Areas of simple shapes
The following table gives a summary of formulas for the areas of triangles, quadrilaterals (squares, rectangles, parallelograms and trapezia), sectors and circles. Being familiar with each will be useful when finding the area of composite shapes.
| Triangle |  |
Area $=\frac{1}{2}bh$=12bh |
| Square |  |
Area $=s^2$=s2 |
| Rectangle |  |
Area $=lw$=lw |
| Parallelogram |  |
Area $=bh$=bh |
| Trapezium |  |
Area $=\frac{1}{2}\left(a+b\right)h$=12(a+b)h |
| Circle |  |
Area $=\pi r^2$=πr2 |
| Sector |  |
Area =$\frac{\theta}{360}\pi r^2$θ360πr2 |
Practice using the formula for these building block shapes before moving on to composite shapes.
Practice questions
Question 1
Find the area of the solid triangle with base length $8$8 m and perpendicular height $10$10 m shown below.
Question 2
Find the area of this trapezium using the rule
$\text{Area }=\frac{1}{2}\times\left(\text{side a}+\text{side b}\right)\times\text{height h }$Area =12×(side a+side b)×height h
Question 3
A circle has a radius of $12$12 mm.
What is the exact area of the circle?
What is the area of the circle rounded to two decimal places?
Areas of composite shapes
Composite shapes consist of multiple circles, sectors, or polygons, all patched together or cut out from each other. We can find the areas of composite shapes by adding or subtracting the areas of simple shapes as appropriate.
Worked example
Find the area of the following composite shape.
Think: We can start with a rectangle that covers the whole figure, then look for the simple shapes we can cut away from this rectangle to get back to the composite shape. These will be the areas we will subtract from the area of the rectangle.
The figure below shows that we will be subtracting the areas of two triangles (triangle B and triangle C), and two parallelograms (parallelogram D and parallelogram E) from the area of one large rectangle (rectangle A).
Do: Combining the formulas for the area of each simple shape, we have:
| Total area | $=$= | Area of rectangle $-$− Area of triangles $-$− Area of parallelograms |
| $=$= | $14\times13-\frac{1}{2}\times6\times13-\frac{1}{2}\times6\times13-2\times5-8\times3$14×13−12×6×13−12×6×13−2×5−8×3 | |
| $=$= | $14\times13-6\times13-2\times5-8\times3$14×13−6×13−2×5−8×3 | |
| $=$= | $182-78-10-24$182−78−10−24 | |
| $=$= | $70$70 |
So the area of the composite shape is $70$70 m2.
Reflect: Labelling each constituent shape can be useful to keep track of which shape is being added and which shape is being subtracted.
Practice questions
Question 4
Find the area of the given shape.
Question 5
Find the total area of the figure shown.
A composite figure composed of a triangle and a parallelogram. The triangle has a horizontal base that measures $16$16 m as labeled. A vertical dashed line from the apex of the triangle to its base measures $4$4 m. The parallelogram has vertical left and right sides that both measure $13$13 m as labeled. A horizontal dash line that measures $8$8 m connect the vertical sides, indicating the length of the base of the parallelogram. The parallelogram has slanted top and bottom sides. The parallelogram sits on top of the triangle such that the slanted bottom side of the parallelogram is the same as the right side of the triangle.
Question 6
Find the area of the shaded figure shown, correct to two decimal places.
