Year 10+ Plane Geometry

Area of Plane Shapes

Area of Plane Shapes

The area of a flat shape is the amount of space it takes up. In other words, it is the size of a surface.

Let's start with the formulas for the area of some plane shapes, and then look at a few examples.

Area Formulas

Area of Plane Shapes


\(\text{Area} = \dfrac{1}{2} \times b \times h\)

\(b\) is the length of the base, and \(h\) is the height of the triangle.

Area of Plane Shapes


\(\text{Area} = s^2\)

\(s\) is the side-length of the square.

Area of Plane Shapes


\(\text{Area} = b \times h\)

\(b\) is the base (or length) of the rectangle.

\(h\) is the height of the rectangle.

Area of Plane Shapes


\(\text{Area} = b \times h\)

\(b\) is the base (or length) of the parallelogram.

\(h\) is the height of the parallelogram.

Area of Plane Shapes


\(\text{Area} = \dfrac{1}{2}(a + b) \times h\)

\(a\) and \(b\) are the parallel sides of the trapezium.

\(h\) is the height of the trapezium.

Area of Plane Shapes


\(\text{Area} = \pi r^2\)

\(r\) is the radius of the circle.

Area of Plane Shapes


\(\text{Area} = \dfrac{1}{2} \times r^2 \times \theta\)

Note: for this to work, the angle must be measured in radians

Here, \(r\) is the radius and \(\theta\) is the angle at the centre, in radians.

Area of Plane Shapes


\(\text{Area} = \pi ab\)

\(a\) and \(b\) are half the lengths of the axes of the ellipse.

Note: In the above, the height is always perpendicular to the base of the shape.

Example 1

Area of Plane Shapes

What is the area of this rectangle?

Solution: The base has length \(8\) m, and the height is \(4\) m, so the area is

\(8 \times 4 = 32 \text{ m}^2\)

Example 2

Area of Plane Shapes

What is the area of this triangle?

Solution: The base has length \(7\) cm, and the height is \(3\) cm, so the area is

\(\dfrac{1}{2} \times 7 \times 3 = 10.5 \text{ cm}^2.\)

Example 3

Area of Plane Shapes

What is the area of this trapezium?

Solution: The parallel sides have lengths \(4\) m and \(7\) m, and the height is \(2\) m, so the area is

\(\dfrac{1}{2} \times (4 +7) \times 2 = 11 \text{ m}^2.\)

One Last Example

Sam has a part-time job, mowing the oval-shaped park shown below.

Area of Plane Shapes

If he is paid \(\$0.02\) per square metre mown, how much does he earn each time he mows the park?

Solution: We need to find the area of the oval in square metres and multiply it by \(\$0.02\) to find Sam's earnings.

The oval is made up of two straight sides of length \(60\) m, and two semi-circles of radius \(\dfrac{1}{2}(40) = 20 \text{ m}\). So, its area is equal to the area of a circle of radius \(20\) m, plus the area of a rectangle with base length \(60\) m and height \(40\) m:

\( \begin{align*} \text{Area} &= \pi r^2 + b \times h\\ &= \pi (20)^2 + (60)(40)\\ &= 400 \pi + 2,400 \text{ m}^2\\ &\approx 3656.64 \text{ m}^2. \end{align*} \)

So, Sam is paid

\( \text{Wage} = 3656.64 \times 0.02 = \$73.13 \)
for the job.


In these chapters you will learn about plane geometry topics such as 

  • Area (Irregular polygons, plane shapes etc)
  • Perimeter
  • Conic sections (Circle, Ellipse, Hyperbola etc)
  • Polygons (Congruent, polygons, similar, triangles etc)
  • Transformations and symmetry (Reflection, symmetry, transformations etc)


Even though these chapters are marked for Year 10 or higher students, several topics are for students in Year 8 or higher


Year 10 or higher, suitable for Year 8 + students as well.

Learning Objectives

Learn about Plane Geometry

Author: Subject Coach
Added on: 28th Sep 2018

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