Area of triangles and quadrilaterals, and volume with fractional edges
Finding area by composing and decomposing shapes into rectangles and triangles, and extending the volume formula to fractional edge lengths
About four lessons of 45 to 60 minutes
How do you find the area of a shape with no area formula?
There is no memorised formula for a stop-sign shape or an arrow-shaped garden bed, but you can always find one for a rectangle. The trick, used constantly in real measurement, is to break an awkward shape into pieces you DO have formulas for, find each piece's area, and add them up -- or, in reverse, to see a triangle as exactly half of a surrounding rectangle.
This unit uses that composing-and-decomposing idea to find the area of any triangle or special quadrilateral, and then applies the exact same volume formula from Grade 5, V = length x width x height, to boxes whose edges are fractions or mixed numbers rather than whole numbers.
- Triangle base 6, height 4half the surrounding rectangle: 1/2 x 6 x 4 = 12 square units
- Parallelogram base 7, height 5cut a triangle from one end, move it to the other to make a rectangle: 7 x 5 = 35 square units
- Trapezoid, bases 6 and 10, height 4average the two bases, then multiply by height: (6+10)/2 x 4 = 32 square units
- A box 2 and 1/2 x 2 x 3 unitsV = length x width x height still works with fractional edges: 2.5 x 2 x 3 = 15 cubic units
What students will be able to do
Students will find the area of triangles and special quadrilaterals (parallelograms and trapezoids) by decomposing them into rectangles and triangles or composing them into a known shape, and will find the volume of right rectangular prisms with fractional edge lengths using the formula V = length x width x height, connecting the result back to packing fractional unit cubes.
- I can find the area of a triangle using half of a surrounding rectangle.
- I can find the area of a parallelogram by rearranging it into a rectangle.
- I can find the area of a trapezoid by averaging its two parallel bases.
- I can find the volume of a rectangular prism with a fractional edge length using V = length x width x height.
- I can explain why the volume formula still works when an edge length is a fraction.
Standards this unit teaches
- 6.G.A.1Common Core (US)Area of triangles and quadrilaterals
Find the area of triangles and special quadrilaterals by composing or decomposing them into known shapes.
- 6.G.A.2Common Core (US)Volume with fractional edges
Find the volume of right rectangular prisms with fractional edge lengths using unit cubes and the formula.
- AC9M6M02Australian Curriculum v9 (ACARA)Perimeter and area of shapes (Year 6)
Solve practical problems involving the perimeter and area of regular and irregular shapes using suitable metric units.
- AC9M7M02Australian Curriculum v9 (ACARA)Volume of prisms (Year 7)
Use formulas and suitable units to solve problems about the volume of right prisms, including rectangular and triangular prisms. Australia's fractional-edge volume work sits within this Year 7 descriptor, so the volume half of this unit runs about a year ahead of the ACARA placement.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 3 area and perimeter teaching unitthe rectangle area formula every shape in this unit is decomposed back into
- Grade 5 volume teaching unitthe whole-number volume formula this unit extends to fractional edges
- Grade 5 fractions as division and multiplying fractions teaching unitmultiplying fractions and mixed numbers, needed for fractional volume
- How to teach geometrymeasuring and comparing shapes
Words to teach and display
- Decompose
- to split a shape into smaller, simpler pieces (such as rectangles and triangles) to find its area
- Compose
- to rearrange or combine pieces of a shape into a known shape, such as a rectangle
- Parallelogram
- a quadrilateral with two pairs of parallel sides
- Trapezoid
- a quadrilateral with exactly one pair of parallel sides, called the bases
- Base and height
- for a triangle or parallelogram, a chosen side (the base) and the perpendicular distance to the opposite side or vertex (the height)
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. A triangle is half of a rectangle
ConcreteDraw a rectangle, then draw a diagonal splitting it into two identical triangles. Each triangle is exactly half the rectangle's area. That relationship gives the triangle area formula: A = 1/2 x base x height, where the base and height are the same two measurements you would use for the surrounding rectangle.
For a triangle with base 6 and height 4: the surrounding rectangle would have area 6 x 4 = 24, so the triangle is half of that, 12.
Find the area of a triangle with base 8 and height 5.
- A = 1/2 x base x height = 1/2 x 8 x 5.
- 8 x 5 = 40.
- 1/2 x 40 = 20.
Answer: The triangle's area is 20 square units.
- Why is a triangle's area formula 1/2 x base x height?
- If a rectangle has area 40, what is the area of a triangle with the same base and height?
2. Parallelograms and trapezoids by rearranging
PictorialA parallelogram can be cut and rearranged into a rectangle: slice a triangle off one slanted end and slide it to the other end, and the parallelogram becomes a rectangle with the same base and height. So a parallelogram's area is simply base x height, the same formula as a rectangle.
A trapezoid has only one pair of parallel sides (the two bases, often different lengths), so it decomposes differently: split it into a rectangle and one or two triangles, or use the shortcut formula A = (base 1 + base 2) / 2 x height, which averages the two parallel bases before multiplying by the height.
Find the area of a parallelogram with base 9 and height 6, and the area of a trapezoid with bases 6 and 10 and height 4.
- Parallelogram: A = base x height = 9 x 6 = 54.
- Trapezoid: average the bases first: (6 + 10) / 2 = 8.
- Multiply by the height: 8 x 4 = 32.
Answer: The parallelogram's area is 54 square units. The trapezoid's area is 32 square units.
- Why does rearranging a parallelogram into a rectangle not change its area?
- Why does a trapezoid's area formula average the two bases first?
