Polygons Examples

How many distinct triangles can a dodecagon be divided into?

For a dodecagon, n = 12. All we need to do is apply the formula n – 2 to find the number of distinct triangles in a polygon. Plugging in n = 12, we get 12 – 2 = 10, which means a dodecagon can be divided into 12 distinct triangles.

What is the total measure of the interior angles of a dodecagon? If the dodecagon is regular, what is the measure of one of these interior angles?

We can apply the formula (n – 2) × 180° to find the sum of all interior angles. In a dodecagon, n = 12, so the total number of interior angles add up to (12 – 2) × 180° = 10 × 180° = 1800°. That's a lot of degrees.

In a regular (and therefore equilateral) dodecagon, all 12 angles are congruent. If they all add up to 1800°, we know that each one will have a measure of 1800° ÷ 12 = 150°.

What's the perimeter of this pentagon?

The perimeter we're looking for is the distance around the outside of the shape. Even though it's drawn in, we don't wanna include the bottom side of that triangle.

We know the bottom section of the pentagon is a square because of the congruent sides and right angles, so the sum of three of those sides is 5 units + 5 units + 5 units = 15 units.

The fourth side of the square is the hypotenuse of a right triangle. Since both legs of the triangle are equal, we can set both their lengths to equal x and use the Pythagorean Theorem.

a² + b² = c²
x² + x² = 5²
2x² = 25
x² = 12.5
x ≈ 3.54

Since each leg of the triangle contributes to the perimeter, the total perimeter is given by 3.54 units + 3.54 units + 15 units = 22.08 units.

ΔABD is a right triangle, and EC || AB. What is the measure of ∠1?

We know ΔABD has angles that add to 180°. This means ∠ABC measures 60° (because 180° – (90° + 30°) = 60°).

Since EC || AB, we can see that AD is a transversal that crosses the two of 'em. In that case, the Corresponding Angle Postulate tells us that ∠BAE ≅ ∠CED, and both have measures of 90°. Since ∠CED is supplementary to ∠CEA, we know that ∠CEA also has a measure of 90° (because 180° – 90° = 90°).

Finally, we know that the interior angles of all quadrilaterals sum to 360°. Of course, ABCE is no exception to this rule. So let's set up an equation, plug in the measures we know, and solve for the measure of ∠1.

m∠ABC + m∠BCE + m∠CEA + m∠EAB = 360°
60° + m∠1 + 90° + 90° = 360°
m∠1 + 240° = 360°
m∠1 = 360° – 240°
m∠1 = 120°