as the set of points forming a two-dimensional figure (e.g., circle), combined with the set of points that lie on the segments, starting at the perimeter of the figure, and ending in a point. This definition is true, if the only common point of the segments (apex of cone) does not lie in the same plane with the two-dimensional figure (the base). Perpendicular to the base of the segment connecting the vertex and the base of the cone, is called its height.
Assume in calculating the amount of different types of cones to the General rule: the sought value should be equal to one third of the base area of this figure by the height. For a traditional cone, the base of which is a circle, its area is calculated by multiplying the number PI of squared radius. This implies that the formula for calculating the volume (V) must include the product of PI (π) on the square of the radius (r) and height (h), which should be reduced three times: V = ⅓*π*r2*h.
To calculate the volume of a cone with base an elliptic shape will need a good knowledge both of its radii (a and b), as the area of this round figure is the multiplication of their works on the PI. Replace this expression footprint in the formula from the previous step and you get this equation: V = ⅓*π*a*b*h.
If the base of the cone lies with a polygon, this particular type is called a pyramid. However, the principle of calculating the volume of the figure remains the same - start in this case with the formula for finding the area of a polygon. For example, for a rectangle it is enough to multiply the lengths of two adjacent sides (a and b), and the triangle is the value to multiply the even and the sine of the angle between them. Replace the formula for the area of the base of the equation from the first step and have the formula calculate the volume of a figure.
Find the area of both bases, if you need to find out the volume of a truncated cone. Fewer of them (S₁) is called a section. Calculate his work in the area of the long base (S₀), add to the resulting value of both squares (S₀ and S₁) and extract from the result of the square root. The resulting value can be used in the formula from the first step instead of the square base: V = ⅓ *V(S₀*S₁+S₀+S₁)*h.