Chapter 11 · Area of polygons

Triangle Area: Word Problems and Mistakes

Solve triangle area word problems by pairing a base with its perpendicular height. Then check finished work for a wrong height, a missing half or wrong units.

13 min read

A real problem about a triangle rarely hands you a formula. It gives you a story, such as a flower bed or a sail, with some lengths in it. Your first job is to find the lengths that matter.

The area of a triangle is the amount of flat space inside it, counted in square units such as square meters. To find it, you need two lengths that go together:

The base is any one side of the triangle that you choose.
The height is the distance from the line through that base to the opposite corner. You measure it perpendicular to the base, which means at a right angle to it.

In this lesson, you use that pair to solve word problems. Then you check finished work and find where it went wrong.

Key concept

A base and its perpendicular height

The area of a triangle is half of the base times the height. In this rule, $A$ is the area, $b$ is the length of the side you choose as the base, and $h$ is the height for that base.

The rule works when two things are true. First, $h$ is measured perpendicular to the base. Second, $b$ and $h$ are in the same unit of length. Then $A$ comes out in square units. If $b$ and $h$ are in meters, $A$ is in square meters. Every problem and every check in this lesson comes back to this pair.

A = \frac{1}{2} \times b \times h

A gardener draws a plan of a triangular flower bed on a coordinate grid. On every grid in this lesson, the x-axis runs left to right and the y-axis runs bottom to top. A point such as $(3, 6)$ is 3 steps right and 6 steps up from where the axes cross. The first number is its x-coordinate and the second is its y-coordinate.

The figure shows the plan. Each grid step stands for 1 meter. The corners of the flower bed are at $(1, 1)$ , $(9, 1)$ and $(3, 6)$ . No lengths are written on the plan. You read them from the corners.

The flower bed has corners at

(1, 1)

(9, 1)

and

(3, 6)

. Each grid step is 1 meter.

Worked example

Area of a flower bed

The flower bed on the plan has corners at $(1, 1)$ , $(9, 1)$ and $(3, 6)$ . Each grid step is 1 meter. How many square meters of ground does the flower bed cover?

Decide what the question asks. The ground that the flower bed covers is its area, so you need a base and its height. Choose the side from (1, 1) to (9, 1) as the base. Both corners have y-coordinate 1, so this side lies on a horizontal grid line. Let b be its length in meters. It is the difference of the x-coordinates.
$b = 9 - 1 = 8$
Find the height for that base. The base is at y-coordinate 1, and the third corner, (3, 6), is at y-coordinate 6. So that corner is 5 steps above the base, counted along a vertical grid line. Vertical grid lines are perpendicular to horizontal ones, so this distance is the height. Let h be the height in meters.
$h = 6 - 1 = 5$
Put the pair into the area rule. The rule applies because h is perpendicular to the base and both lengths are in meters. Let A be the area in square meters.
$A = \frac{1}{2} \times 8 \times 5$
Multiply. Half of 8 is 4, and 4 times 5 is 20. Meters times meters gives square meters.
$A = 4 \times 5 = 20$

Answer

The flower bed covers $20$ square meters of ground.

Do not leave out the half. For the flower bed, a common mistake is to write $8 \times 5 = 40$ square meters. That is the area of the whole 8 by 5 rectangle with corners $(1, 1)$ , $(9, 1)$ , $(9, 6)$ and $(1, 6)$ , and the flower bed does not fill it.

The figure shows why. The vertical grid line through $(3, 1)$ and $(3, 6)$ cuts the flower bed into two right triangles (triangles with a right angle). The slanted side of each piece cuts a smaller rectangle from corner to corner into two matching halves. The left piece is half of a 2 by 5 rectangle, so its area is 5 square meters. The right piece is half of a 6 by 5 rectangle, so its area is 15 square meters. Together they make $5 + 15 = 20$ square meters, which is half of 40.

Left piece: corners

(1, 1)

(3, 1)

(3, 6)

. Right piece: corners

(3, 1)

(9, 1)

(3, 6)

. Together they fill half of the 8 by 5 rectangle.

Worked example

How tall is the sail?

