Table of Contents

- 1 How do you find the area of a circumscribed quadrilateral?
- 2 Is there a circumscribed circle for every quadrilateral?
- 3 Which figures can be circumscribed by a circle?
- 4 How is the area of a circle related to the area of a quadrilateral?
- 5 What is a quadrilateral inscribed in a circle?
- 6 What is the cyclic quadrilateral theorem?
- 7 How to calculate the inscribed angle of a quadrilateral?
- 8 How to calculate the area of a quadrilateral in a circle?

## How do you find the area of a circumscribed quadrilateral?

area=½(AB+CD)h. A quadrilateral is cyclic if it can be inscribed in a circle, that is, if its four vertices belong to a single, circumscribed, circle. This is possible if and only if the sum of opposite angles is 180°. If R is the radius of the circumscribed circle, we have (in the notation of Figure 1, left):

## Is there a circumscribed circle for every quadrilateral?

In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. All triangles have a circumcircle, but not all quadrilaterals do.

**What is a circumscribed angle?**

A circumscribed angle is the angle made by two intersecting tangent lines to a circle. A tangent line is a line that touches a curve at one point. This angle is equal to the arc angle between the two tangent points on the circumference of the circle.

### Which figures can be circumscribed by a circle?

The circumscribed circle is the circle drawn outside of any other shapes such as polygon, touching all the vertices of the polygon, and is termed as circumcircle.

As we showed in an earlier section we can divide a square or a quadrilateral into two triangles. This gives us that the area of a triangle is half the area of the quadrilateral with the same base and height. We get the area of a circle by multiplying π by the radius, r, squared.

**What conclusions can you make about the angles of a quadrilateral inscribed in a circle?**

Inscribed Quadrilateral Theorem If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

## What is a quadrilateral inscribed in a circle?

An inscribed quadrilateral is any four sided figure whose vertices all lie on a circle. (The sides are therefore chords in the circle!) This conjecture give a relation between the opposite angles of such a quadrilateral. It says that these opposite angles are in fact supplements for each other.

## What is the cyclic quadrilateral theorem?

The ratio between the diagonals and the sides can be defined and is known as Cyclic quadrilateral theorem. If there’s a quadrilateral which is inscribed in a circle, then the product of the diagonals is equal to the sum of the product of its two pairs of opposite sides.

**Where do the vertices of a quadrilateral lie in a circle?**

All the four vertices of a quadrilateral inscribed in a circle lie on the circumference of the circle. The sum of two opposite angles in a cyclic quadrilateral is equal to 180 degrees (supplementary angles) The measure of an exterior angle is equal to the measure of the opposite interior angle.

### How to calculate the inscribed angle of a quadrilateral?

If a, b, c, and d are the inscribed quadrilateral’s internal angles, then a + b = 180˚ and c + d = 180˚. a + b = 180˚. Join the vertices of the quadrilateral to the center of the circle. Recall the inscribed angle theorem (the central angle = 2 x inscribed angle). a + b = 180 o.

### How to calculate the area of a quadrilateral in a circle?

The area of a quadrilateral inscribed in a circle is given by the Bret Schneider’s formula as: Area = √ [s (s-a) (s-b) (s – c) (s – c)] where a, b, c and d are the side lengths of the quadrilateral. s = Semi perimeter of the quadrilateral = 0.5 (a + b + c + d)

**How is the sum of two angles in a quadrilateral equal to?**

The sum of two opposite angles in a cyclic quadrilateral is equal to 180 degrees (supplementary angles) The measure of an exterior angle is equal to the measure of the opposite interior angle. The product of the diagonals of a quadrilateral inscribed in a circle is equal to the sum of the product of its two pairs of opposite sides.