Measuring Angles of Two- & Three-Dimensional Figures

Instructor: Matthew Bergstresser

Matthew has a Master of Arts degree in Physics Education

Both two-dimensional and three-dimensional figures contain angles. In this lesson, we'll investigate the tools we need to measure these angles through some practice problems.

Two- and Three-Dimensional Shapes

Can you think of any two-dimensional shapes? A two-dimensional shape has to fit on an x-y axis and cannot include a z-axis. Some two-dimensional shapes include a rectangle, triangle and parallelogram. Three-dimensional shapes fit on all three axes and include a cube and pyramid. Let's look at a few tools that can be used to measure angles in both two-dimensional and three-dimensional shapes.

Measuring Tools

The most basic of measuring tools is a ruler. But, wait! How can you measure an angle with a ruler? You can't, but you can measure the sides of triangles and then calculate their angles using those measurements.


A protractor is a tool that measures angles. It consists of a straight edge and an arc with progressively larger degree marks. To measure an angle with a protractor, place the vertex in the small circle in the middle of the straight edge. The vertex of a triangle is where two sides of the triangle meet.

Rulers and protractors can be used to measure angles.

Angle Finder

For three-dimensional objects, you can use an angle finder, which has two rotatable arms and a circular grid. Place the arms of the angle finder on either side of the three-dimensional object and locate the angle measurement on the grid.

The black arrow shows the angle between the two blue arms on an angle finder.

These tools are commonly used in carpentry when complex angles have to be cut. Let's put these tools to use and measure some angles!

Measuring Angles: Examples

Two-Dimensional Figures

Let's measure the angle of a triangle. First, put the vertex of the triangle in the small circle and follow the hypotenuse to the arc of the protractor. Now, identify the line where the hypotenuse intersects with the arc of the protractor. In this example, the number circled in red is the angle of the triangle's vertex, or 20°.

Measuring the angle of a right triangle

Now let's look at the vertex of a trapezoid. Notice that the angle of this vertex is obtuse, which means it's greater than 90°. So, you have to add 90° to the 10° measured with the protractor to get the full angle: 90° + 10° = 100°.

Measuring the angle of a vertex of a trapezoid

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