Volume and Geometric Modeling
Aligned to G-GMD.A.1 — Common Core State Standards for Mathematics.
What this lesson teaches
Volume formulas tell you how much space a solid holds. A cone has exactly one-third the volume of a cylinder with the same base and height. You can apply these formulas to model and solve real design problems with containers and objects.
Worked examples
Example 1
Find the volume of a cylinder with radius 2 and height 5. Multiply pi times radius squared times height: about 3.14 times 4 times 5, which is 62.8 cubic units.
Example 2
A rectangular pool is 25 m long, 10 m wide and 2 m deep. Its volume is 25 × 10 × 2 = 500 cubic meters. Since 1 cubic meter holds 1,000 liters, the pool holds about 500,000 liters of water.
Example 3
Common mistake: mixing units inside one volume. A can 10 cm across and 0.2 m tall is not 3.14 × 25 × 0.2 — the radius is in centimeters and the height in meters. Convert first: 0.2 m = 20 cm, so V = 3.14 × 5² × 20 = 1,570 cm³. Volume units also cube, so multiplying every length by 2 multiplies the volume by 8, not by 2.
Practice questions
- Find the volume of a cylinder with radius 3 and height 10 (use 3.14).
- How does a cone's volume compare to a matching cylinder's?
- Why is volume measured in cubic units?
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Volumen y modelado geométrico
Las fórmulas de volumen te indican cuánto espacio ocupa un sólido. Un cono tiene exactamente un tercio del volumen de un cilindro con la misma base y altura. Puedes aplicar estas fórmulas para modelar y resolver problemas reales de diseño con recipientes y objetos.
Ejemplo 1: Encuentra el volumen de un cilindro con radio 2 y altura 5. Multiplica pi por el radio al cuadrado por la altura: aproximadamente 3.14 por 4 por 5, que es 62.8 unidades cúbicas.
Ejemplo 2: Una piscina rectangular mide 25 m de largo, 10 m de ancho y 2 m de profundidad. Su volumen es 25 × 10 × 2 = 500 metros cúbicos. Como 1 metro cúbico contiene 1,000 litros, la piscina contiene unos 500,000 litros de agua.
Ejemplo 3: Error común: mezclar unidades dentro de un mismo volumen. Una lata de 10 cm de diámetro y 0.2 m de altura no da 3.14 × 25 × 0.2 — el radio está en centímetros y la altura en metros. Convierte primero: 0.2 m = 20 cm, así V = 3.14 × 5² × 20 = 1,570 cm³. Las unidades de volumen también se elevan al cubo: multiplicar cada longitud por 2 multiplica el volumen por 8, no por 2.
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