Q1)
the lateral area of the pyramid is the total area of all the lateral faces excluding the base.
In this regular octagonal pyramid, the lateral sides are triangles. As there are 8 triangles we need to find the area of all 8 sides.
Area of one lateral triangle face = 1/2 * base * slant height
slant height is the hypotenuse of the right angled triangle formed from the base of the pyramid with the perpendicular height.
slant height - l
l² = 7² + 7² = 49 *2
l² = 98
l = √98
l = 9.9
Area = 1/2 * 5.8 cm * 9.9 cm
= 28.71 cm²
There are 8 sides
total lateral area = 8 * 28.71 = 229.68 rounded off is 229.7 cm²
third option is correct - 229.7 cm²
Q2)
in the triangular face, the lateral edge makes a 60° angle with the base edge. Therefore 2 of the angles are 60° each, since the sum of the interior angles of a triangle is 180°, the third angle too is 60°. this makes the triangle an equilateral triangle with equal angles, hence equal sides.
since lateral edge is 8 cm,base edge too is 8 cm.
since this is an equilateral triangle, the perpendicular line cuts the base edge at its midpoint, bisecting the line forming 2 right angled triangles.
in the right angled triangle, height of triangle is x slant height ,
base = 8 /2 = 4 cm
hypotenuse = 8 cm
We need to find x, use Pythogoras' theorem
4² + x² = 8²
16 + x² = 64
x² = 62 - 16
x = √48
x = √4x√4x√3
= 2x2√3
= 4√3 cm
Q3)
surface area of the square pyramid
surface area of the base + surface area of triangular faces
square area = length x length
= 6.2 x 6.2
= 38.44 yd²
triangular face area = 1/2 * length * height
since the angle between lateral edge and base edge is 60°, its an equilateral triangle where all sides are equal. in this case each side is 6.2 yd.
to find the perpendicular height, use pythogoras' theorem
the perpendicular line(slant height ) cuts the base edge at its midpoint, therefore length of the right angled triangle is = 6.2/2 = 3.1 yd
slant height - l
l² + 3.1² = 6.2²
l² = 38.44 -9.61
l² = 28.83
l = 5.37
area = 1/2 *length *height
= 1/2 * 6.2 * 5.37
= 16.64 yd²
there are 4 triangles = 4 * 16.64 = 66.58 yd²
total area = 38.44 + 66.58 = 105 yd²
correct answer is 3rd option - 105 yd²
