Do all quadrilaterals add up to 360?

Properties of quadrilaterals, interior angles of quadrilaterals

Table of Contents Show

What do the Angles in a Quadrilateral Add Up to?
How to Find a Missing Angle in a Quadrilateral
How to Find Angles in a Parallelogram
Sum of Angles in a Quadrilateral Formula
Why do Angles in a Quadrilateral Sum to 360 Degrees?
Exterior Angles of a Quadrilateral
Do all angles in a quadrilateral add up to 360?
Do all squares have 360 degrees?
Do concave quadrilaterals have 360 degrees?

Polygons: Properties of Quadrilaterals

Sum of the Interior Angles of Quadrilaterals:

To find the sum of the interior angles of a quadrilaterals, divide it up into triangles... There are two triangles... Because the sum of the angles of each triangle is 180 degrees... We get

So, the sum of the interior angles of a quadrilateral is 360 degrees.

Regular Quadrilaterals - Squares:

The properties of squares:

All sides are the same length (congruent) and all interior angles are the same size (congruent).

To find the measure of the interior angles, we know that the sum of all the angles is 360 degrees (from above)... And there are four angles...

So, the measure of the interior angle of a square is 90 degrees.

The measure of the central angles of a square:

To find the measure of the central angle of a square, make a circle in the middle... A circle is 360 degrees around... Divide that by four angles...

So, the measure of the central angle of a square is 90 degrees.

What do the Angles in a Quadrilateral Add Up to?

The four angles in a quadrilateral always add up to 360°. This rule is true for all quadrilaterals.

A quadrilateral is any 4-sided shape. Quadrilaterals also always have 4 angles.

For example, the 4 angles in a rectangle are all 90°. Four lots of 90° is 360°.

Here are some more examples of quadrilaterals where the angles are shown adding up to 360°.

How to Find a Missing Angle in a Quadrilateral

The 4 angles in a quadrilateral always add up to 360°. To find a missing angle in a quadrilateral, add up the 3 known angles and subtract this result from 360°.

For example, here is a quadrilateral containing 3 known angles of 75°, 85° and 140°.

The first step is to add up the 3 known angles. 75° + 85° + 140° = 300°.

The second step is to subtract this result from 360°.

360° – 300° = 60° and so, the missing angle is 60°.

How to Find Angles in a Parallelogram

If one angle in a parallelogram is known, all other angles can be calculated. To find missing angles in a parallelogram, use the following rules:

Opposite angles are equal.
Adjacent angles add to 180°.

In the example below, one angle is known to be 120°.

The angle opposite to this angle is the same size and so, angle a = 120°.

Angles b and c are both next to the 120° angle.

Therefore 120° + b = 180° and angle b = 60°.

120° + c = 180° and angle c = 60°.

b and c are opposite to each other, so we can see that they too are equal.

We can check the angles in our parallelogram by making sure that all 4 angles add to 360°.

120° + 120° + 60° + 60° = 360° and so, the result is correct.

Sum of Angles in a Quadrilateral Formula

The formula for the sum of the angles in a polygon is (n-2) × 180°, where n is the number of sides. A quadrilateral has 4 sides and so, n = 4. The formula (n-2) × 180° becomes 2 × 180° = 360°. Therefore the sum of the angles in a quadrilateral is 360°.

The formula for the sum of the angles in any polygon is (n-2) × 180°.

The (n-2) part of the formula tells us how many triangles can be drawn inside any polygon.

n is the number of sides the shape has. Every time a new side is added, a new triangle can be formed.

In a quadrilateral, we can only form 2 triangles. For a quadrilateral, n = 4 and so the (n-2) becomes 4 – 2 which equals 2.

We have 2 triangles, so 2 lots of 180°.

2 × 180° = 360° and so, all 4 angles in a quadrilateral add to 360°.

Why do Angles in a Quadrilateral Sum to 360 Degrees?

Angles in a quadrilateral add to 360° because two triangles can be made inside any quadrilateral by drawing straight lines from one corner to the other corners. Each triangle contains 180° and so, two triangles contain 360°.

In the example below, we see the four-sided shape divided into two triangles.

The 3 angles in each triangle add up to 180°.

The combination of both triangle angles makes the full angles of the quadrilateral.

2 lots of 180° is 360°.

Therefore the angles in a quadrilateral sum to 360°.

All quadrilaterals sum to 360° because we can always draw two triangles inside them by connecting two opposite corners.

Exterior Angles of a Quadrilateral

The exterior angles of a quadrilateral always add up to 360°. If the quadrilateral is shrunk down to the size of a point, all that is left are the exterior angles around the outside of it. Angles around a point add to 360° and so, the exterior angles add to 360°.

In the image below, we can see the exterior angles of a quadrilateral marked.

We can see that these exterior angles encircle a point completely.

Angles around a point add to 360° because 360° is a full turn.

All 4 exterior angles on a quadrilateral add to 360°.

This rule is true for all polygons. The exterior angles of any polygon always add up to 360°.

Do all angles in a quadrilateral add up to 360?

What are angles in a quadrilateral? Angles in a quadrilateral are the four angles that occur at each vertex within a four-sided shape; these angles are called interior angles of a quadrilateral. The sum of the interior angles of any quadrilateral is 360 ° 360° 360°. We can prove this using the angle sum of a triangle.