The base of the pyramid is a right-angled triangle with legs 12 cm and 16 cm. Each side edge of the pyramid is 2v41 cm. Find the volume of the pyramid.
From the right-angled triangle ABC, we determine, according to the Pythagorean theorem, the length of the hypotenuse AC.
AC ^ 2 = AB ^ 2 + BC ^ 2 = 12 ^ 2 + 16 ^ 2 = 144 + 256 = 400.
AC = 20 cm.
Since all the side faces of the pyramid are equal to each other, the segment DH is the height of the pyramid, and point H is the center of the circle described near the right-angled triangle ABC, and the radius of this circle is equal to half the length of the hypotenuse AC.
AH = AC / 2 = 20/2 = 10 cm.
From the right-angled triangle АНD, according to the Pythagorean theorem, we determine the length of the leg DH.
DH ^ 2 = AD ^ 2 – AH ^ 2 = (2 * √41) ^ 2 – 10 ^ 2 = 164 – 100 = 64.
DН = 8 cm.
Determine the area of the base of the pyramid.
Sbn = AB * BC / 2 = 12 * 16/2 = 96 cm2.
Let’s define the volume of the pyramid.
V = Sbase * DH / 3 = 96 * 8/3 = 256 cm3.
Answer: The volume of the pyramid is 256 cm3.