Area & Perimeter of Triangles & Quadrilaterals

Perimeter measures the distance around a figure — walk the boundary and add up every side. Area measures the number of square units inside the figure. They answer different questions: perimeter tells you how much fencing you need, area tells you how much grass you're fencing in.

Every area formula in this lesson comes from the rectangle. A parallelogram is a rectangle with its side pushed over, a triangle is half a parallelogram, and a trapezoid is the average of its two bases times the height. Once you see that chain, the formulas stop being four random facts.

The formulas

Rectangle: $A = \ell w$ — length times width. Parallelogram: $A = bh$ — base times height. Triangle: $\displaystyle A = \frac{1}{2}bh$ — half of base times height, because a triangle is half of a parallelogram with the same base and height. Trapezoid: $\displaystyle A = \frac{1}{2}(b_1 + b_2)h$ — average the two parallel bases, then multiply by the height.

Perimeter has no shortcut formula for most figures: just add all the sides. The one worth memorizing is the rectangle, $P = 2\ell + 2w$ , because its opposite sides match.

Height means perpendicular height

The single biggest mistake in this topic is using a slanted side as the height. In every area formula, $h$ is the perpendicular distance from the base to the opposite side or vertex — it must meet the base at a right angle.

In a diagram, the height is usually drawn as a dashed segment with a small square marking the right angle. If a parallelogram has a base of $10$ , a slant side of $6$ , and a dashed height of $5$ , the area is $10 \cdot 5 = 50$ , not $10 \cdot 6 = 60$ . The slant side is for perimeter; the perpendicular height is for area. The figure below shows exactly that setup.

Which number is the base?

Any side can be the base — the height just has to be perpendicular to whichever side you pick. In a triangle problem, the base and height are the pair that meet at a right angle. If a problem gives you three numbers, find the two that are perpendicular to each other and ignore the third for area.

Worked examples

Example 1: rectangle, area and perimeter

A rectangle is $8$ units by $5$ units. Find its area and perimeter.

Area is length times width

A = 8 \cdot 5 = 40

Perimeter doubles the length and the width

P = 2(8) + 2(5)

Simplify

P = 16 + 10 = 26

Answer: $A = 40$ square units, $P = 26$ units

Example 2: triangle

A triangle has a base of $10$ and a height of $6$ . Find its area.

Start with the triangle area formula

A = \dfrac{1}{2}bh

Substitute the base and height

A = \dfrac{1}{2}(10)(6)

Simplify

A = 30

Answer: $A = 30$ square units

Example 3: trapezoid

A trapezoid has bases $6$ and $10$ and a height of $4$ . Find its area.

Start with the trapezoid area formula

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

Substitute both bases and the height

A = \dfrac{1}{2}(6 + 10)(4)

Add the bases first

A = \dfrac{1}{2}(16)(4)

Simplify

A = 32

Answer: $A = 32$ square units

Try one yourself

Common questions

What's the difference between the height and a side?

The height is the perpendicular distance from the base to the opposite side or vertex — it always meets the base at a right angle. A slanted side is only the height in special cases, like the legs of a right triangle or the sides of a rectangle.

Why is a triangle's area half of base times height?

Two copies of any triangle fit together to make a parallelogram with the same base and height. The parallelogram's area is $bh$ , so each triangle gets half: $\displaystyle A = \frac{1}{2}bh$ .

What units do area and perimeter use?

Perimeter is a length, so it uses plain units like feet or centimeters. Area counts unit squares, so it uses square units like square feet or $\text{cm}^2$ . If an answer choice has the wrong kind of unit, it can't be right.

Which two bases go in the trapezoid formula?

The two parallel sides. A trapezoid has exactly one pair of parallel sides, and those are $b_1$ and $b_2$ . The other two sides never appear in the area formula — they only matter for perimeter.

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Area & Perimeter of Triangles & Quadrilaterals

The formulas

Height means perpendicular height

Which number is the base?

Worked examples

Example 1: rectangle, area and perimeter

Example 2: triangle

Example 3: trapezoid

Try one yourself

Common questions

Keep learning

Perimeter & Area on the Coordinate Plane

Area of Regular Polygons

Composite & Shaded-Region Area

Scale Factor: Perimeter, Area & Volume