Modeling Objects with Shapes
Aligned to G-MG.A.1 — Common Core State Standards for Mathematics.
What this lesson teaches
Real objects can be approximated by geometric shapes so their measurements can be computed. A can is a cylinder, a tent may be a triangular prism, a scoop of ice cream a sphere. Choosing a close shape lets you estimate area, volume, or capacity.
Worked examples
Example 1
Model a grain silo as a cylinder to estimate its capacity. A silo is 4 m across and 10 m tall, so radius = 2 m. Volume = pi*r^2*h = pi*(2)^2*(10) = pi*4*10 = 40*pi cubic meters, about 125.7 cubic meters. This models the storage capacity even though a real silo has a slightly domed top.
Example 2
Model a tree trunk as a cylinder to estimate its board feet. A trunk is 18 inches across and 20 feet long, so radius = 9 in = 0.75 ft. Volume = pi*(0.75)²*20 = pi*0.5625*20 which is about 35.3 cubic feet. A real trunk tapers, so this overestimates — the model is the starting point, and knowing which way it errs is part of using it.
Example 3
Common mistake: picking a model that is easy rather than one that fits. Modelling an ice cream cone with a cylinder makes the arithmetic simpler but overstates the volume by a factor of three. A good model matches the shape's key feature: something that tapers to a point wants a cone, something with a flat top and bottom and a constant cross-section wants a prism or cylinder. State which feature you matched and which you ignored.
Practice questions
- What geometric solid best models a soup can, and what two measurements do you need for its volume?
- A traffic cone is about 0.2 m in base radius and 0.7 m tall. Estimate its volume as a cone.
- Name a geometric shape that models a basketball and give the formula for its volume.
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
Modelar objetos con formas
Los objetos reales se pueden aproximar con formas geométricas para calcular sus medidas. Una lata es un cilindro, una tienda de campaña puede ser un prisma triangular, una bola de helado una esfera. Elegir una forma cercana permite estimar área, volumen o capacidad.
Ejemplo 1: Modela un silo de grano como cilindro para estimar su capacidad. Un silo mide 4 m de ancho y 10 m de alto, así que el radio = 2 m. Volumen = pi*r^2*h = pi*(2)^2*(10) = pi*4*10 = 40*pi metros cúbicos, unos 125.7 metros cúbicos. Esto modela la capacidad de almacenamiento aunque un silo real tenga una tapa algo abovedada.
Ejemplo 2: Modela el tronco de un árbol como un cilindro para estimar su volumen. Un tronco mide 18 pulgadas de diámetro y 20 pies de largo, así que el radio = 9 pulg = 0.75 pies. Volumen = pi*(0.75)²*20 = pi*0.5625*20, unos 35.3 pies cúbicos. Un tronco real se afina, así que esto sobreestima — el modelo es el punto de partida, y saber hacia qué lado se equivoca es parte de usarlo.
Ejemplo 3: Error común: elegir un modelo fácil en vez de uno que ajuste. Modelar un cono de helado con un cilindro simplifica las cuentas pero sobreestima el volumen en un factor de tres. Un buen modelo respeta el rasgo clave de la forma: algo que se afina hasta un punto pide un cono, algo con tapa y base planas y sección constante pide un prisma o un cilindro. Di qué rasgo respetaste y cuál ignoraste.
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