Prisms and Cylinders Volume

Key Concepts

• Use volume of a prism.
• Find the volume of an oblique cylinder.
• Solve a real-world problem.

Introduction

Right cylinder vs oblique cylinder

Cavalieri’s Principle

Both the shapes given below have equal heights h and equal cross-sectional areas B. Mathematician Bonaventura Cavalieri’s (1598–1647) claimed that both the solids have the same volume.  

If two solids have the same height and the same cross-sectional area at every level, then they have the same volume. 

Use volume of a prism 

Example 1: 

Use the measurements given to solve for x. 

Solution: 

A side length of the cube is x feet. 

V = S³ 

100 = x³ 

4.64 = x 

So, the height, width, and length of the given cube are about 4.64 feet. 

Find the volume of an oblique cylinder 

Example 2: 

Find the volume of the oblique cylinder. Round to the nearest tenth if necessary. 

Solution: 

Cavalieri’s Principle allows you to use Theorem to find the volume of the oblique cylinder.  

V = πr²h (Formula for the volume of a cylinder)  

    = π(8²)(30) (Substitute known values)  

    = 1920π (Simplify)  

    » 6028.8 (Use a calculator)  

The volume of the oblique cylinder is about 6028.8 cm³. 

Example 3: 

Find the volume of the oblique cylinder. Round to the nearest tenth if necessary. 

Solution: 

Cavalieri’s Principle allows you to use Theorem to find the volume of the oblique cylinder.  

r = 7.5 mm and h = 15.2 mm.  

V = πr²h (Formula for the volume of a cylinder)  

    = π(7.5²)(15.2) (Substitute known values) 

    = 855π (Simplify)  

    » 2684.7 (Use a calculator)  

The volume of the oblique cylinder is about 2684.7 mm³. 

Solve a real-world problem 

Example 4: 

The base of a rectangular swimming pool is sloped; so, one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool.  

Solution: 

The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two prisms. 

The volume of water it takes to fill the pool is 1575 ft³. 

Exercise

• As per _____________________, if two solids have the same height and the same cross-sectional area at every level, then they have the same volume.
• In what type of units is the volume of a solid measured?
• Use the measurements given to solve for x.

• Use the measurements given to solve for x.

• Find the volume of each oblique cylinder. Round to the nearest tenth if necessary.

• The volume of a right cylinder is 684π cubic inches and the height is 18 inches. Find the radius.

Find the volume of the oblique prism shown below.

• Use Cavalieri’s Principle to find the volume of the oblique prism or cylinder. Round your answer to two decimal places.

• Use Cavalieri’s Principle to find the volume of the oblique prism or cylinder. Round your answer to two decimal places.

• In the concrete block shown, the holes are 8 inches deep. Find the volume of the block using the Volume Addition Postulate.

Concept Map

What have we learned

• Use volume of a prism and find an unknown value.
• Find the volume of an oblique cylinder using the formula.
• Solve a real-world problem in terms of volume.