How do you find the area of a triangle?

There are several formulas depending on what you know. The most common is Area = ½ × base × height, where height is the perpendicular distance from the base to the opposite vertex. If you know all three sides, use Heron's formula: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2 is the semi-perimeter. If you know two sides and the included angle, use Area = ½ab·sin(C). All three methods for Area: • Base and height: A = ½ × b × h (simplest method) • Heron's formula (all 3 sides): s=(a+b+c)/2, A=√(s(s−a)(s−b)(s−c)) • Two sides + included angle (SAS): A = ½ × a × b × sin(C) • For a right triangle: A = ½ × leg₁ × leg₂ (the two legs are base and height) • Example — 3,4,5 triangle: s=6, A=√(6×3×2×1)=√36=6 square units ✓

What is the Law of Cosines and when do you use it?

The Law of Cosines generalises Pythagoras's theorem to any triangle. It states: c² = a² + b² − 2ab·cos(C), where C is the angle opposite side c. Use it when you have SSS (all three sides) to find angles, or SAS (two sides + included angle) to find the third side. Forms of the Law of Cosines: • Finding side c: c² = a² + b² − 2ab·cos(C) • Finding angle C: cos(C) = (a²+b²−c²)/(2ab) • When C = 90°: cos(90°)=0, so c²=a²+b² — this is just Pythagoras! Examples: • SAS: a=5, b=7, C=60° → c²=25+49−2(35)(0.5)=39 → c=6.245 • SSS: a=3,b=4,c=5 → cos(C)=(9+16−25)/24=0 → C=90° (right angle confirmed) • Use Law of Sines when you have ASA or AAS — it's simpler in those cases

What is the Law of Sines?

The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C) — the ratio of each side to the sine of its opposite angle is the same for all three sides. This equals the diameter of the circumscribed circle (2R). Use it when you have ASA (two angles + included side) or AAS (two angles + non-included side). Using Law of Sines: • Given ASA: A=60°, c=8, B=60° → C=60°, ratio = 8/sin(60°) = 9.238, a=b=9.238×sin(60°)=8 • Given AAS: A=30°, B=60°, a=5 → C=90°, ratio=5/sin(30°)=10, b=10×sin(60°)=8.660, c=10 • Ambiguous case (SSA): two angles and a non-included side can sometimes give two valid triangles Law of Sines vs Law of Cosines: • Two angles known (ASA/AAS) → use Law of Sines • All sides known (SSS) → use Law of Cosines for angles • Two sides + included angle (SAS) → use Law of Cosines for third side

How do you identify the type of triangle?

Triangles are classified by their angles and by their sides. By angles: acute (all angles 90°). By sides: equilateral (all sides equal), isosceles (two sides equal), or scalene (all sides different). By angles: • Acute triangle: all three angles less than 90° — e.g. 60°-60°-60° (equilateral) • Right triangle: one angle exactly 90° — e.g. 30°-60°-90°, 45°-45°-90°, 3-4-5 • Obtuse triangle: one angle greater than 90° — e.g. 100°-40°-40° By sides: • Equilateral: all three sides equal → all angles = 60° • Isosceles: two sides equal → two base angles equal • Scalene: all three sides different → all three angles different Pythagoras test for right triangles: • If a²+b²=c² (where c is the longest side) → right triangle • If a²+b²>c² → acute triangle • If a²+b²<c² → obtuse triangle

What is Heron's formula and how do you use it?

Heron's formula calculates the area of a triangle using only its three side lengths — no angles or height needed. Given sides a, b, c: first compute the semi-perimeter s = (a+b+c)/2, then Area = √(s(s−a)(s−b)(s−c)). Step-by-step for a 3-4-5 triangle: • s = (3+4+5)/2 = 6 • s−a = 6−3 = 3 • s−b = 6−4 = 2 • s−c = 6−5 = 1 • Area = √(6×3×2×1) = √36 = 6 ✓ Step-by-step for a 5-6-7 triangle: • s = (5+6+7)/2 = 9 • Area = √(9×4×3×2) = √216 = 14.697 Why Heron's formula works: it is derived from the formula Area = ½ab·sin(C) combined with the Law of Cosines — both methods give identical results. Heron's formula is preferred when you know all three sides and do not want to compute any angles first.

