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11. Modeling in Two & Three Dimensions
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United States of AmericaFL
Grade 912

11.01 Area of 2D figures

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Concept summary

A dimension is a measure of length in one direction such as height, width, or depth. We often denote dimension with a single capital D.

It is common for an object to be said to be 2D, which means that the object has two dimensions (such as height and width). If we say an object is 3D, then it has three dimensions (such as height, width, and length).

One of the key properties we consider about 2D objects is their area. We have encountered many area formulas already:

Area of triangle

The area of triangle is half the product of the base times height, where b is the base and h is the height.

A = \dfrac{1}{2}bh

A triangle drawn with its base as the horizontal side labelled b and the height drawn as a dashed segment from one of the vertices perpendicular to the base and labelled as h.
Area of rectangle

The area of a rectangle is the product of the base times height, where b is the base and h is the height.

A=bh

A rectangle in which the vertical side on the right is labelled h and the bottom horizontal side at the bottom is labelled b.
Area of parallelogram

The area of a parallelogram is the product of the base times height, where b is the base and h is the height.

A=bh

A parallelogram with its height drawn as a dashed segment from one of the vertices perpendicular to the horizontal side labelled b.
Area of trapezoid

The area of a trapezoid is one half of the product of the height and the sum of the lengths of the bases, where h is the height and a and b are the bases.

A = \dfrac{1}{2}h(a+b)

A trapezoid with the top base side labelled as a and bottom base side labelled as b. A dashed segment from the top base angle perpendicular to the bottom base b is drawn and labelled with h.
Area of a square

The area of a square is the side length squared, where s is the side length.

A=s^2

A square with side length labelled s.
Area of a regular polygon

The area of a regular polygon is one half of the product of the perimeter and apothem, where P is the perimeter and a is the apothem. An apothem is the distance from the center to a side of a regular polygon.

A=\dfrac{1}{2}Pa

A regular six sided polygon with segment called apothem and labelled as a drawn from the center to a side of hte polygon. The sides are marked showing the lengths as distances from one vertex to another and labelled with P for perimeter.
Area of rhombus

The area of a rhombus is the product of the base and height, where b is the base and h is the height.

A=bh

The area of a rhombus can also be determined by one half of the product of the diagonals, where d_1 and d_2 are the diagonals.

A=\dfrac{1}{2}d_1d_2

A rhombus with diagonals drawn as dashed segments intersecting each other. The diagonals are labelled d sub 1 and d sub 2 respectively.

A composite figure is a figure that can be decomposed into smaller figures.

We can work out the area of composite figures by breaking them down into simpler shapes. After we find the area of the simpler shapes, we can add the areas to get the area of the composite shape.

Once we have the area of a shape, then we can work out the population density.

Population density

The number of people (or other living things) living in one unit of area.

\text{Population density} = \dfrac{\text{Number of people}}{\text{Area}}

Worked examples

Example 1

Find the area.

A quadrilateral with sides of length 11, 8, 4, and 13. Between sides 11 and 8 is an included right angle. The included angle between side lengths 11 and 8 as well as the angle between side lengths 4 and 13 is a right angle.

Approach

We don't have a formula for the area of this type of quadrilateral. Instead, we can decompose the shape into two right-triangles as follows:

A quadrilateral with sides of length 11, 8, 4, and 13. The included angle between side lengths 11 and 8 as well as the angle between side lengths 4 and 13 is a right angle. A dashed segment from the angle between sides of length 11 and 13 and the angle between sides of length 8 and 4 serves as the hypotenuse of the right triangles. The triangle with side lengths 11 and 8 is labelled 1 and the the triangle with side lengths 4 and 13 is labelled 2

Solution

Area of triangle 1:

\displaystyle A_1\displaystyle =\displaystyle \dfrac{1}{2}\cdot b\cdot hFormula
\displaystyle {}\displaystyle =\displaystyle \dfrac{1}{2}\cdot 8\cdot11Substitution
\displaystyle {}\displaystyle =\displaystyle \dfrac{1}{2} \cdot 88Simplify
\displaystyle {}\displaystyle =\displaystyle 44Simplify

Area of triangle 2:

\displaystyle A_2\displaystyle =\displaystyle \dfrac{1}{2}\cdot b\cdot hFormula
\displaystyle {}\displaystyle =\displaystyle \dfrac{1}{2}\cdot 4\cdot13Substitution
\displaystyle {}\displaystyle =\displaystyle \dfrac{1}{2} \cdot 52Simplify
\displaystyle {}\displaystyle =\displaystyle 26Simplify

Area of the composite shape:

\displaystyle A\displaystyle =\displaystyle A_1 + A_2Formula
\displaystyle {}\displaystyle =\displaystyle 44+26Substitution
\displaystyle {}\displaystyle =\displaystyle 70Simplify

Area = 70 \text{ units}^2

Example 2

A theatre is showing the ballet The Nutcracker. If the theatre has 400 seats for the audience and if the seating area is 300 \, \text{yd}^2, what is the population density in the seating area if every seat is filled?

Approach

We know that every seat is filled, so we know the number of people in the audience. We also know the area of the seating area, so we can use the population density formula to calculate the population density.

Solution

\displaystyle \text{Population density}\displaystyle =\displaystyle \dfrac{\text{Number of people}}{\text{Area}}Formula
\displaystyle {}\displaystyle =\displaystyle \dfrac{400}{300}Substitution
\displaystyle {}\displaystyle =\displaystyle \dfrac{4}{3}Simplify

For every 3 square yards of space, there are 4 people.

Outcomes

MA.912.GR.4.4

Solve mathematical and real-world problems involving the area of two-dimensional figures.

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