VOLUME OF SQUARE PYRAMID

To calculate volume

• Find the base area
• Multiply it by height
• Multiply this by 1/3

For the answer use cubic units.

Find the volume of the square pyramids given below.

Problem 1 :

Solution :

Base is in the shape of square 

Base area = area of square

Area of square = (Side)²

= (32)²

Height = 50 ft

Volume of pyramid = (1/3) x (32)² x 50

= 17066.6 ft³

Problem 2 :

Solution :

Base area = area of square

height = 6 cm

Area of square = 5²

Volume = (1/3) x 5² x 6

= 50 cm³

Problem 3 :

Solution :

Base area = area of square

height = 8 cm

Area of square = 6²

Volume = (1/3) x 6² x 8

= 96 cm³

Problem 4 :

Solution :

Base area = area of square

Let height = h cm

Slant height = 8 cm

Using Pythagorean theorem h2 + 62 = 82 h2 + 36 = 64 h2 = 64 - 36 h2 = 28 h = 5.29

Area of square = 12²

= 144

Volume = (1/3) x 144 x 5.29

= 253.92 cm³

Problem 4 :

Find the volume of the square pyramid with a height of 15 in and a slant height of 17 in. The square base measures 16 by 16 inches. 

Solution :

Base length of square = 16 inches

height = 15 in

slant height = 17 in

Volume of square pyramid = (1/3) x 16² x 15

= 1280 cubic inches

Problem 5 :

Solution :

Base length = 32 in

Slant height = 34 in

height = h 

h² + 16² = 34²

h² + 256 = 1156

h² = 1156 - 256

h² = 900

h = 30

Volume = (1/3) x 32² x 30

 = 1024 x 10

= 10240 cubic inches

Problem 6 :

Solution :

When we fold this shape, we will get square base pyramid.

Volume of the square pyramid = (1/3) x base area x height

Base area = 18²

= 324 square feet

Considering the equilateral triangle.

h² + 9² = 18²

h² + 81 = 324

h² = 324 - 81

h² = 243

h = 15.58

Volume = (1/3) x 324 x 15.58

= 108 x 15.58

= 1682.64 cubic ft

Problem 7 :

Solution :

Area of the base = 6² ==> 36

height = 9 in

Volume = (1/3) x 36 x 9

= 108 cubic inches

Problem 8 :

Originally, Khafre’s Pyramid had a height of about 144 meters and a volume of about 2,218,800 cubic meters. Find the side length of the square base.

Solution :

Volume of pyramid = 2,218,800 cubic meters

Height = 144 m

(1/3) x base area x height = 2,218,800

Let x be the side length of base of square.

(1/3) ⋅ x² ⋅ 144 = 2,218,800

x² = 2,218,800 (3) / 144

x² = 46225

x = √46225

x = 215 meter

Problem 9 :

A pyramid with a square base has a volume of 128 cubic inches and a height of 6 inches. Find the side length of the square base.

Solution :

(1/3) x base area x height = 128 cubic inches

Let x be the side length of base of square.

Height = 6 inches

(1/3) ⋅ x² ⋅ 6 = 128

x² = (128/6)(3)

x² = 64

x = √64

x = 8 inches

Problem 10 :

A pyramid with a rectangular base has a volume of 6 cubic feet. The length of the rectangular base is 3 feet and the width of the base is 1.5 feet. Find the height of the pyramid.

Solution :

(1/3) x base area x height = 128 cubic inches

Let x be the side height of hte pyramid.

Length = 3 ft, width = 1.5 ft and height = x

(1/3) ⋅ 3 ⋅ 1.5 ⋅ x = 128

x = [ 128/(3 ⋅ 1.5) ] ⋅ 3

x = 85.33 feet

Problem 11 :

A pyramid with a triangular base has a volume of 18 cubic centimeters. The height of the pyramid is 9 centimeters and the height of the triangular base is 3 centimeters. Find the width of the base.

Solution :

Volume of the triangular base = 18 cubic centimeter

Height of the pyramid = 9 cm

height of triangular base = 3 cm

(1/3) x base area x height = 18

1/3 x 1/2 x base x height x height of prism = 18

1/6 x base x 3 x 9 = 18

base = (18 x 6)/(3 x 9)

= 4 cm

So, the measure of width of the base is 4 cm.

Problem 12 :

The pyramids are similar. Find the volume of pyramid B.

Solution :

Since the two shapes are similar, their corresponding sides will be in the same ratio. The ratio of cubes of corresponding sides will be equal to the ratio of cubes of volume of these figures.

(15/10)³ = Volume of pyramid A/Volume of pyramid B

(15/10)³ = 125/Volume of pyramid B

125/1000 = 125/Volume of pyramid B

Volume of pyramid B = 125/0.125

= 1000 in³

So, volume of pyramid B is 1000 in³.