Find the total surface area of the cone if its height is 10 s, and the cone is inclined to the plane
Find the total surface area of the cone if its height is 10 s, and the cone is inclined to the plane of its base at an angle of 30 degrees.
A cone is a geometric body created by rotating a right-angled triangle around its leg.
The total surface area of the cone can be calculated using the formula:
Sp.p. = πrL + πr ^ 2, where:
Sp.p. – the total surface area of the cone;
r is the radius of the base;
L – generator;
Π – number ≈ 3.14.
To do this, you need to calculate the radius of the base and the generatrix of the cone.
Consider a triangle formed by the height, generatrix and radius of the cone. This triangle is right-angled, therefore, to calculate the generatrix, we apply the theorem of sines. The sine of an acute angle of a right triangle is the ratio of the opposite leg to the hypotenuse:
sin α = h / L;
L = h / sin α;
sin 30 ° = 1/2 = 0.5;
L = 10 / 0.5 = 20 cm.
To calculate the radius, we use the Pythagorean theorem:
L ^ 2 = h ^ 2 + r ^ 2;
r ^ 2 = L ^ 2 – h ^ 2;
r ^ 2 = 20 ^ 2 – 10 ^ 2 = 400 – 100 = 300;
r = √300 = 17.32 cm.
Sp.p. = (3.14 * 13.32 * 20) + (3.14 * 300) = 1087.7 + 942 = 2029.7 cm2.
Answer: the total surface area of the cylinder is 2029.7 cm2.
