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Calculate the sum of interior angles for polygons with 6 and 10 sides, and find the number of sides for a polygon with an angle sum of 540 degrees.

Mathematics

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Answers

3 parts
a

720∘\text{}720^\circ\text{}

b

1440∘\text{}1440^\circ\text{}

c

5\text{}5\text{}

Step-by-step solution

7 steps

The formula for the sum of the interior angles of a polygon with nn sides is given by S=(n−2)×180∘S = (n-2) \times 180^\circ.

Part a· 2 steps

Find the sum of the interior angles of a regular polygon with 6 sides.

  1. Identify the number of sides, nn.

    For a polygon with 6 sides, n=6n=6.

  2. Substitute nn into the formula for the sum of interior angles.

    S=(6−2)×180∘S = (6-2) \times 180^\circ S=4×180∘S = 4 \times 180^\circ S=720∘S = 720^\circ

    The sum of the interior angles is 720∘\boxed{\text{}720^\circ\text{}}

Part b

1440∘\text{}1440^\circ\text{}

A polygon has 10 sides. Find the sum of interior angles.

  1. Identify the number of sides, nn.

    For a polygon with 10 sides, n=10n=10.

  2. Substitute nn into the formula for the sum of interior angles.

    S=(10−2)×180∘S = (10-2) \times 180^\circ S=8×180∘S = 8 \times 180^\circ S=1440∘S = 1440^\circ

    The sum of the interior angles is 1440∘\boxed{\text{}1440^\circ\text{}}

Part c

5\text{}5\text{}

If the sum of interior angles of a regular polygon is 540∘540^\circ, find the number of sides.

  1. Set the sum of interior angles formula equal to the given sum.

    (n−2)×180∘=540∘(n-2) \times 180^\circ = 540^\circ

  2. Divide both sides by 180∘180^\circ.

    n−2=540∘180∘n-2 = \frac{540^\circ}{180^\circ} n−2=3n-2 = 3

  3. Solve for nn.

    n=3+2n = 3 + 2 n=5n = 5

    The number of sides is 5\boxed{\text{}5\text{}}

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