find the area and volume of the figure developed by an equilateral triangle with sides *s* if it is revolved about one of its sides.

May 8, 2015 - 7:23am

#1
equilateral triangle revolved about one of its sides

trigonometry formula derivation derive quadratic formula from standard form formula for finding the surface area of a sphere conical formula bisector in triangle definite integration formulas newtons law of cooling formula equations for parabola integration of trigonometric substitution volume of frustum of cone torsional shearing stress shear stress in t beam examples of rectilinear motion degree of indeterminacy examples angle formulas geometry cantilever diagram strength of materials book poisson ratio steel arithmetic sequence examples inverse trigonometric functions problems solutions trigonometry summary present value of a deferred annuity formula inscribed and circumscribed inscribed quadrilaterals integral calculus problems with solutions pdf moment equations for beams cylindrical shells method surface area of a pentagonal prism what does coplanar formula of shear modulus oblique triangle word problems length of arc calculus frustum of cone volume volume formula for pentagonal prism area of a polygon inscribed in a circle slant height of a pyramid formula regular pyramid calculator projectile motion dynamics restained couple statics newtons law of cooling heron's formula how to find n in arithmetic series altitude right triangle difference between geometric and arithmetic sequences properties of circumcenter of a triangle formulas for solids how to solve angle of elevation find the volume of a circular cylinder cos40 prove cyclic quadrilateral how to find the slant height of a triangular pyramid maxima solve equation beam deflection pdf equation of a hemisphere in 3d standard laplace transforms algebra money word problems steel density 7850 kg m3 equation for sum of arithmetic series volume of an isosceles trapezoid shell method calculus formula arithmetic series formulas how to calculate slant height surface area formula calculus sample problem of hyperbola with solution engineering mathematics laplace transform stress strain curve for plastics pythagoras pythagorean theorem torsional force naperian logarithm laplace transformation calculator inscribed circle formula properties of parabola volume of a tetrahedron calculator what does the word reciprocal mean in math maxima and minima in calculus compound interest solved problems median triangle formula formula for a right circular cylinder algebraic substitution shear stress meaning formula of a pyramid surface area volume for pyramid formula

Use Pappus theorem:

V = volume generated

A = surface area generated

A

_{t}= generating areaL = generating length of curve

C = distance traveled by the centroid

$h = \sqrt{s^2 - (\frac{1}{2}s)^2} = \frac{\sqrt{3}}{2}s$

$\bar{y} = \frac{1}{3}h = \frac{1}{3}(\frac{\sqrt{3}}{2}s) = \frac{\sqrt{3}}{6}s$ ← centroid of area

$A_t = \frac{1}{2}sh = \frac{1}{2}s(\frac{\sqrt{3}}{2}s) = \frac{\sqrt{3}}{4}s^2$

$V = A_t \times C = A_t \times 2\pi \bar{y}$

$V = \frac{\sqrt{3}}{4}s^2 \times 2\pi (\frac{\sqrt{3}}{6}s)$

$V = \frac{1}{4}\pi s^3$

$L = 2s$ ← the third side, being the axis of rotation, cannot generate a surface

$\bar{y} = \frac{1}{2}h = \frac{1}{2}(\frac{\sqrt{3}}{2}s) = \frac{\sqrt{3}}{4}s$ ← centroid of lines

$A = L \times C = L \times 2\pi \bar{y}$

$A = 2s \times 2\pi (\frac{\sqrt{3}}{4}s)$

$A = \sqrt{3}\pi s^2$

You can consider the volume generated as a cone, there will be two cones involved with common bases. The radius of each cone is h and the height is s/2.

$V = 2 \times \frac{1}{3}\pi h^2 (\frac{1}{2}s)$

$V = 2 \times \frac{1}{3}\pi (\frac{\sqrt{3}}{2}s)^2 (\frac{1}{2}s)$

$V = \frac{1}{4}\pi s^3$

For the surface area generated, it is the lateral area of two cones

$A = 2 \times \pi hs = 2 \times \pi (\frac{\sqrt{3}}{2}s)s$

$A = \sqrt{3}\pi s^2$