Triangle Calculator — Area, Angles, Sides & Type

The Triangle Calculator solves any triangle from any combination of known values. Choose from five input modes: SSS (three sides, uses Heron's formula and Law of Cosines), SAS (two sides and included angle, uses Law of Cosines), ASA (two angles and included side, uses Law of Sines), AAS (two angles ...

TRIANGLE CALCULATOR

KNOWN VALUES

Side a

Side b

Side c

AREA

square units

PERIMETER

units

SIDES

ANGLES

A36.8699°

B53.1301°

C90°

Type (angle)Right

Type (side)Scalene

Inradius1 units

Circumradius2.5 units

Height to a4 units

Height to c2.4 units

STEP-BY-STEP

Given sides: a=3, b=4, c=5

Semi-perimeter: s = (3+4+5)/2 = 6

Area (Heron's): √(s(s−a)(s−b)(s−c)) = √(6×3×2×1) = 6

Angle A (Law of Cosines): arccos((b²+c²−a²)/(2bc)) = 36.8699°

Angle B: arccos((a²+c²−b²)/(2ac)) = 53.1301°

Angle C: 180° − 36.8699° − 53.1301° = 90°

QUICK EXAMPLES

LIVE DIAGRAM

TRIANGLE PROPERTIES

Angle sum check36.8699° + 53.1301° + 90° = 180° ✓

Triangle typeScalene / Right

Semi-perimeter6 units

Created with❤️byeaglecalculator.com

TRIANGLE FORMULAS

ALL TRIANGLE FORMULAS

NAME	FORMULA	USE WHEN
Area (base×height)	A = ½ × b × h	Base and height known
Area (SAS)	A = ½ab·sin(C)	Two sides and included angle
Heron formula	A = √(s(s−a)(s−b)(s−c)), s=(a+b+c)/2	All three sides known
Law of Cosines	c² = a²+b²−2ab·cos(C)	SAS: find third side; SSS: find angles
Law of Sines	a/sin(A) = b/sin(B) = c/sin(C)	ASA/AAS: find remaining sides
Perimeter	P = a + b + c	Sum of all three sides
Inradius	r = Area / s	Radius of inscribed circle
Circumradius	R = abc / (4×Area)	Radius of circumscribed circle
Altitude to side a	h_a = 2×Area / a	Perpendicular height from a to opposite vertex

HOW TO USE

1
Select your input mode based on what you know: SSS for three sides, SAS for two sides and the angle between them, ASA for two angles and the side between them, AAS for two angles and any one side, or Base×Height for area only.
2
Enter your known values. For angle inputs, enter degrees. The calculator accepts any valid triangle — it checks the triangle inequality for SSS and angle sum for angle-based modes.
3
Area and perimeter appear instantly in the black result box. All sides, angles, inradius, circumradius, and altitudes are shown below.
4
Read the step-by-step solution to see which law was applied — Heron's formula for SSS, Law of Cosines for SAS, Law of Sines for ASA/AAS.
5
Check the live diagram — the actual triangle is drawn to scale with all sides and angles labelled. A right angle marker appears for right triangles.

WORKED EXAMPLE

Example 1 (SSS 3-4-5): s=6, Area=√(6×3×2×1)=6. Angles: A=36.87°, B=53.13°, C=90°. Right triangle. Example 2 (SAS a=5,C=60°,b=7): c=√(25+49−35)=√39=6.245. Area=½×5×7×sin60°=15.155. Example 3 (equilateral 6): Area=(√3/4)×36=15.588, all angles=60°.

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Last updated: April 29, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.