Area of triangles, parallelograms and composite shapes
Deriving the triangle and parallelogram area formulas from a rectangle, then combining them to find the area of L-shaped and composite figures
About three to four lessons of 45 to 60 minutes
How do you find the area of a shape that is not a rectangle?
Every area formula you will ever use for a straight-sided shape traces back to one idea: the area of a rectangle, length times width. A triangle is exactly half of a rectangle that encloses it. A parallelogram can be cut and shifted into a rectangle of the exact same base and height. And an L-shaped room, an oddly shaped block of land, or a floor plan with a bay window, a 'composite' shape, is just several rectangles (or rectangle-minus-rectangle) added or subtracted together.
Once you can see any straight-sided shape as rectangles in disguise, area stops being a list of formulas to memorise and becomes one idea, applied a few different ways.
- A triangular sail or a warning signalways exactly half the area of the rectangle that encloses it
- A parallelogram-shaped garden bedcut and shift one end to make a rectangle of the same base and height
- An L-shaped house floor planthe total area is a big rectangle with a smaller rectangle removed, or two rectangles added
- A paved patio with a rectangular garden bed cut out of one cornersubtract the garden bed's area from the whole paved rectangle
What students will be able to do
Students will find the area of a triangle using A = 1/2 x base x height, find the area of a parallelogram using A = base x height, and find the area (and perimeter) of a composite shape by adding or subtracting the areas of simpler rectangles that make it up.
- I can find the area of a triangle using A = 1/2 x base x height, and explain why the formula includes the 1/2.
- I can find the area of a parallelogram using A = base x height, and explain why it does NOT need a 1/2.
- I can find a missing base or height when the area of a triangle or parallelogram is given.
- I can find the area of a composite (L-shaped or joined) figure by splitting it into rectangles and adding, or by subtracting a cut-out rectangle from a larger one.
- I can find the perimeter of a composite shape, tracing every outside edge including the ones created by a notch.
Standards this unit teaches
- AC9M7M01Australian Curriculum v9 (ACARA)Area of triangles and parallelograms
Use established formulas and suitable units to solve problems about the area of triangles and parallelograms.
- AC9M8M01Australian Curriculum v9 (ACARA)Area and perimeter of composite shapes
Solve problems about the area and perimeter of irregular and composite shapes using suitable units.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 3 area & perimeter teaching unitarea as covering, and the rectangle formula (length x width) every formula in this unit builds on
- Area in the glossarya refresher on what area measures
- Triangle in the glossarynaming the base and the height (the perpendicular distance to the opposite vertex) of a triangle
Words to teach and display
- Base
- the side of a triangle or parallelogram chosen as the reference side for the area formula
- Height (of a triangle or parallelogram)
- the PERPENDICULAR (right-angle) distance from the base to the opposite vertex or side, not a slanted side length
- Parallelogram
- a four-sided shape with two pairs of parallel sides
- Composite shape
- a shape made up of two or more simpler shapes (usually rectangles) joined together or with a piece removed
- Perimeter
- the total distance around the outside edge of a shape
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. The area of a triangle: half of a rectangle
ConcreteDraw a rectangle around any triangle so the triangle's base matches the rectangle's width and the triangle's highest point touches the top: the triangle always covers exactly HALF of that rectangle. That is why the triangle area formula is A = 1/2 x base x height.
A rectangle with base 6 and height 4 has area 6 x 4 = 24 square units. Any triangle sharing that same base and height, no matter how it leans, has exactly half that area: 24 / 2 = 12 square units. This works whether the triangle is a neat right triangle or a slanted, 'obtuse-looking' one, because the rectangle always encloses it the same way.
Find the area of a triangle with base 12 and height 7.
- The enclosing rectangle would have area 12 x 7 = 84 square units.
- The triangle is exactly half of that rectangle.
- Area = 84 / 2 = 42, or directly: A = 1/2 x 12 x 7 = 42.
Answer: Area = 42 square units.
- Why does the triangle area formula include a 1/2 but the rectangle formula does not?
- If a triangle's base is 10 and its area is 20, what must its height be?
