Math Core

How much fence goes around a yard, and how much sod covers it? The fence is the perimeter, the distance around a shape. The sod is the area, the space inside it. In this lesson you'll use formulas for both, and then turn those formulas into equations to find a missing length.

Perimeter

To find the perimeter of any polygon, add the lengths of all its sides. For a rectangle with length $\ell$ and width $w$ , two sides of each length give

P = 2\ell + 2w.

Perimeter is a length, so it's measured in plain units: cm, ft, m.

Area formulas

Area counts square units. Every formula below comes from the rectangle.

Area formulas

Shape	Area
Rectangle	$A = \ell w$
Parallelogram	$A = bh$
Triangle	$A = \dfrac{1}{2}bh$
Trapezoid	$A = \dfrac{1}{2}(b_1 + b_2)h$

Here $b$ is a base and $h$ is the height, the perpendicular distance from the base to the opposite side or corner.

Why these work:

Parallelogram: slice a right triangle off one end and slide it to the other end. You get a rectangle with the same base and height.
Triangle: two copies of the same triangle fit together to make a parallelogram. The triangle is half of it.
Trapezoid: two copies of the same trapezoid, one flipped, make a parallelogram with base $b_1 + b_2$ . The trapezoid is half of it.

A triangle with base 8 cm. The dashed segment is the height, 4 cm.

Worked example: Area of a triangle

Find the area of the triangle above.

The base is $8$ cm and the height is $4$ cm. The slanted sides are not the height.

A = \frac{1}{2}bh = \frac{1}{2}(8)(4) = 16 \text{ cm}^2

Common mistake

The height must meet the base at a right angle. Using a slanted side as the height is the most common mistake with triangles and parallelograms. In the triangle above, $\dfrac{1}{2}(8)(5) = 20$ would be wrong.

A trapezoid with bases 9 m and 4 m and height 4 m.

Worked example: Area of a trapezoid

Find the area of the trapezoid above.

Add the two parallel bases, multiply by the height, and take half:

A = \frac{1}{2}(9 + 4)(4) = \frac{1}{2}(13)(4) = 26 \text{ m}^2

Composite shapes

A composite shape is made of simpler pieces. Split it into rectangles and triangles, find each area, then add. Sometimes it's easier to find a big area and subtract the part that's missing.

Worked example: An L-shaped room

An L-shaped room is a $12$ ft by $10$ ft rectangle with a $4$ ft by $5$ ft rectangle cut out of one corner. Find its area.

Whole rectangle: $12 \times 10 = 120 \text{ ft}^2$ . Missing corner: $4 \times 5 = 20 \text{ ft}^2$ .

Area: $120 - 20 = 100 \text{ ft}^2$ .

Working backward with equations

A formula is an equation. If you know the area or perimeter and all but one length, substitute what you know and solve for the missing length.

Worked example: Finding a missing side

A rectangular garden has perimeter $46$ m and length $15$ m. How wide is it? What is its area?

\begin{aligned} 2\ell + 2w &= P \\ 2(15) + 2w &= 46 \\ 30 + 2w &= 46 \\ 2w &= 16 \\ w &= 8 \end{aligned}

The garden is $8$ m wide, and its area is $15 \times 8 = 120 \text{ m}^2$ .

Tip

Check units at the end. Perimeter is in units (m); area is in square units ( $\text{m}^2$ ). If you get square units for a perimeter, you multiplied when you should have added.