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Quizzes > Mathematics

Decagon practice problems: interior angle sum quiz

Quick quiz on decagon angle sum to test your geometry skills. Instant results.

Editorial: Review CompletedCreated By: Paras WaghmareUpdated Aug 27, 2025
Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art decagon with angle marks and number labels on teal background quiz theme interior angle sum challenge

This quiz helps you practice finding the interior angle sum of a decagon and apply the formula on quick, clear problems. Get instant results and build speed with focused questions. When you want more, review angle pairs in our angle relationships quiz, or broaden skills with an angles quiz.

What is the sum of the interior angles of a decagon?
1080°
1440°
1620°
1260°
The sum of interior angles in any n-sided polygon is given by (n-2)×180°. For a decagon, n=10 so (10?2)×180°=8×180°=1440°. This formula applies regardless of whether the polygon is regular or irregular.
Which formula gives the sum of interior angles for any n-sided polygon?
n×180°
(n+2)×180°
(n-2)×180°
(n-1)×180°
The general derivation divides a polygon into (n-2) triangles, each summing to 180°. Thus the total interior angle sum is (n-2)×180°.
What is the measure of each interior angle of a regular decagon?
160°
144°
168°
126°
A regular decagon has all interior angles equal, so divide the total sum 1440° by 10 to get 144° per angle. Regular polygons split the sum evenly.
A decagon has one interior angle measuring 120°. What is the sum of the remaining nine interior angles?
1420°
1240°
1320°
1280°
Subtract the known angle from the total: 1440° ? 120° = 1320°. The remaining nine angles must sum to 1320°.
What is the measure of each exterior angle of a regular decagon?
72°
144°
36°
18°
The sum of exterior angles of any convex polygon is 360°, so each is 360°/10 = 36° in a regular decagon.
In any convex decagon, what is the relationship between an interior angle and its adjacent exterior angle?
They sum to 360°
They are supplementary
They are complementary
They are equal
For convex polygons, the interior and its adjacent exterior angle share a straight line, summing to 180°. Thus they are supplementary.
One interior angle of a decagon is 150°. What is its corresponding exterior angle?
180°
210°
150°
30°
An exterior angle is supplementary to its interior, so 180° ? 150° = 30°. This applies at each vertex of a convex polygon.
In an irregular decagon, nine interior angles are each 140°. What is the tenth interior angle?
100°
60°
80°
180°
Nine angles sum to 9×140°=1260°, so the tenth is 1440°?1260°=180°. Total must equal the decagon's interior sum.
A concave decagon has one interior angle of 200°; the other nine are equal. What is the measure of each of the nine equal angles?
122.22° (approx)
137.78° (approx)
1240°
144°
Total interior sum is 1440°, subtract the concave angle: 1440°?200°=1240°. Divide by 9 gives ?137.78° each. The formula holds for concave polygons.
What is the sum of the exterior angles of a concave decagon, taken one at each vertex?
180°
360°
1080°
1440°
The sum of one exterior angle per vertex is 360° for any simple polygon, convex or concave, when angles are measured consistently.
How many triangles can a decagon be divided into by drawing non-overlapping diagonals from one vertex?
7
9
10
8
Drawing diagonals from one vertex divides an n-gon into (n-2) triangles. For n=10, that's 10?2=8 triangles.
How many diagonals does a decagon have?
25
35
40
30
A polygon has n(n?3)/2 diagonals. For n=10: 10×7/2=35 diagonals in a decagon.
The interior angles of a decagon form an arithmetic progression with smallest 120° and largest 200°. What is the common difference?
12°
8°
16/3°
5°
Sum of an AP: n/2[2a?+(n?1)d]=1440°. Here 10/2[2×120+9d]=1440 gives 5(240+9d)=1440 ?240+9d=288 ?d=48/9=16/3°.
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Study Outcomes

  1. Calculate the Interior Angle Sum of a Decagon -

    Participants will use the formula (n − 2) × 180° to find the interior angle sum of a decagon, mastering the steps required for decagon angle calculations.

  2. Apply the Interior Angle Sum Formula to Polygons -

    Readers will extend the interior angle sum of a polygon formula to pentagons, hexagons, and beyond, gaining confidence in general polygon angle sums.

  3. Analyze Relationships Between Angles and Sides -

    Users will interpret how the number of sides affects each interior angle and total sum, enhancing their geometric reasoning skills.

  4. Solve an Interactive Polygon Quiz -

    By taking the quiz, learners will actively test their knowledge of what the angle sum of a decagon is under engaging, timed conditions.

  5. Verify Calculation Accuracy -

    Participants will develop techniques to check and confirm their interior angle sum calculations, reducing errors in geometry problems.

  6. Compare Angle Sums of Different Polygons -

    Readers will contrast interior angle sums across various polygons, observing patterns and reinforcing their understanding of polygon geometry.

Cheat Sheet

  1. General Polygon Angle Sum Formula -

    Every simple polygon's interior angle sum follows the formula (n − 2)×180°, where n is the number of sides. This interior angle sum of a polygon rule holds for both convex and concave shapes. University mathematics departments around the world reference this as the foundational angle-sum theorem.

  2. Decagon Interior Angle Sum -

    If you've ever asked "what is the angle sum of a decagon," the answer is (10 − 2)×180°=1,440°. This decagon interior angle sum calculation shows why a ten-sided figure always totals 1,440°. Many academic sources, including MIT OpenCourseWare, demonstrate this exact example.

  3. Regular Decagon Individual Angles -

    In a regular decagon, each of the ten interior angles is equal, so each angle measures 1,440°÷10=144°. Knowing this makes angle-finding easy when all sides and angles are congruent. This fact is commonly used in competition problems and geometry textbooks.

  4. Triangulation Method -

    Visualize the interior angle sum of a decagon by dividing it into eight triangles from one vertex, since 10−2=8. Each triangle has 180°, so 8×180°=1,440° neatly explains the decagon interior angle sum. This triangulation trick is taught in many university geometry courses.

  5. Mnemonic Sequence for Quick Recall -

    Memorize the sequence: triangle 180°, quadrilateral 360°, pentagon 540°, …, decagon 1,440° by thinking "add 180° each time." This mnemonic helps you instantly recall what is the angle sum of a decagon without re-deriving the formula. It's a favorite technique in math contests and high-school review guides.

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