2. The area of a parallelogram: cut and shift into a rectangle
PictorialA parallelogram looks like a 'pushed-over' rectangle, and it turns out to have exactly the same area as one. Cut a triangle off one slanted end and slide it around to the other end: the parallelogram rearranges perfectly into a rectangle with the same base and the same height, so A = base x height (no 1/2 needed this time).
A parallelogram has base 9 and height 6. Rearranged, it forms a rectangle with the same base and height: area = 9 x 6 = 54 square units, exactly the parallelogram's area, because cutting and shifting a piece never changes the total area, only its shape.
A parallelogram has area 72 square units and height 8. Find its base.
- Area = base x height, so 72 = base x 8.
- Base = 72 / 8 = 9.
Answer: Base = 9 units.
- Why does cutting a triangle off one end of a parallelogram and sliding it to the other end not change its total area?
- How is finding a missing base from area = base x height similar to solving a simple division problem?
3. Composite shapes: rectangles added or subtracted
AbstractA composite shape, like an L-shaped room, is built from rectangles. Two methods always work: SPLIT the shape into two or more rectangles and add their areas, or find the area of the smallest RECTANGLE that fully contains the shape and SUBTRACT the piece that is missing.
An 8 by 6 metre room has a 3 by 4 metre rectangular alcove missing from one corner, forming an L-shape. Using the subtraction method: the full rectangle would be 8 x 6 = 48 square metres, and the missing corner is 3 x 4 = 12 square metres, so the L-shaped room's actual area is 48 - 12 = 36 square metres. Perimeter needs care too: tracing around an L-shape's outside edge (including the two new edges the notch creates) for this room gives the same total distance as the original 8-by-6 rectangle's perimeter, 2 x (8 + 6) = 28 metres, because cutting a right-angled notch from a corner does not change the total perimeter (the two edges removed exactly equal the two edges the notch adds back in).
A composite shape is made of a 6 by 4 rectangle and a 3 by 5 rectangle joined edge to edge. Find its total area.
- Rectangle 1: 6 x 4 = 24 square units.
- Rectangle 2: 3 x 5 = 15 square units.
- Total area = 24 + 15 = 39 square units.
Answer: Total area = 39 square units.
- What are the two general methods for finding a composite shape's area?
- Why does cutting a rectangular notch out of a corner NOT change a shape's perimeter?
Common misconceptions and how to address them
MisconceptionThe 'height' of a triangle or parallelogram is the length of a slanted side.
Why it happens: Students measure whichever side looks like it goes 'up', rather than the perpendicular distance to the base.
How to address it: Height is always the PERPENDICULAR (right-angle) distance from the base to the highest point or opposite side, often shown as a dashed line with a small right-angle mark. A slanted side is almost always longer than the true height, and using it gives an area that is too big.
MisconceptionForgetting the 1/2 when finding the area of a triangle, or adding an unnecessary 1/2 to a parallelogram.
Why it happens: The two formulas (1/2 x base x height for a triangle, base x height for a parallelogram) are easy to mix up because they use the same two measurements.
How to address it: Say the reason every time: a triangle is HALF of its enclosing rectangle, hence 1/2; a parallelogram rearranges into a WHOLE rectangle of the same base and height, hence no 1/2. Attaching the reason to the rule prevents them blurring together.
MisconceptionA composite shape's perimeter can be found the same way as its area, by adding or subtracting the perimeters of the rectangles it is made from.
Why it happens: Since area combines by adding or subtracting rectangle areas, students assume perimeter must work identically.
How to address it: Perimeter must be found by tracing the ACTUAL outside boundary of the composite shape, edge by edge, not by combining the original rectangles' separate perimeters (which double-counts internal edges that are no longer on the outside).
MisconceptionFor a composite shape's area, only the subtraction method (big rectangle minus a piece) works.
Why it happens: The first method taught often sticks as 'the' method, and students do not recognise other composite shapes as the same idea.
How to address it: Both splitting into separate rectangles (adding) and enclosing-rectangle-minus-cutout (subtracting) always give the same correct answer for the same shape; which one is easier depends on how the shape is drawn. Practise both on the same shape to see they agree.
