6th Grade · 6.G.A.1

Area of triangles in 6th grade

Finding the area of triangles is one of the main geometry skills of 6th grade. Your child learns that every triangle is half of a rectangle or a parallelogram, which gives the rule $\frac{1}{2} \times b \times h$ . The same cut-and-rearrange thinking then finds the area of other four-sided shapes and odd-shaped polygons.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

What it means, in plain words

Area is the amount of flat space inside a shape. It is measured in square units, such as square inches or square centimeters. In 5th grade your child found the area of a rectangle by multiplying its length by its width. In 6th grade the rectangle becomes the tool for every other shape.

Every triangle is exactly half of a rectangle or a parallelogram (a four-sided shape whose opposite sides are parallel, like a rectangle pushed over) with the same base and height. That is why the area of a triangle is $\frac{1}{2} \times b \times h$ . The base ( $b$ ) can be any side. The height ( $h$ ) is the straight-up distance from the base to the opposite corner, and it meets the base at a square corner. In many triangles the height is not one of the sides, and it can even fall outside the triangle.

The same idea works for other shapes. Cut a triangle off one end of a parallelogram and slide it to the other end to make a rectangle, so a parallelogram's area is base times height. A trapezoid (a four-sided shape with one pair of parallel sides) or an L-shaped room gets cut into rectangles and triangles, and the areas of the parts are added. With the area of triangles and rectangles, children work out real things, such as how much carpet a room needs.

Your child should be able to:

Find the area of a triangle with $\frac{1}{2} \times b \times h$
Pick out the height that goes with a chosen base, even when the height is not a side
Find the area of a parallelogram or a trapezoid by cutting it into rectangles and triangles
Split an odd-shaped polygon into simple parts and add their areas

Example

What is the area of a triangle with a base of 12 centimeters and a height of 5 centimeters?

A rectangle with the same base and height has an area of $12 \times 5 = 60$ square centimeters.
The triangle is half of that rectangle: $\frac{1}{2} \times 12 \times 5 = 30$ .
The area of the triangle is 30 square centimeters.

Example

A parallelogram has a base of 10 meters and a height of 7 meters. A line from one corner to the opposite corner cuts it into two triangles. What is the area of each triangle?

The area of the parallelogram is base times height: $10 \times 7 = 70$ square meters.
The two triangles are the same size, so each one has half of the area.
$70 \div 2 = 35$ , so each triangle has an area of 35 square meters.

Where children usually get stuck

On a triangle with a base of 10 cm, a slanted side of 7 cm and a height of 6 cm, your child writes $\frac{1}{2} \times 10 \times 7 = 35$ .
The slanted side looks like the side that goes up, so it gets used as the height. The height has to meet the base at a square corner, and the slanted side does not. The right answer is 30 square centimeters.
What helps. Put the corner of an index card on the base so one edge points straight up to the top corner of the triangle. That edge shows where the height is.
For a triangle with a base of 12 cm and a height of 5 cm, your child writes $12 \times 5 = 60$ square centimeters.
Base times height is the rectangle rule from 5th grade. The child does not yet see the triangle as half of that rectangle.
What helps. Draw the triangle on graph paper and draw the rectangle around it. Shade the triangle. Your child can see that the shaded part fills half the rectangle.
Your child adds the three sides of a triangle and calls the total the area.
Area and perimeter (the distance around a shape) both measure a shape, and both use the side lengths. The child mixes up which one the question asks for.
What helps. Ask whether the question is about the fence around a garden or the grass inside it. The fence is perimeter. The grass is area, counted in squares.

Try four practice questions

Real 6th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up

The composite figure below has side lengths measured in centimeters.

Liam's work is shown below.

Step 1	$\displaystyle \text{Rectangle area} = 6 \times 4 = 24\text{ cm}^2$
Step 2	$\displaystyle \text{Triangle area} = \frac{1}{2} \times 6 \times 2 = 6\text{ cm}^2$
Step 3	$\displaystyle \text{Total area} = 24 + 6 = 30\text{ cm}$

Select the statement that best describes Liam's work.

The number is correct, but the unit should be square centimeters.

The triangle part should have been halved.

The triangle area was left out of the total.

Liam's work is correct.

Show the answer and how to solve it

The number 30 is correct, but the unit should be square centimeters. The total area is 30 square centimeters.

1The figure shows a rectangle on the bottom and a triangle on top.
2Check Step 1 and Step 2. The rectangle area is 24 square centimeters and the triangle area is 6 square centimeters. Both part calculations are correct.
3Step 3 shows the right total but labels it cm instead of square centimeters. Area is always measured in square units, so the unit must be square centimeters.

Common mistake. A common mistake is accepting a line with the right number without checking the unit. Area is always measured in square units, so the answer must be labeled square centimeters, not centimeters.

The idea underneath. After you decompose a composite figure, decide whether each part is included or removed before choosing to add or subtract the areas.

2On level
A garden bed is shaped like the trapezoid shown below. All dimensions are in feet.
Select the area of the garden bed in square feet.
A
36
B
48
C
72
D
144
Show the answer and how to solve it
The area of the garden bed is 72 square feet because 12 and 6 add to 18, half of 18 is 9, and 9 times 8 is 72.
1. 1Look at the trapezoid. The parallel bases are 12 ft and 6 ft, and the dashed segment shows the height is 8 ft.
2. 2For a trapezoid, add the bases, multiply by the height, and divide by 2.
3. 3Now simplify the expression.
4. 4So 72 is the correct choice.
Common mistake. A common mistake is choosing 144 by adding both bases and multiplying by 8 without dividing by 2. Another mix-up is choosing 48 by using only the top base of 6 instead of both bases.
The idea underneath. A trapezoid's area is the height times the average of its two parallel bases, so both bases must be used.
3On level
Find the area of the right triangle below.
Enter the area.
square units
Show the answer and how to solve it
Answer. 40
The area is 40 square units because $10 \times 8 = 80$ and half of 80 is 40.
1. 1The two legs of the right triangle meet at the right angle, so they are the base and the perpendicular height. Use 10 as the base and 8 as the height.
2. 2Two copies of this right triangle fit together along a slanted side to make a rectangle with the same base and height. Multiply the base by the height to find the rectangle area.
3. 3The triangle is one of the two equal copies, so divide the area by 2. The area is 40 square units.
Common mistake. A common mistake is multiplying $10 \times 8$ and stopping at $80$ without halving, which gives the rectangle area instead of the triangle area.
The idea underneath. Two copies of this right triangle fit together along a slanted side to make a rectangle with the same base and height. Each triangle has half the rectangle area: base times perpendicular height, divided by 2.
4Stretch
Find the total area of the composite figure below.
Enter the total area.
Total area:square units
Show the answer and how to solve it
Answer. 75
The total area is 75 square units because the figure splits into three rectangles with areas 30, 30, and 15.
1. 1The Bridge width is labeled 5. Its height is the pillar height minus the gaps above and below it: 10 - 4 - 3 = 3. The dashed lines split the figure into three rectangles: the Left Pillar (3 by 10), the Right Pillar (3 by 10), and the Bridge (5 by 3).
2. 2Find the area of the Left Pillar and the Right Pillar.
3. 3Find the area of the Bridge rectangle.
4. 4Add the 3 rectangle areas.
Common mistake. Treating the H-shape as one large rectangle counts the open spaces above and below the Bridge.
The idea underneath. A composite figure does not need a new formula. Split it into non-overlapping shapes with known area formulas and add those areas.