a.) slant edge,Originally Posted byphgao

Slant edge is the hypotenuse of a right triangle whose vertical leg is 15cm and whose horizontal leg is (1/2)[10sqrt(2)] or 5sqrt(2).

So, slant edge = sqrt[15^2 +(5sqrt(2))^2] = sqrt[225 +50] = sqrt[275] = sqrt[25*11] = 5sqrt(11) cm. --------answer.

b.) inclination of slant edge from the base.

Let us call that angle alpha.

sin(alpha) = 15/[5sqrt(11)] = 3/sqrt(11)

alpha = arcsin[3/sqrt(11)] = 64.76 degrees. ------answer.

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c.) inclination of a sloping face from the base.

Let us call that angle beta.

Imagine a right triangle, with these:

---hypotenuse = sloping face

---vertical leg = 15cm

---horizontal leg = 10/2 = 5cm

---angle between hypotenuse and horizontal leg = angle beta.

tan(beta) = 15/5 = 3

beta = arctan(3) = 71.565 degrees. -------answer.

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d.) magnitude of angle between two adjacent sloping faces.

Let us call that angle gamma.

In the right triangle where a slant edge is the hypotenuse, project a line segment from the intersection of the vertical and horizontal legs (actually, the center of the square base of the pyramid) to the slant edge, such that this line segment is perpendicular to the slant edge. Let us call this line segment, x. Line segment x intersects the slant edge at, say, point E.

Another right triangle is formed, with these:

--hypotenuse = 5sqrt(2) cm -------half of diagonal.

--one leg = x

--the other leg = unknown ----a portion of the slant edge, say, line segment DE, where DO is the slant edge.

--angle opposite leg x is 64.76 degrees.

sin(64.76deg) = x/5sqr(2)

x = [5sqrt(2)]sin(64.76deg) = 6.396 cm.

Now, in the whole pyramid, project a line segment from a corner of the square base, say, corner A, to point E.

Yet another right triangle is formed, with these:

---hypotenuse = AE

---one leg = x = 6.396 cm

---the other leg = 5sqrt(2) ---------half of diagonal AC of the square base.

---angle between x and hypotenuse is half of gamma.

tan(gamma/2) = [5sqrt(2)]/6.396 = 1.105545

gamma/2 = arctan(1.105545) = 47.87 degrees.

Therefore, gamma = 2(47.87deg) = 95.74 degrees --------answer.