Geometric Design Problems
Aligned to G-MG.A.3 — Common Core State Standards for Mathematics.
What this lesson teaches
Design problems ask you to choose dimensions or shapes to meet constraints such as minimizing cost, fitting a space, or maximizing volume. Set up an equation from the geometry, then solve for the value that satisfies the requirement.
Worked example
Design an open-top box from a 12-by-12 cm sheet by cutting squares of side x from each corner and folding. The base becomes (12 - 2x) by (12 - 2x) with height x, so volume V = x(12 - 2x)^2. Test x = 2: V = 2(12 - 4)^2 = 2(8)^2 = 2(64) = 128 cubic cm. Trying values shows x = 2 gives the maximum whole-number volume for this sheet.
Practice questions
- A rectangular pen must have area 48 square meters with one side 6 m. Find the other side.
- You have 20 m of fencing for a rectangular garden. Give the dimensions of a square that uses it all, and its area.
- A cylindrical can must hold about 500 cubic cm. If its radius is 5 cm, find the required height (use pi about 3.14).
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
Problemas de diseño geométrico
Los problemas de diseño piden elegir dimensiones o formas para cumplir restricciones como minimizar costo, caber en un espacio o maximizar el volumen. Plantea una ecuación a partir de la geometría, luego resuelve el valor que satisface el requisito.
Ejemplo: Diseña una caja sin tapa con una hoja de 12 por 12 cm cortando cuadrados de lado x en cada esquina y doblando. La base queda (12 - 2x) por (12 - 2x) con altura x, así que el volumen V = x(12 - 2x)^2. Prueba x = 2: V = 2(12 - 4)^2 = 2(8)^2 = 2(64) = 128 cm cúbicos. Al probar valores se ve que x = 2 da el volumen máximo en números enteros para esta hoja.
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