8.3: Area of Polygons and Circles

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You may use a calculator for most of this module as needed.

$a rectangular sign indicating that Area 51 viewer's guide is sold here.$

We have seen that the perimeter of a polygon is the distance around the outside. Perimeter is a length, which is one-dimensional, and so it is measured in linear units (feet, centimeters, miles, etc.). The area of a polygon is the amount of two-dimensional space inside the polygon, and it is measured in square units: square feet, square centimeters, square miles, etc.

You can always think of area as the number of squares required to completely fill in the shape.

Exercises $\PageIndex{1}$

1. Find the area of this rectangle.

$diagram of a rectangle marked of in 4 by 5 squares (20 squares).$

2. Find the area of this square.

$diagram of a square marked off in 4 by 4 units.$

Answer

1. $20\text{ cm}^2$

2. $16\text{ cm}^2$

Rectangles and Squares

There are of course formulas for finding the areas of rectangles and squares; we don’t have to count little squares.

Area of a Rectangle

$A=lw$^[1] or $A=bh$

Area of a Square

\[A=s^2 \nonumber \]

Exercises $\PageIndex{1}$

Find the area of each figure.

3. $diagram of a rectangle with a height of 1.8 meters and a width of 2.7 meters$

4. $a diagram of a square with a side length of 3.5 feet.$

Answer

3. $4.86\text{ m}^2$

4. $12.25\text{ ft}^2$

Parallelograms

Another common polygon is the parallelogram, which looks like a tilted rectangle. As the name implies, the pairs of opposite sides are parallel and have the same length. Notice that, if we label one side as a base of the parallelogram, we have a perpendicular height which is not the length of the other sides.

$a diagram of 2 parallelograms each one showing a different base.$

A parallelogram with a base of $7$ units and a vertical height of $6$ units is transformed into a $7$ by $6$ rectangle, with an area of $42$ square units.

$a diagram that shows we can cut off part of a parallelogram and rearrange the pieces into a rectangle with the same base and height as the original parallelogram.$

Therefore, the formula for the area of a parallelogram is identical to the formula for the area of a rectangle, provided that we are careful to use the base and the height, which must be perpendicular.

Area of a Parallelogram

$A=bh$

Exercises $\PageIndex{1}$

Find the area of each parallelogram.

5. $a diagram of a parallelogram with a height of 10 meters and a base of 12 meters.$

6. $a diagram of a parallelogram with a height of 15 meters and a base of 24 meters.$

Answer

5. $120\text{ in}^2$

6. $360\text{ m}^2$

Triangles

When we need to find the area of a triangle, we need to identify a base and a height that is perpendicular to that base. If the triangle is obtuse, you may have to imagine the height outside of the triangle and extend the base line to meet it.

$a diagram of two triangles with the height labeled on each perpendicular to the base.$

As shown below, any triangle can be doubled to form a parallelogram. Therefore, the area of a triangle is one half the area of a parallelogram with the same base and height.

$This diagram shows that any triangle can be doubled to form a parallelogram. Therefore, the area of a triangle is one half the area of a parallelogram with the same base and height$

Area of a Triangle

$A=\dfrac{1}{2}bh$ or $A=bh\div2$

As with a parallelogram, remember that the height must be perpendicular to the base.

Exercises $\PageIndex{1}$

Find the area of each triangle.

7. $a diagram of a triangle with a base of 28 feet and a height of 15 feet and sides of 17 ft and 25 feet.$

8. $a diagram of a triangle with a base of 21 centimeters and a height of 12 centimeters with the other 2 sides measuring 20 centimeters and 13 centimeters.$

9. $a diagram of a triangle with a base of 11 centimeters and a height of 7 centimeters.$

10. $a diagram of a triangle that has a base of 17 feet and a height of 24 feet.$

Answer

7. $210\text{ ft}^2$

8. $126\text{ cm}^2$

9. $38.5\text{ cm}^2$

10. $204\text{ ft}^2$

Trapezoids

A somewhat less common quadrilateral is the trapezoid, which has exactly one pair of parallel sides, which we call the bases. The first example shown below is called an isosceles trapezoid because, like an isosceles triangle, its two nonparallel sides have equal lengths.

$a diagram of three trapezoids. The first is an isosceles trapezoid that has one set of parallel sides called bases and the other two the same length. The second trapezoid has 2 bases and one side perpendicular to the base. The third trapezoid has 2 bases and neither is perpendicular to a base.$

There are a number of ways to show where the area formula comes from, but the explanations are better in video because they can be animated.^[2]^[3]^[4]

Area of a Trapezoid

\[A=\dfrac{1}{2}h(b_1+b_2) \nonumber \]

\[A=(b_1+b_2)h\div2 \nonumber \]

Don’t be intimidated by the subscripts on $b_1$ and $b_2$; it’s just a way to name two different measurements using the same letter for the variable. (Many people call the bases $a$ and $b$ instead; feel free to write it whichever way you prefer.) Whatever you call them, you just add the two bases, multiply by the height, and take half of that.

Exercises $\PageIndex{1}$

Find the area of each trapezoid.

11. $a diagram of a trapezoid with 2 bases (6 meters and 12 meters), 2 sides (5 meters each) and a height of 4 meters.$

12. $a diagram of a trapezoid with 2 bases (5 centimeters and 26 centimeters), 2 sides (17 centimeters and 10 centimeters each) and a height of 8 centimeters.$

13. $a diagram of a trapezoid with 2 bases (19 centimeters and 29 centimeters), 2 sides (13 centimeters and 13 centimeters each) and a height of 12 centimeters.$

Answer

11. $36\text{ m}^2$

12. $124\text{ cm}^2$

13. $288\text{ cm}^2$

Circles

The area of a circle is $\pi$ times the square of the radius: $A=\pi{r^2}$. The units are still square units, even though a circle is round. (Think of the squares on a round waffle.) Because we can’t fit a whole number of squares—or an exact fraction of squares—inside the circle, the area of a circle will be an approximation.

$a circular waffle with a square grid pattern$

Area of a Circle

\[A=\pi{r^2}\)]

Remember that $\pi\approx3.1416$.

Exercises $\PageIndex{1}$

Find the area of each circle. Round to the nearest tenth or to three significant figures, whichever seems appropriate.

14. $a circle with radius labeled 3 cm$

15. $a circle with radius labeled 4 cm$

16. $a circle with diameter labeled 14 in$

17. $a circle with diameter labeled 9 in$

Each figure is a fraction of a circle. Calculate each area.

18. The radius of the quarter circle is $5$ meters.

$a diagram of a quarter circle.$

19. A quarter circle has been removed from a circle with a diameter of $7$ feet.

$A diagram of a 3 quarter circle (or a circle with 1 quarter removed).$

Answer

14. $28.3\text{ cm}^2$

15. $50.3\text{ cm}^2$

16. $153.9\text{ in}^2$

17. $63.6\text{ in}^2$

18. $19.6\text{ m}^2$

19. $28.9\text{ ft}^2$

You might choose to use capital letters for the variables here because a lowercase letter "l" can easily be mistaken for a number "1". ↵
https://youtu.be/yTnYRpcZA9c ↵
https://youtu.be/WZtO3oERges ↵
https://youtu.be/uLHc6Br2veg ↵

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Rectangles and Squares

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Parallelograms

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Triangles

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Trapezoids

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Circles

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