AC9M7M01 · Year 7 · Measurement

Area of triangles and parallelograms

ACARA v9 CONTENT DESCRIPTION “solve problems involving the area of triangles and parallelograms using established formulas and appropriate units”

You already know that the area of a rectangle is its length times its width. This year you extend that idea to two more shapes, the parallelogram and the triangle. The beautiful thing is that you do not need to memorise unrelated formulas: both new formulas come directly from the rectangle you already understand, just by cutting and rearranging.

Area measures the space inside a shape, counted in square units. Learning to find the area of triangles and parallelograms, using their established formulas, opens the door to working out the area of almost any straight-sided shape, since complex shapes can be split into these simpler pieces.

The area of a parallelogram

A parallelogram looks like a pushed-over rectangle. To find its area, imagine cutting a triangular piece off the slanted end and sliding it across to the other side. The shape becomes a perfect rectangle, with exactly the same base and the same height. Because rearranging the pieces does not change the area, the parallelogram has the same area as that rectangle: base times height.

A parallelogram is a rectangle
Change the slant: the shape skews but the area never moves, because area is base times perpendicular height.
Sliding the slanted end across rebuilds the same 7 by 4 rectangle, so the area stays 28 square units. The slant changes the shape, not the area, because only the straight-up height counts.

The one subtlety is which length to use as the height. It must be the perpendicular height, measured straight up from the base, not the length of the slanted side. The slanted side is longer, but it is the straight-up distance that determines how much space the shape encloses. Getting this right is the most common place to slip with parallelograms.

The area of a triangle

A triangle is even simpler once you see it the right way. Take any triangle and make an identical copy of it. Rotate the copy and fit the two together, and they form a parallelogram. Since the triangle is exactly half of that parallelogram, its area must be half of base times height. One triangle, half the parallelogram, half the area.

A triangle is half a parallelogram
Step in the rotated copy to complete the parallelogram and see the triangle is exactly half of it.
One triangle with base 8 and height 5. On its own its area is half of 8 times 5, which is 20 square units.

So the area of a triangle is one half times base times height. A triangle with base 10 and height 6 has area one half of 10 times 6, which is half of 60, giving 30 square units. As with the parallelogram, the height is the perpendicular distance from the base to the opposite corner, measured straight up. With these two formulas, both grown from the rectangle, you can find the area of any triangle or parallelogram, and by splitting them apart, the area of far more complicated shapes as well.

Teaching tip: scissors and paper make these formulas obvious. Cut a parallelogram from paper, snip the slanted end, and slide it to form a rectangle, so the student sees the area is unchanged. For the triangle, cut two identical ones and fit them into a parallelogram. Building the formulas by hand makes them impossible to forget.

The recurring trap is using the slanted side as the height. Keep returning to the idea that height means straight up from the base, perpendicular to it. A quick sketch with the perpendicular height marked, every time, trains the eye to find the right measurement.

Builds on: Area of a rectangle (AC9M6M02). That unit found the area of rectangles; this unit extends area to triangles and parallelograms.

Half the squares, every time

Here is the same half idea, counted directly. A triangle is drawn inside the rectangle that has its base and its height, and it covers exactly half of the unit squares. Change the base and the count stays half, which is why the triangle area is half of base times height.

A triangle fills half its rectangle
Count the unit squares: a triangle always covers exactly half of its base-by-height rectangle.
Inside a 6 by 4 rectangle of area 24, the triangle fills exactly half the unit squares. So its area is half of 6 times 4, which is 12 square units.

Building areas from pieces

Many real shapes are not a single triangle or parallelogram, but they can be cut into ones that are. A house outline splits into a rectangle and a triangle. Find the area of each piece with the formula that suits it, then add the pieces together for the total area.

A complex shape is pieces added
Split an awkward shape into a rectangle and a triangle, find each area, and add them.
The rectangular body is 8 wide and 5 tall, so its area is 8 times 5 = 40 square units.

Using appropriate units

The descriptor asks for appropriate units, and area always has a square unit. If the sides are in centimetres the area is in square centimetres; if they are in metres the area is in square metres. The number comes from the formula, but the unit must match the lengths.

Area needs the right square unit
The area value stays 15, but the unit changes with what the sides were measured in.
With base 5 and height 3, the area is always 15. But the unit depends on what was measured: 5 centimetres by 3 centimetres gives 15 square cm. Area is always in square units that match the lengths.
Quick self-check
1. What is the area of a parallelogram with base 8 cm and height 5 cm?
2. The area of a triangle is found using
3. A triangle has base 10 cm and height 6 cm. What is its area?
4. When finding the area of a parallelogram, which height do you use?
5. Why does a triangle have half the area formula of a parallelogram?
Teaching pack: free to printReady-to-teach plans, student sheets, cut-outs and answers for this unit. Print or save as PDF.