A triangular sail must have an area of $54$ square feet. Its bottom edge is $12$ feet long. How tall must the sail be, measured perpendicular to the bottom edge?

Name the unknown and write what you know. The question gives the area and asks for a length. Let h be the height of the sail in feet, measured perpendicular to the bottom edge. The bottom edge is the base, so b is 12 feet. The area A is 54 square feet. Put these into the area rule.
$54 = \frac{1}{2} \times 12 \times h$
Multiply the two known numbers first. Half of 12 is 6. When you multiply three numbers, you may multiply any two of them first and the result does not change, so the equation still says the same thing.
$54 = 6 \times h$
Undo the multiplication. Dividing by 6 undoes multiplying by 6, so it leaves h alone on the right. Do the same to the left side so that the two sides stay equal.
$h = 54 \div 6 = 9$
Check by putting 9 back into the area rule. Half of 12 is 6, and 6 times 9 is 54. That matches the area in the problem.
$\frac{1}{2} \times 12 \times 9 = 6 \times 9 = 54$

Answer

The sail must be $9$ feet tall. The answer is in feet, not square feet, because a height is a length.

Now switch roles. Someone else has finished a triangle area problem, and you check the work. A new answer alone is not enough. You also find the exact place where the work went wrong.

Each check tests one part of the area rule: the pair of lengths, the half and the units. You already met the second one with the flower bed. The table lists all three. After the table, you use them on a triangle that Mia measured, which is drawn in the next figure.

Three checks for triangle area work

Check	Ask yourself	What a mistake looks like
The pair	Is the height measured perpendicular to the side used as the base?	A side that is not perpendicular to the base is used as the height.
The half	Is base times height multiplied by $\frac{1}{2}$ exactly once?	The work stops at base times height, so the answer is twice the area.
The units	Are the base and height in the same unit? Is the area in square units?	An area is written as $20$ meters instead of $20$ square meters.

Mia's triangle has corners at

(0, 0)

(14, 0)

and

(5, 12)

. It is drawn as two pieces that share the edge from

(5, 0)

(5, 12)

Worked example

Find the mistake in Mia's work

Mia's triangle has corners at $(0, 0)$ , $(14, 0)$ and $(5, 12)$ on a grid where each step is 1 centimeter. The side from $(0, 0)$ to $(5, 12)$ is $13$ centimeters long. Mia writes $A = \frac{1}{2} \times 14 \times 13 = 91$ square centimeters. Find her mistake and the correct area.

Read the base and height from the corners. The corners (0, 0) and (14, 0) both lie on the x-axis, so that side is horizontal. Let b be its length in centimeters. The corner (5, 12) is 12 steps above the x-axis along a vertical grid line, which is perpendicular to the base. Let h be that height in centimeters. In the figure, it is the edge the two pieces share.
$b = 14 - 0 = 14, \quad h = 12 - 0 = 12$
Check the pair. Mia used 14 and 13. The 13 is the length of the side from (0, 0) to (5, 12). That side moves 5 steps right while it rises 12 steps, so it does not follow a vertical grid line. It is not perpendicular to the base, so 13 is not the height. This is the mistake.
Check the half and the units. Mia multiplied by one half exactly once, and she wrote square centimeters. Those parts of her work are right.
Redo the area with the matching pair, 14 and 12. Let A be the area in square centimeters. Half of 14 is 7, and 7 times 12 is 84.
$A = \frac{1}{2} \times 14 \times 12 = 7 \times 12 = 84$
Make sense of the two answers. The shortest path from the corner (5, 12) to the line through the base is the perpendicular one, which is the height. The 13-centimeter side also runs from that corner to the base, but it is not perpendicular, so it is longer than the height. Using it made the area too large.
$13 > 12, \quad 91 > 84$

Answer

Mia used the 13-centimeter side as the height, but that side is not perpendicular to the base. With the true height of 12 centimeters, the area is $84$ square centimeters.

Takeaway: every triangle area problem rests on one matching pair: a base and the height measured perpendicular to it, both in the same unit. In a word problem, find that pair in the story, use $A = \frac{1}{2} \times b \times h$ , and answer in the unit the question asks for. When you check work, test the pair, the half and the units.

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