Guided practice (with answers)
1. Find the area of a triangle with base 10 and height 4.
Answer: 20 square units, because A = 1/2 x 10 x 4 = 20.
2. Find the area of a triangle with base 16 and height 5.
Answer: 40 square units, because A = 1/2 x 16 x 5 = 40.
3. Find the area of a parallelogram with base 11 and height 7.
Answer: 77 square units, because A = 11 x 7 = 77.
4. A parallelogram has area 42 square units and base 14. Find its height.
Answer: 3 units, because 42 = 14 x height, so height = 42 / 14 = 3.
5. A composite shape is a 12 by 9 rectangle with a 4 by 3 rectangle cut from one corner. Find its area.
Answer: 96 square units, because 12 x 9 = 108, and 108 - (4 x 3) = 108 - 12 = 96.
6. A composite shape is made of a 6 by 4 rectangle joined to a 9 by 2 rectangle. Find its total area.
Answer: 42 square units, because 6 x 4 = 24, 9 x 2 = 18, and 24 + 18 = 42.
Independent practice worksheets
Practise triangle, parallelogram and composite-shape area with computed, never-wrong answer keys.
Differentiation
- Physically fold a paper rectangle in half diagonally to show a triangle is exactly half of it, before introducing the formula.
- For a parallelogram, cut a paper one and physically slide the triangular piece to the other end to form a rectangle, making the 'no 1/2 needed' rule tangible.
- For composite shapes, use squared or grid paper so students can count squares to check their formula answer.
- Always mark the perpendicular height with a small right-angle symbol on every diagram, so it is never confused with a slanted side.
- Introduce composite shapes combining a triangle AND a rectangle (e.g. a rectangle with a triangular roof), requiring both formulas in one problem.
- Ask students to design a floor plan with a target total area (e.g. exactly 60 square metres) made from at least two rectangles.
- Investigate composite shapes where a piece is removed from the MIDDLE (not a corner), and discuss how the perimeter calculation changes when this happens.
- Compare two different composite shapes with the same area but different perimeters, and discuss which is more efficient to build a fence around.
Assessment: exit ticket
A three-question exit ticket sampling triangle area, parallelogram area, and composite-shape area.
1. Find the area of a triangle with base 18 and height 6.
Answer: 54 square units, because A = 1/2 x 18 x 6 = 54.
2. Find the area of a parallelogram with base 13 and height 5.
Answer: 65 square units, because A = 13 x 5 = 65.
3. A composite shape is a 20 by 12 rectangle with a 7 by 5 rectangle removed from one corner. Find its area.
Answer: 205 square units, because 20 x 12 = 240, and 240 - (7 x 5) = 240 - 35 = 205.
Teacher notes and timings
- Rough timing across three to four lessons: Lesson 1 triangle area (section 1), Lesson 2 parallelogram area (section 2), Lesson 3-4 composite shapes plus the exit ticket (section 3 and assessment).
- This unit assumes comfort with the rectangle area formula (the Grade 3 area/perimeter unit). Revisit that first if 'area = length x width' is not yet automatic.
- Language to keep repeating: height is always PERPENDICULAR to the base, never a slanted side; a triangle is half its enclosing rectangle; a parallelogram rearranges into a whole rectangle of the same base and height; a composite shape's area is rectangles added or subtracted, but its perimeter must be traced around the actual outside edge.
- The array-model figures in sections 1 and 2 deliberately show the ENCLOSING rectangle, not the triangle or parallelogram itself (this site's figure engine draws rectangular grids, not arbitrary polygons), so the caption does the work of connecting the rectangle shown to the sloped shape it represents; make this link explicit when presenting live.
- Curriculum note: AC9M7M01 (Australian Curriculum v9) covers triangle and parallelogram area at Year 7; AC9M8M01 covers composite shapes one year later at Year 8. This unit is deliberately sequenced to teach both together, since composite-shape problems are usually built from the same triangle/rectangle pieces.
- Present mode and print both work: project the array-model figures to build the 'enclosing rectangle' idea live with the class, then print the worksheets for independent practice.