Basic Geometry

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Quick recall

• Use Pythagoras only for a right triangle.
• Distinguish an angle sum from one angle in a regular polygon.

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Formula Cards

1. Triangle angle sum

$$A+B+C=180^\circ$$

2. Polygon interior angle sum

$$(n-2)180^\circ$$

3. One interior angle of a regular polygon

$$\frac{(n-2)180^\circ}{n}$$

4. Pythagoras

$$a^2+b^2=c^2$$

Only for right triangles; c is the hypotenuse.

5. Angle in a semicircle

$$\theta=90^\circ$$

The angle subtended by a diameter at the circumference is a right angle.

6. Triangle exterior angle

$$\text{Exterior angle}=\text{Sum of two remote interior angles}$$

Worked Examples

Example 1

Question: A right triangle has legs 3 and 4. Find its hypotenuse.

Step 1: Use Pythagoras.

$$c^2=3^2+4^2=25$$

Step 2: Take the positive square root.

$$c=\sqrt{25}=5$$

Answer: 5 units.

Example 2

Question: Find the interior angle sum of a hexagon.

Step 1: Identify the number of sides.

$$n=6$$

Step 2: Apply the sum formula.

$$(6-2)\times180^\circ=720^\circ$$

Answer: 720°.

Example 3

Question: Find each interior angle of a regular pentagon.

Step 1: Find the total interior angle sum.

$$(5-2)\times180^\circ=540^\circ$$

Step 2: Divide equally among five angles.

$$\frac{540^\circ}{5}=108^\circ$$

Answer: 108°.

Example 4

Question: Two angles of a triangle are 50° and 60°. Find the third.

Step 1: Add the given angles.

$$50^\circ+60^\circ=110^\circ$$

Step 2: Subtract from 180°.

$$180^\circ-110^\circ=70^\circ$$

Answer: 70°.

Drill

Try each question before opening its answer.

Question 1

Find hypotenuse of right triangle with legs 6, 8.

• Show answer

  10

Question 2

Sum of interior angles of an octagon.

• Show answer

  1080°

Question 3

Each interior angle of a regular hexagon.

• Show answer

  120°

Question 4

Two angles of a triangle are 90° and 35°. Find the third.

• Show answer

  55°

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