Perimeter, Area, and Volume

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infoWhy this? Perimeter and area describe different measurable properties of shapes and require careful selection of dimensions, formulae, and units. Deriving and applying formulae develops spatial reasoning and connects geometry with multiplication and algebra.

scheduleWhy now? Following work on shapes and angles, we can identify bases, perpendicular heights, and boundaries more precisely. These foundations are needed before calculating with trapezia, circles, compound shapes, surface area, and volume.

The perimeter of a polygon is the total distance around its outside edges.
The perimeter of any polygon is found by adding the lengths of all its sides.
A regular polygon has all sides equal, so its perimeter is the number of sides multiplied by one side length.
A rectangle has two pairs of equal opposite sides, so its perimeter is `2 \times (\text{length} + \text{width})`.
A parallelogram has two pairs of parallel sides, and its opposite sides are equal in length.
Area is the amount of surface inside a 2D shape, and it is measured in square units such as `\text{cm}^2` or `\text{m}^2`.
Perimeter is measured in linear units such as cm or m, while area is measured in square units such as `\text{cm}^2`.
The area of a rectangle is `\text{length} \times \text{width}`.
The area of a rectangle can also be described as base multiplied by perpendicular height.
The area of a parallelogram is `\text{base} \times \text{perpendicular height}`.
In a parallelogram, the perpendicular height is the shortest distance between the base and the opposite side, not the sloping side length.
The area of a triangle is `\frac{1}{2} \times \text{base} \times \text{perpendicular height}`.
A triangle’s perpendicular height is the distance from the chosen base to the opposite vertex measured at a right angle.
The area of a triangle is half the area of a parallelogram with the same base and perpendicular height.
When calculating area, all dimensions must be in the same unit before using a formula.
Two shapes can have the same perimeter but different areas, and two shapes can have the same area but different perimeters.

rocket_launchYou must be able to

Calculate the perimeter of a basic polygon by identifying every side length, adding them correctly, and stating the answer in linear units.
Calculate the perimeter of a regular polygon by multiplying one side length by the number of sides.
Calculate the perimeter of a rectangle by using its equal opposite sides and applying `2 \times (\text{length} + \text{width})`.
Calculate the area of a rectangle by multiplying its length by its width and giving the answer in square units.
Identify the perpendicular height in a parallelogram by choosing the distance that meets the base at a right angle, then calculate the area using `\text{base} \times \text{perpendicular height}`.
Identify a suitable base and matching perpendicular height in a triangle, then calculate the area using `\frac{1}{2} \times \text{base} \times \text{perpendicular height}`.
Check whether an answer should be written in linear units or square units by deciding whether the calculation is for perimeter or area.
Convert dimensions into the same unit before calculating area or perimeter, so the final answer is valid.

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