Question

Q 40 What is the area of lateral surface of a right
pyramid with a square base of side 32 cm as
its base and having a height of 12 cm?
03
C
XXX
Ops: A.
1220 cm?
N
B.
1250 cm?
om
NNNNN
C.
WO
1280 cm2
D.
1240 cm2​

Answer 1

Answer:

you are given either the area or the height of the pyramid and its apothem, you can find the volume of a pyramid, using the Pythagorean Theorem. The apothem is the height of the sloping triangular face of the pyramid, drawn from the vertex of the triangle to its base. To find the apothem, use the base of the pyramid and its height. Apothem divides the base in half and intersects it at a right angle.

Now study the right-angled triangle formed with apothem, height, and the segment connecting the center of the base and the middle of its side. In such a triangle, apothem is the hypotenuse, which can be found using the Pythagorean Theorem. The segment connecting the center of the base and the middle of its side is equal to half of the base (this segment is one of the legs; the second leg is the height of the pyramid).

Recall that the Pythagorean Theorem is written as follows: a2 + b2 = c2, where a and b are legs, and c is the hypotenuse of a right triangle.

For example, there is a pyramid with a base of 4 cm and the apothem of 6 cm. In order to find the height of the pyramid, put these values in the Pythagorean Theorem: a2 + b2 = c2; a2 + (4/2)2 = 62; a2 = 32; a = √32 = 5,66 cm. So, you have found the second leg of a right triangle, which is the height of the pyramid (likewise, if there was a value of the apothem and the height of the pyramid, you could have found the half of the base of the pyramid).

Use the values to find the volume of a square pyramid by the formula: a2 x (1/3) h. In our example, you have calculated that the height of the pyramid is equal to 5.66 cm. Substitute the values into the formula: a2 x (1/3) h; 42 x (1/3) (5,66); 16 x 1,89 = 30,24 cm3.

If don’t have a value of the apothem, use the edge of the pyramid. The edge is a segment connecting the top of the pyramid with the top of the square at the base of it. In this case, you will get a right-angled triangle, legs of which are the height of the pyramid and the half of the diagonal of the square at the base of the pyramid, and the hypotenuse is the edge of the pyramid. Since the diagonal of a square is equal to √2 x side of the square, you can find a side of the square (base) by dividing the diagonal by √2. Then you’ll find the volume of a square pyramid by the formula described above.

For example, there is a square pyramid with the height of 5 cm and the edge of 11 cm. Calculate the half of the diagonal as follows: 52 + b2 = 112; b2 = 96; b = 9,80 cm.

You have found the half of the diagonal, a diagonal is therefore: 9.80 cm x 2 = 19.60 cm.

The side of the square (base) is equal to √2 × diagonal, so 19,60 / √2 = 13,90 cm. Now find the volume of a square pyramid according to the formula: a2 x (1/3) h; 13,902 x (1/3) (5); 193,23 x 5/3 = 322,05 cm3.

Advice

In a square pyramid, its height, the base, and the apothem are linked with the Pythagorean Theorem: (side ÷ 2)2 + (height)2 = (apothem)2.

In any right pyramid, the apothem, the base, and the edge are connected with the Pythagorean Theorem: (side ÷ 2)2 + (apothem)2 = (rib)2.