Learn on PengiReveal Math, Course 3Module 10: Volume

Lesson 10-2: Volume of Cones

In this Grade 8 lesson from Reveal Math, Course 3 (Module 10: Volume), students learn how to calculate the volume of a cone using the formula V = ⅓πr²h, building on the relationship that a cone's volume is one-third that of a cylinder with the same base and height. Students practice applying the formula given a cone's radius or diameter and height, including real-world problems such as finding the volume of a cone-shaped paper cup. The lesson also challenges students to compare the volumes of cylindrical and conical containers to solve cost-based problems.

Section 1

Relationship Between Cone and Cylinder Volumes

Property

A cone has exactly one-third the volume of a cylinder with the same base and height: Vcone=13⋅VcylinderV_{cone} = \frac{1}{3} \cdot V_{cylinder}

If cylinder volume is Vcylinder=πr2hV_{cylinder} = \pi r^2 h, then cone volume is Vcone=13πr2hV_{cone} = \frac{1}{3}\pi r^2 h

Section 2

Volume of a Cone Using Base Area

Property

The volume VV of a cone is one-third the product of its base area BB and its perpendicular height hh.

V=13BhV = \frac{1}{3}Bh

Section 3

Volume of a Cone

Property

A cone is a three-dimensional shape with a circular base and an apex. The volume of a right circular cone is one-third the product of the area of the base and the height. If the height is hh and the radius of the base is rr, then V=13πr2hV = \frac{1}{3} \pi r^2 h.

Examples

  • A party hat is a cone with a height of 8 inches and a base radius of 3 inches. Its volume is V=13π(32)(8)=24πV = \frac{1}{3} \pi (3^2)(8) = 24\pi cubic inches.
  • A cylinder has a volume of 9090 cubic cm. A cone with the same base and height would have a volume of V=13(90)=30V = \frac{1}{3} (90) = 30 cubic cm.
  • An hourglass cone holds 15π15\pi cubic inches of sand and has a base radius of 3 inches. Its height is found by solving 15π=13π(32)h15\pi = \frac{1}{3} \pi (3^2)h, which gives h=5h=5 inches.

Explanation

A cone's volume is directly related to a cylinder's. For a cone and cylinder with the same base radius and height, the cone's volume is exactly one-third of the cylinder's. You could fit three full cones of water into one cylinder.

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Relationship Between Cone and Cylinder Volumes

Property

A cone has exactly one-third the volume of a cylinder with the same base and height: Vcone=13⋅VcylinderV_{cone} = \frac{1}{3} \cdot V_{cylinder}

If cylinder volume is Vcylinder=πr2hV_{cylinder} = \pi r^2 h, then cone volume is Vcone=13πr2hV_{cone} = \frac{1}{3}\pi r^2 h

Section 2

Volume of a Cone Using Base Area

Property

The volume VV of a cone is one-third the product of its base area BB and its perpendicular height hh.

V=13BhV = \frac{1}{3}Bh

Section 3

Volume of a Cone

Property

A cone is a three-dimensional shape with a circular base and an apex. The volume of a right circular cone is one-third the product of the area of the base and the height. If the height is hh and the radius of the base is rr, then V=13πr2hV = \frac{1}{3} \pi r^2 h.

Examples

  • A party hat is a cone with a height of 8 inches and a base radius of 3 inches. Its volume is V=13π(32)(8)=24πV = \frac{1}{3} \pi (3^2)(8) = 24\pi cubic inches.
  • A cylinder has a volume of 9090 cubic cm. A cone with the same base and height would have a volume of V=13(90)=30V = \frac{1}{3} (90) = 30 cubic cm.
  • An hourglass cone holds 15π15\pi cubic inches of sand and has a base radius of 3 inches. Its height is found by solving 15π=13π(32)h15\pi = \frac{1}{3} \pi (3^2)h, which gives h=5h=5 inches.

Explanation

A cone's volume is directly related to a cylinder's. For a cone and cylinder with the same base radius and height, the cone's volume is exactly one-third of the cylinder's. You could fit three full cones of water into one cylinder.