Uses of Surface Area Volumes

Key Concepts

• Finding volume of a cone.
• Applying Pythagoras theorem to solve volume problems.
• Finding the volume of a cone given the circumference of the base.

Introduction:

Volume of a Cone:

A cone is a three-dimensional figure with a circular base. A curved surface connects the base and the vertex.

The cylinder and cone given below have the same height and their bases are congruent.

1. Predict how the volume of the cone compares to the volume of the cylinder.

The volume of a cylinder is 3 times the volume of a cone having the same base area and height or that the volume of a cone is 1/3 the volume of a cylinder having the same base area and height.

V=  1/3πr ²h

• If you fill the cone with water or other filling material, predict how many cones of water will fit into the cylinder.

Three cones of water will fit into the cylinder.

8.3.1 Finding Volume of a Cone

Example 1:

A water tank is shaped like the cone shown.

How much water can the tank hold? Use 3.14 for π, and round to the nearest tenth.

Solution:

Find the volume of the water tank.

               V =  Bh

               V=  1/3πr ² × h

               V=  1/3π(1.5)² × (4)

               V=  1/3× 3.14 × (2.25)(4)

               V= 9.42 in.³

The water tank can hold about 9.42 in.³ of water.

8.3.2 Applying Pythagoras Theorem to Solve Volume Problems

Example 2:

A city engineer determines that 5,500 cubic meters of sand will be needed to combat erosion at the city’s beach. Does the city have enough sand to combat the erosion? Use 22/7 for π. Explain.

Solution:

Step 1:

Use Pythagorean Theorem to find the height of the cone.

               r ² + h ² = l ²

               r ² + 35² = 37²

               r ² + 1225 = 1369

               r ² = 144

               r = 12

The radius of the cone is 12 meters.

Step 2:

Find the volume of the cone.

Use 22/7 for π.

               V=  1/3 πr ²h

               V=  1/3π(12)²(35)

               V=   1/3(22/7) (144)(35)

               V= 5280

The volume of the cone is 5280 cubic meters.

8.3.3 Finding the Volume of a Cone Whose Base Circumference is Given

Example 3:

The circumference of the base of a cone is 16π feet. What is the volume of the cone in terms of π?

Solution:

Step 1:

Use the circumference to find the radius of the base of the cone.

               C = 2πr

               16π/2π = r

               8 = r

The radius of the cone is 8 ft.

Step 2:

Find the volume of the cone.

               V=  1/3π(8)²(21)

               V=  1/3π(64)(21)

               V= 448π

The volume of the cone is 448π feet.

Example 4:

An ice cream cone is filled exactly level with the top of a cone. The cone has a 9-centimeter depth and a base with a circumference of 9π centimetres. How much ice cream is in the cone in terms of π?

Solution:

Step 1:

Use the circumference to find the radius of the base of the cone.

               C = 2πr

               9π/2π = r

               4.5 = r

The radius of the cone is 4.5 centimeters.

Step 2:

Find the volume of the cone.

               V=  1/3π(4.5)²(9)

               V=  1/3π(20.25)(9)

               V= 60.75π

The volume of the cone is 60.75π feet.

Exercise:

1. The volume of a _______________ is 3 times the volume of a cone having the same base area and height.
2. Find the volume of the following cone. Write your answer in terms of π.

• Mary found a cone-shaped seashell on the beach. The shell has a height of 63 millimeters and a base radius of 8 millimeters. What is the volume of the seashell? Estimate using 22/7 for π.

• A cone has a height of 14 inches and a volume of 769.3 cubic inches. What is the radius of the cone? Use 3.14 as an approximation for π.

Ans: The radius of the cone is 7 inches.

• A water dispenser in an office comes with cone-shaped paper cups. Each paper cup has a height of 4 inches and a diameter of 6 inches. If the water dispenser contains 2,826 cubic inches of water, how many paper cups can be filled with the water? Use 3.14 as an approximation for π.
• If a cone-shaped hole is 6 feet deep and the circumference of the base of the hole is 22 feet, what is the volume of the hole? Use 22/7 for π.

• The volume of the cone is 462 cubic yards. What is the radius of the cone? Use 22/7 for π.

• What is the volume, in cubic inches, of a cone that has a radius of 8 inches and a height of 12 inches? Use 3.14 for π and round to the nearest hundredth.
• Find the exact volume of the cone. Use 3.14 as an approximation for π to find the approximate volume of the cone. Round your answer to the nearest tenth.

• Find the exact volume of the cone. Use 3.14 as an approximation for π to find the approximate volume of the cone. Round your answer to the nearest tenth.

Concept Map:

What have we learned:

• Finding volume of a cone given height and radius.
• Applying Pythagoras theorem to solve volume problems.
• Finding the volume of a cone given the circumference of the base.