E2.3 Solve problems involving the perimeter, circumference, area, volume, and surface area of composite two-dimensional shapes and three-dimensional objects, using appropriate formulas.
Skill: Solving Problems Associated with Different Attributes of Composite Plane Figures and Solids, Using Appropriate Formulas
Two-dimensional shapes and three-dimensional objects can be decomposed into measurable parts.
The attributes of length (including distance, perimeter, and circumference), area (including surface area), volume, capacity, and mass all have the property of additivity. Measures of parts can be combined to determine the measure of the whole.
For some attributes and for some shapes, relationships exist that can be expressed as formulas. To apply these formulas to composite shapes and objects, the shapes and objects are decomposed into parts that have known formulas. For example, an L-shaped area could be decomposed into two rectangles, and the smaller areas added together to calculate the whole. (Note: This does not hold true for its perimeter.)
Applying formulas to real-world contexts requires judgement and thoughtfulness. For example, to apply the formula for the area of a rectangle to a garden:
- determine whether the garden is rectangular;
- determine whether the garden is close enough to a rectangle that, for the needs of the moment, the formula can be applied;
- if not rectangular, determine whether the garden can be broken into smaller rectangles (e.g., if it is an L-shaped garden) and the areas combined;
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Measurement and Geometry
Measurement and geometry are at the heart of our daily activities, and, by the same token, they help us understand the world around us. Observing and describing an object requires mastering concepts related to these two fields. The applications of measurement in geometry are many and varied, whether it is to explain the shape of a soap bubble, determine the popularity of a web page, or estimate the population in certain districts. Measurement involves making explicit the conjectures made when solving problems; for example, determining the importance of a web page is as simple as relying on the number of links associated with other web pages. The use of measurements in geometry helps to determine an unknown length, area or volume by using known measurements of two-dimensional shapes and three-dimensional objects. Measurements can be calculated from attributes such as side length, surface area, or coordinates of a shape in a Cartesian plane.
- To construct and measure the perimeter or area of complex two-dimensional shapes, students will need experiences with decomposing them or rearranging their parts into other simple shapes or polygons.
- To calculate the perimeter, area, or volume of three-dimensional objects, students will need experiences with decomposing and rearranging them into spheres, cones, pyramids, or prisms, and by otherwise assembling them into parts of new objects. Activities for estimating measurements of three-dimensional objects provide students with a useful strategy for calculating the dimensions of irregular three-dimensional objects.
Example
The volume of Prism A is the same as that of Prism B, because if we break Prism A into two identical prisms and put them on top of each other, we get Prism B.
Image Prisms "A" and "B" are rectangular based, made of paper, and then represented with interlocking blocks of similar sizes. A version of every prism is placed next to each other.
In summary, the essential concepts that support measurement learning through geometry include:
- Calculating the area of two-dimensional shapes and the volume of three-dimensional objects by breaking them down into simpler geometric shapes and objects or rearranging their parts.
- Estimating measurements using a variety of tools, including technology.
- Calculating the perimeter, area or volume of the image of a shape following a transformation, using ratios and proportions.
Source: translated from Guide d’enseignement efficace des mathématiques de la 7e à la 10e année, Mesure et géométrie, Fascicule 3, p. 54-55.
Knowledge: Perimeter, Circumference, Area, Volume
Known length formulas at this grade level include:
- Perimeter = side + side + side + ...
- Diameter = 2 × radius or 2r
- Circumference = π × diameter or πd
Known area formulas at this grade level include:
- Area of a rectangle = base × height
- Area of a parallelogram = base × height
- Area of a triangle = 1/2 (base × height)
- Area of a trapezoid = 1/2(base 1 + base 2) × height (or its equivalent)
- Area of a circle = π × radius × radius or πr2
Known volume formulas at this grade level include:
- Volume of a prism = (area of the base) × height
- Volume of a cylinder = (area of the base) × height
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.