3. Volume with fractional edge lengths
AbstractThe Grade 5 volume formula, V = length x width x height, still works exactly when an edge length is a fraction or mixed number -- multiply the three edge lengths together the same way. A box 2 and 1/2 units long, 2 units wide, and 3 units tall: V = 2.5 x 2 x 3 = 15 cubic units.
This still connects back to packing unit cubes: imagine filling the box with tiny cubes, each 1/2 unit on a side, so 8 of them (2 x 2 x 2) would exactly fill one full unit cube. A 2.5 x 2 x 3 box needs 5 x 4 x 6 = 120 of these little cubes, each worth 1/8 of a full unit cube, giving 120 x 1/8 = 15 cubic units -- exactly matching the formula.
A prism measures 1 and 1/2 units long, 4 units wide, and 2 units tall. Find its volume.
- Change the mixed number to a decimal or improper fraction: 1 and 1/2 = 1.5 (or 3/2).
- V = length x width x height = 1.5 x 4 x 2.
- 1.5 x 4 = 6. Then 6 x 2 = 12.
Answer: The prism's volume is 12 cubic units.
- Does the volume formula change at all when an edge length is a fraction?
- Why does packing little 1/2-unit cubes give the same answer as multiplying the fractional edge lengths directly?
Common misconceptions and how to address them
MisconceptionA triangle's area formula is base x height, the same as a parallelogram, without the 1/2.
Why it happens: Students remember 'multiply base by height' from parallelograms and rectangles and drop the halving step that is unique to triangles.
How to address it: Show the triangle sitting inside its surrounding rectangle every time: the triangle is visibly half the rectangle, so its area must be half of base x height, never the full amount.
MisconceptionThe height of a triangle or parallelogram is the length of one of its slanted sides.
Why it happens: Students pick whichever labelled side looks like it is 'going up' rather than checking for a true perpendicular measurement.
How to address it: The height is always the perpendicular (straight up-and-down) distance from the base to the opposite side or vertex, which is often a dashed line drawn inside the shape, not one of its actual sides, especially for a slanted parallelogram or triangle.
MisconceptionThe volume formula does not work once an edge length is a fraction, and you must count fractional cubes by hand instead.
Why it happens: Students trust the formula only for whole-number examples and revert to slower counting once a fraction appears.
How to address it: The formula V = length x width x height applies exactly the same way whether the edges are whole numbers or fractions -- multiply all three, same as always. Show the fractional-cube-packing picture once to prove why, then trust the formula for every case afterward.
Guided practice (with answers)
1. Find the area of a triangle with base 8 and height 5.
Answer: 20 square units, since 1/2 x 8 x 5 = 20.
2. Find the area of a parallelogram with base 9 and height 6.
Answer: 54 square units, since 9 x 6 = 54.
3. Find the volume of a prism 3 units long, 1 and 1/2 units wide, and 4 units tall.
Answer: 18 cubic units, since 3 x 1.5 x 4 = 18.
4. Find the area of a trapezoid with bases 5 and 9 and height 4.
Answer: 28 square units, since (5 + 9) / 2 x 4 = 7 x 4 = 28.
Independent practice worksheets
Set the matching ChalkBee geometry worksheets for independent work. The answer keys are computed in code, so they are never wrong. Start with triangle and parallelogram area before moving to fractional-edge volume.
Differentiation
- Physically cut a paper rectangle along its diagonal to show the two triangles are identical and each is half the rectangle.
- Physically cut and rearrange a paper parallelogram into a rectangle to make the base x height formula visible, not just stated.
- Keep early fractional-volume practice to a single fractional edge (with the other two edges whole numbers) before mixing more than one fractional edge.
- Provide a labelled diagram distinguishing a slanted side from the true perpendicular height for every shape before computing its area.
- Find the area of an irregular polygon by decomposing it into a combination of rectangles and triangles.
- Derive the trapezoid area formula from scratch by splitting a trapezoid into a rectangle and two triangles and adding the pieces.
- Find a missing edge length given the volume and the other two edges, including when the missing edge is a fraction.
- Compare the volumes of two prisms with the same total edge-length sum but different individual fractional edges, to see volume is not determined by the sum alone.
Assessment: exit ticket
A three-question exit ticket for the last five minutes, sampling triangle area and fractional-edge volume.
1. Find the area of a triangle with base 10 and height 7.
Answer: 35 square units, since 1/2 x 10 x 7 = 35.
2. Find the volume of a prism 2 units long, 2 and 1/2 units wide, and 5 units tall.
Answer: 25 cubic units, since 2 x 2.5 x 5 = 25.
3. Find the area of a parallelogram with base 7 and height 5.
Answer: 35 square units, since 7 x 5 = 35.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 triangle area (section 1), Lesson 2 parallelograms and trapezoids (section 2), Lesson 3 fractional-edge volume (section 3), Lesson 4 mixed practice and the exit ticket.
- Language to keep saying: a triangle is half its surrounding rectangle, height is the perpendicular distance not a slanted side, the volume formula works the same with fractional edges. These target the three main misconceptions directly.
- This unit assumes the Grade 5 whole-number volume formula and comfortable fraction-to-decimal conversion for common fractions (halves, quarters); revisit those units first if either is shaky.
- Curriculum note: ACARA v9's perimeter-and-area-of-shapes descriptor sits at Year 6 (AC9M6M02), matching the area half of this unit closely. The formal volume-of-prisms descriptor, which covers fractional and decimal edge lengths, sits at Year 7 (AC9M7M02), so the volume half of this unit runs about a year ahead of that placement.
- Present mode and print both work: use Present to physically rearrange a projected parallelogram into a rectangle, then print for independent area and volume practice.