What do I do if I am not given the perpendicular height?

If the perpendicular height is not given, you have two main options: use Heron's Formula if you know all three side lengths of the triangle, or use the Pythagorean Theorem to calculate the height from other given information in the problem.

Can I use any side of a triangle as the base?

Yes, any side can be your base. The key requirement is that the height must be perpendicular to whichever side you choose as the base. Different choices of base will pair with different heights, but the final area will always be the same.

Why does the formula have a \frac{1}{2} in it?

A triangle with a given base and height is exactly half of a rectangle (or parallelogram) with the same base and height. That is why the area of a triangle is \frac{1}{2} \times \text{base} \times \text{height}.

What units should I use for the area?

The area is always expressed in square units. If your base and height are in centimeters, the area is in square centimeters (\text{cm}^2); if they are in inches, use square inches (\text{in}^2); if they are in feet, use square feet (\text{ft}^2).

Area of a Triangle

Calculates the area enclosed by a triangle given its base and perpendicular height; essential for geometry, SAT, and ACT math problems involving triangles.

A = \frac{1}{2}bh

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What each symbol means

What Area of a Triangle takes

Symbol	Meaning
$A$	The area of the triangle, measured in square units (such as square centimeters or square inches); this is the total space enclosed by the three sides of the triangle.
$b$	The base of the triangle, which can be any of the three sides; it is measured in linear units (centimeters, inches, meters, etc.) along that chosen side.
$h$	The perpendicular height from the base to the opposite vertex, always measured at a right angle (90 degrees) to the base; if you measure along a slant side instead of perpendicular to the base, you will get an incorrect area.

When to use it

Use this formula whenever you know a triangle's base and its perpendicular height.

Level

Usually taught in: Geometry · Appears on: SAT, ACT

Worked examples

1. Find the area with simple integer dimensions

Problem

A triangle has a base of 8 cm and a perpendicular height of 5 cm. Find the area.

$A = \frac{1}{2}bh$
Write the formula for the area of a triangle.
$A = \frac{1}{2} \cdot 8 \cdot 5$
Substitute the given values: $b = 8$ cm and $h = 5$ cm.
$A = \frac{1}{2} \cdot 40$
Multiply the base and height: $8 \times 5 = 40$ .
$A = 20$
Divide by 2: $\frac{40}{2} = 20$ .

Answer: $A = 20 \text{ cm}^2$

This is a straightforward application of the formula where both the base and height are whole numbers. We substitute directly and perform the arithmetic in order: multiply the base and height, then divide by 2.

2. Find the area when height is a fraction

Problem

A triangle has a base of 10 cm and a height of

\frac{3}{5}

cm. Find the area.

$A = \frac{1}{2}bh$
Write the formula.
$A = \frac{1}{2} \cdot 10 \cdot \frac{3}{5}$
Substitute $b = 10$ cm and $h = \frac{3}{5}$ cm.
$A = \frac{1}{2} \cdot \frac{30}{5}$
Multiply the base and height: $10 \times \frac{3}{5} = \frac{30}{5}$ .
$A = \frac{1}{2} \cdot 6$
Simplify the fraction: $\frac{30}{5} = 6$ .
$A = 3$
Multiply by $\frac{1}{2}$ : $\frac{6}{2} = 3$ .

Answer: $A = 3 \text{ cm}^2$

When the height is a fraction, multiply the base and fractional height first, then simplify before applying the $\frac{1}{2}$ factor. This problem requires careful handling of fractions at each step to avoid errors.

3. Find the area of a basketball court logo

Problem

During a basketball season, a team's logo on the court is a triangle with a base of 14 feet and a perpendicular height of 9 feet. What is the area of the logo?

$A = \frac{1}{2}bh$
Use the triangle area formula.
$A = \frac{1}{2} \cdot 14 \cdot 9$
Substitute the base $b = 14$ feet and height $h = 9$ feet from the logo dimensions.
$A = \frac{1}{2} \cdot 126$
Multiply the base and height: $14 \times 9 = 126$ .
$A = 63$
Divide by 2: $\frac{126}{2} = 63$ .

Answer: $A = 63 \text{ sq ft}$

Word problems give you real-world context but require the same process: identify the base and perpendicular height, substitute them into the formula, and work through the arithmetic step by step.

Common mistakes

Where Area of a Triangle usually goes wrong

The mistake	Why it is wrong	The fix
Writing $A = bh$ without the $\frac{1}{2}$ factor	The formula requires the $\frac{1}{2}$ multiplier because a triangle is exactly half of a rectangle (or parallelogram) with the same base and height.	Always write out the complete formula $A = \frac{1}{2}bh$ before substituting numbers, and perform the division by 2 as a separate step at the end of your calculation.
Using a slant side of the triangle as the height instead of the perpendicular distance	The formula specifically requires the perpendicular (90-degree) height from the chosen base to the opposite vertex; a slant side is not the same as perpendicular height and will give you an incorrect area.	Always verify that your height measurement is perpendicular to the base. If you are given a slant side instead, use the Pythagorean Theorem to find the actual perpendicular height, or switch to using Heron's Formula if you know all three sides.
Forgetting to multiply by $\frac{1}{2}$ after computing the product of base and height	Students often compute $bh$ correctly but then write down that product as the final area, skipping the crucial step of dividing by 2.	Write $A = \frac{1}{2}(bh)$ to remind yourself that after you compute the product of base and height, you must multiply by $\frac{1}{2}$ (or equivalently, divide by 2) to get the final area.

Tips and when to use something else

If you know all three side lengths but no perpendicular height, use Heron's Formula instead, which calculates area directly from the three sides.
You can choose any of the triangle's three sides as the base—just make sure you use the perpendicular height to that base, not to a different one.
For right triangles, the two legs are perpendicular to each other, so you can treat one leg as the base and the other as the height without any extra steps.
Always check that your units are consistent: if base and height are in centimeters, your area will be in square centimeters ( $\text{cm}^2$ ).

Related formulas

Heron's FormulaHeron's Formula finds the area of a triangle using only its three side lengths, making it useful when the height is unknown or hard to measure.Area of a RectangleMultiply the length and width of a rectangle to find the total area of flat space it covers, measured in square units.Pythagorean TheoremFind the length of any side of a right triangle given the other two sides, or check whether a triangle is a right triangle.Area of a TrapezoidCalculates the exact area of any trapezoid when you know the lengths of both parallel bases and the perpendicular height between them.

Frequently asked questions

What do I do if I am not given the perpendicular height?: If the perpendicular height is not given, you have two main options: use Heron's Formula if you know all three side lengths of the triangle, or use the Pythagorean Theorem to calculate the height from other given information in the problem.
Can I use any side of a triangle as the base?: Yes, any side can be your base. The key requirement is that the height must be perpendicular to whichever side you choose as the base. Different choices of base will pair with different heights, but the final area will always be the same.
Why does the formula have a $\frac{1}{2}$ in it?: A triangle with a given base and height is exactly half of a rectangle (or parallelogram) with the same base and height. That is why the area of a triangle is $\frac{1}{2} \times \text{base} \times \text{height}$ .
What units should I use for the area?: The area is always expressed in square units. If your base and height are in centimeters, the area is in square centimeters ( $\text{cm}^2$ ); if they are in inches, use square inches ( $\text{in}^2$ ); if they are in feet, use square feet ( $\text{ft}^2$ ).

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Reviewed 2026-09-18