Properties of Triangles
Properties of Triangles covers the three essential formulas every pre-algebra student needs: angles of any triangle sum to 180°, perimeter equals the sum of all three sides (P = a + b + c), and area equals one-half base times height (A = ½bh). Students also distinguish isosceles triangles (two equal sides) from equilateral triangles (three equal sides). From OpenStax Prealgebra 2E, this skill connects geometric intuition — a triangle's area is exactly half a matching rectangle — to algebraic formula use for solving missing dimensions.
Key Concepts
Property For any triangle $\triangle ABC$, the sum of the measures of the angles is $180^\circ$. $$m\angle A + m\angle B + m\angle C = 180^\circ$$ The perimeter of a triangle is the sum of the lengths of the sides. $$P = a + b + c$$ The area of a triangle is one half the base, $b$, times the height, $h$. $$A = \frac{1}{2}bh$$ An isosceles triangle has two sides of the same length. An equilateral triangle has three sides of equal length.
Examples A triangle has a base of 14 cm and a height of 10 cm. Its area is $A = \frac{1}{2}(14)(10) = 70$ square cm.
A triangle has a perimeter of 30 feet. Two of its sides are 8 feet and 12 feet. The third side is found by solving $30 = 8 + 12 + c$, which gives $c = 10$ feet.
Common Questions
What do the angles of a triangle always add up to?
The three interior angles of any triangle always sum to 180°. If two angles are 60° and 80°, the third must be 40°.
What is the formula for the area of a triangle?
Area = ½ × base × height, or A = ½bh. A triangle with base 14 cm and height 10 cm has area ½(14)(10) = 70 cm².
How do you find the perimeter of a triangle?
Add all three side lengths: P = a + b + c. If a triangle has perimeter 30 ft with two sides of 8 ft and 12 ft, the third side is 30 − 8 − 12 = 10 ft.
What is the difference between isosceles and equilateral triangles?
An isosceles triangle has exactly two sides of equal length. An equilateral triangle has all three sides equal.
Why is the triangle area formula ½bh?
A triangle is exactly half of a parallelogram (or rectangle) with the same base and height, so its area is half the product of base and height.
How do you find a missing angle in a triangle?
Subtract the known angles from 180°. If two angles are 45° and 75°, the third is 180° − 45° − 75° = 60°.