Pyramids and Cones

Key Concepts

• Find the volume of a solid.
• Use volume of a pyramid.
• Use trigonometry to find the volume of a cone.

Introduction

Volume of a Pyramid

The volume V of a pyramid is, 

V = 1/3 x Bh, where B is the area of the base and h is the height.  

Volume of a Cone

The volume of a cone is, 

V = 1/3 x Bh = πr²h, where B is the area of the base, h is the height and r are the radius of the base. 

Find the volume of a solid 

Example 1: 

Find the volume of the following solid. 

Solution: 

The base can be divided into six equilateral triangles. Using the formula of an equilateral triangle,  

¼ √3 • s², the area of the base B can be found as follows: 

6.1/4√3.s² = 6.1/4√3.3² = 27/2√3cm²

Use theorem to find the volume of the pyramid.  

V = 1/3Bh (Formula for volume of pyramid) 

= 1/3(27/2 √3)(4) (Substitute) 

= 18√3 (Simplify) 

So, the volume of the pyramid is 18√3, or about 31.2 cubic centimeters. 

Example 2: 

Find the volume of the following solid. 

Solution

Use the formula for the volume of a cone. 

V = 1/3Bh  (Formula for volume of cone) 

V= 1/3(πr²)h (Base area equal πr²)

V= 13(π 1.5²)(4) (Substitute) 

V=3π (Simplify) 

So, the volume of the cone is 3𝝅, or about 9.42 cubic inches. 

Use volume of a pyramid 

Example 3: 

A greenhouse has the shape of a square pyramid. The height of the greenhouse is 18 yards and volume is 5400 yd³. What is the side length of the square base of the greenhouse? 

Solution: 

V = 1/3 Bh à Write the formula. 

5400 = 1/3 (x²)(18) (Substitute) 

16200 = 18x² (Multiply each side by 3) 

900 = x² (Divide each side by 18) 

30 = x  (Find the square root) 

Use trigonometry to find the volume of a cone 

Example 4: 

Caron made a teepee for her math project. Her teepee had a diameter of 6 feet. The angle the side of the teepee made with the ground was 65. What was the volume of the teepee? Round your answer to the nearest hundredth. 

Solution: 

To find the radius r of the base, use trigonometry. 

tan 65⁰ = opp./adj. (Write ratio)

tan 65⁰ = h/3 (Substitute)

h =  tan 65⁰ × 3 ≈ 6.43 (Solve for r) 

Use the formula for the volume of a cone. 

V  = 1/3 (πr²) 

     = 1/3 π (3²)(6.43) 

≈ 60.57 ft³ 

Exercise

1. A ________________________ is a pyramid having a triangular base.
2. A ________________________ is a pyramid having a square base.
3. Find the volume of the following solid. Round your answer to two decimal places.

4. Find the volume of the following solid. Round your answer to two decimal places.

5. Find the volume of the following solid. Round your answer to two decimal places.

6. The volume of a pyramid is 45 cubic feet and the height is 9 feet. What is the area of the base?
7. A ________________________ is a pyramid having a triangular base.

Find the value of x.

8. A ________________________ is a pyramid having a square base.

Find the value of x.

9. Find the volume of the following solid. Round your answer to two decimal places.
10. The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth.

Concept Map

What have we learned

• Finding the volume of a solid.
• Use volume of a pyramid to find an unknown value.
• Apply trigonometry to find the volume of a cone.