Triangle

Triangle

Triangle with vertices B (top), C (bottom-left), A (bottom-right); sides labeled a, b, c; height h shown.

Key Formula

A = \frac{1}{2}\,b\,h

Where:

$A$ = Area of the triangle
$b$ = Length of the base (any one side)
$h$ = Height (perpendicular distance from the base to the opposite vertex)

Worked Example

Problem: A triangle has a base of 10 cm and a height of 6 cm. Find its area.

Step 1: Write the area formula for a triangle.

A = \frac{1}{2}\,b\,h

Step 2: Substitute the given base and height.

A = \frac{1}{2}\times 10 \times 6

Step 3: Multiply the base by the height.

10 \times 6 = 60

Step 4: Divide by 2 to get the area.

A = \frac{60}{2} = 30 \text{ cm}^2

Answer: The area of the triangle is 30 cm².

Another Example

This example focuses on the angle sum property and perimeter rather than area, showing two other fundamental triangle calculations.

Problem: A triangle has angles of 50° and 65°. Find the third angle. Then, given that the triangle has sides of 7 cm, 8 cm, and 9 cm, find its perimeter.

Step 1: Use the angle sum property: the three interior angles of any triangle add up to 180°.

\angle_1 + \angle_2 + \angle_3 = 180°

Step 2: Substitute the two known angles and solve for the third.

50° + 65° + \angle_3 = 180° \implies \angle_3 = 180° - 115° = 65°

Step 3: Find the perimeter by adding all three side lengths.

P = 7 + 8 + 9 = 24 \text{ cm}

Answer: The third angle is 65° and the perimeter is 24 cm.

Frequently Asked Questions

What are the different types of triangles?

Triangles are classified by sides or by angles. By sides: equilateral (all three sides equal), isosceles (exactly two sides equal), and scalene (no sides equal). By angles: acute (all angles less than 90°), right (one angle exactly 90°), and obtuse (one angle greater than 90°). A triangle can belong to one category from each group — for example, a right isosceles triangle.

Why do the angles of a triangle always add up to 180°?

This follows from the properties of parallel lines. If you draw a line through one vertex parallel to the opposite side, the three angles at that vertex form a straight line (180°). Alternate interior angles show that two of those angles equal the triangle's base angles. This proof works for every triangle in flat (Euclidean) geometry.

When do you use Heron's formula instead of the basic area formula?

Use Heron's formula when you know all three side lengths but do not know the height. It computes the area as

A = \sqrt{s(s-a)(s-b)(s-c)}

, where

s

is the semiperimeter. The basic formula

A = \frac{1}{2}bh

is simpler but requires the perpendicular height, which is not always given.

Triangle vs. Quadrilateral

	Triangle	Quadrilateral
Number of sides	3	4
Angle sum	180°	360°
Basic area formula	A = ½ b h	Depends on the specific shape (rectangle, parallelogram, trapezoid, etc.)
Rigidity	Rigid — side lengths fix the shape	Not rigid — can flex even with fixed side lengths
Minimum polygon?	Yes — the simplest polygon	No — one step more complex

Why It Matters

Triangles appear throughout mathematics, engineering, and science because any polygon can be divided into triangles — a process called triangulation. You will use triangle properties extensively in trigonometry, coordinate geometry, and physics (resolving forces into components). In construction and architecture, the triangle's rigidity makes it the most structurally stable shape, which is why you see it in bridges, roof trusses, and support braces.

Common Mistakes

Mistake: Using a slanted side length as the height instead of the perpendicular distance to the base.

Correction: The height must be measured at a right angle (90°) from the base to the opposite vertex. If you use a slanted side, your area calculation will be too large.

Mistake: Assuming the angle sum is 360° (confusing it with quadrilaterals).

Correction: The interior angles of a triangle always sum to exactly 180°, not 360°. A quick check: an equilateral triangle has three 60° angles, and 3 × 60° = 180°.

Related Terms

Polygon — A triangle is the simplest polygon
Area of a Triangle — Formula for computing a triangle's area
Equilateral Triangle — Triangle with all three sides equal
Isosceles Triangle — Triangle with exactly two equal sides
Right Triangle — Triangle containing a 90° angle
Altitude of a Triangle — Perpendicular height used in the area formula
Heron's Formula — Finds area from three side lengths
Trigonometry — Study of triangle angle-side relationships