Curriculum Core Rules & Strategy
Always write out given values and the relevant formula before substituting numbers.
Substitute your final answer back into the original problem to verify correctness.
Double-check negative signs, common denominators, and arithmetic order of operations.
Mastering Types of Triangles with SolveCalc
Triangles are classified two ways simultaneously: by side lengths (equilateral, isosceles, scalene) and by angles (acute, right, obtuse). A triangle can be "isosceles and right" or "scalene and obtuse" — these are independent classifications.
Equilateral, Isosceles, and Scalene Triangles
By side lengths:
- Equilateral: All 3 sides equal. All 3 angles equal 60°. The most symmetric triangle.
- Isosceles: Exactly 2 sides equal. The base angles (angles opposite the equal sides) are also equal. The line of symmetry bisects the apex angle perpendicularly.
- Scalene: All 3 sides different. All 3 angles different. No lines of symmetry.
Note: equilateral is a special case of isosceles (all three sides are equal, so certainly two are equal). Some definitions exclude this; most include it.
Acute, Right, and Obtuse Triangles
By angle types:
- Acute: All three angles are less than 90°. Equilateral triangles are always acute.
- Right: Exactly one angle equals 90°. The hypotenuse is always opposite the right angle. The Pythagorean theorem applies.
- Obtuse: Exactly one angle is greater than 90°. A triangle can have at most one obtuse angle (since three must sum to 180°).
Quick test: in triangle with sides a, b, c (c largest): acute if a²+b²>c², right if a²+b²=c², obtuse if a²+b²
Key Properties of Each Triangle Type
- Equilateral: 3 lines of symmetry; rotational symmetry of order 3; all altitudes, medians, and angle bisectors coincide; circumcentre = incentre = centroid.
- Isosceles: 1 line of symmetry (through apex and midpoint of base); base angles equal; the perpendicular bisector of the base passes through the apex.
- Right triangle: Hypotenuse = √(a²+b²); the median to the hypotenuse = ½ × hypotenuse; trigonometric ratios (sin, cos, tan) are defined for its angles.
Triangle Classification — Four Problems
Two equal sides → isosceles. 5²+5²=50 > 8²=64? No, 50 < 64 → obtuse. Classification: isosceles obtuse.
All different → scalene. 3²+4²=25=5² → right. Classification: scalene right triangle.
Triangle Classification Errors
- Forgetting dual classification: Triangles have BOTH a side-length classification AND an angle classification. An answer of just "isosceles" is incomplete if the question asks for full classification.
- Assuming isosceles means non-right: An isosceles right triangle has legs equal and a 90° angle — perfectly valid. The 45-45-90 triangle is the classic example.
- Using the shortest side as c in the acute/obtuse test: Always use the LONGEST side as c in a²+b²⪌c².
Triangle Types — Never Forget Them
- Side names: "Equal-lateral = Equal-sides" (equilateral). "Iso = same" (isosceles, from Greek "isos" = equal). "Scalenus = unequal" (scalene).
- Always make two classifications: Step 1: check sides (equilateral/isosceles/scalene). Step 2: check angles (acute/right/obtuse).
- The 45-45-90 and 30-60-90 triangles: These special right triangles appear constantly — memorise their side ratios (1:1:√2 and 1:√3:2 respectively).
Use SolveCalc's Geometry Calculator to compute triangle properties for any given dimensions.
Common Student Pitfalls & Exam Traps
- Skipping steps: Even experienced mathematicians write every step. Skipping creates opportunities for errors.
- Not checking answers: Always substitute your answer back to verify correctness.
- Confusing similar concepts: Small differences in definitions can require completely different approaches.
- Unit errors: Always keep track of units and convert consistently when working with real-world problems.
SolveCalc Insight
DEEP DIVESolveCalc provides free, browser-based math tools for students and educators at every level. Use our interactive calculators alongside this curriculum guide to verify your calculations and explore concepts hands-on — no sign-up, no downloads, no limits.
Explore with SolveCalc's Math Tools Library→Frequently Asked Questions
What is an equilateral triangle? +
All 3 sides equal, all 3 angles equal to 60°.
What is a scalene triangle? +
All 3 sides different lengths and all 3 angles different measures.