Equilateral Triangle Calculator — Side, Area, Height & Radii
Calculate all properties of an equilateral triangle from side length, area, perimeter, or height. Includes area, height, perimeter, circumradius, inradius, and angles (always 60°).
Isosceles Triangle Calculator — Sides, Angles & Area
Complete isosceles triangle solver. Enter base + leg, base + height, or leg + apex angle to compute all sides, angles, area, perimeter, altitudes, circumradius, and inradius.
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Planning notes, formulas, and examples
About the Isosceles Triangle Calculator — Sides, Angles & Area
An isosceles triangle has two sides of equal length, called the legs, and a third side called the base. The two angles opposite the equal sides (base angles) are always equal. Isosceles triangles appear everywhere — from the gable of a roof to the cross-section of an A-frame cabin, from the shape of yield signs to the geometry of suspension bridges.
Computing the full set of properties requires only two independent measurements. Given the base b and leg a, the height h = √(a² − (b/2)²), the base angles equal arccos(b / (2a)), and the apex angle is 180° minus twice the base angle. Given the base and height, the leg is √((b/2)² + h²). Given a leg and the apex angle α, the base = 2a·sin(α/2).
Once all three sides and all three angles are known, derived quantities follow: area = ½bh, perimeter = 2a + b, circumradius R = (a²b) / (4·Area), inradius r = Area / s where s is the semi-perimeter, the altitude from a base vertex to the opposite leg, and the medians. Structurally, the altitude from the apex to the base is also the perpendicular bisector and the median — a key symmetry of isosceles triangles.
This calculator supports three input modes (base + leg, base + height, leg + apex angle), a unit selector, presets for common isosceles shapes, and a summary table of all properties.
When This Page Helps
Isosceles Triangle — Sides, Angles & Area problems often require several dependent steps, and a small arithmetic slip can propagate through every derived value. This calculator is tailored to that workflow: you enter base (b), leg (a), height (h), and it returns area, perimeter, height, apex angle in one consistent pass. It is useful for homework checks, worksheet generation, tutoring walkthroughs, and fast field/design estimates where you need reliable geometry results without rebuilding the full derivation each time.
How to Use the Inputs
- Select an input mode: Base + Leg, Base + Height, or Leg + Apex Angle.
- Choose a measurement unit.
- Enter the required values.
- Or click a preset to load common isosceles triangles.
- View all computed properties: sides, angles, area, perimeter, circumradius, inradius.
- Check the summary table for a compact overview.
- Compare dimensions visually with the bar chart.
Formula used
Height from base: h = √(a² − (b/2)²)
Base angles: β = arccos(b / (2a))
Apex angle: α = 180° − 2β
Area: A = ½ × b × h
Perimeter: P = 2a + b
Semi-perimeter: s = (2a + b) / 2
Circumradius: R = (a² × b) / (4A)
Inradius: r = A / sExample Calculation
Result: Height = 12, Area = 60, Perimeter = 36, Base angle ≈ 67.38°, Apex angle ≈ 45.24°
With base 10 and leg 13: height = √(169 − 25) = √144 = 12. Area = ½ × 10 × 12 = 60. Perimeter = 26 + 10 = 36. Base angle = arccos(5/13) ≈ 67.38°. Apex = 180 − 2(67.38) ≈ 45.24°.
Tips & Best Practices
- An equilateral triangle is a special isosceles triangle where the base equals the legs — all angles are 60°.
- The altitude from the apex always bisects the base and the apex angle — a key symmetry.
- If the apex angle is 90°, the isosceles right triangle has base angles of 45° each.
- For a very narrow isosceles triangle (small apex angle), the height approaches the leg length.
How Isosceles Triangle — Sides, Angles & Area Calculations Work
This isosceles triangle — sides, angles & area tool links the entered values (base (b), leg (a), height (h), apex angle) to the target geometry relationships used in class and practice problems. Instead of solving each intermediate step manually, you can validate setup and arithmetic quickly while still tracing which measurements drive the final result.
Formula focus: the calculator formula
Practical Uses for Isosceles Triangle — Sides, Angles & Area
Isosceles Triangle — Sides, Angles & Area shows up in school geometry, technical drafting, construction layout checks, and early engineering design estimates. When values are changed repeatedly, the calculator helps you compare scenarios quickly and see how sensitive the shape is to each dimension.
Interpreting the Results Correctly
Start with the primary outputs (area, perimeter, height, apex angle) and then use the remaining cards/tables to confirm consistency with your diagram. Keep units consistent across inputs, and round only at the end if your assignment or project specifies a fixed precision.
Sources & Methodology
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Frequently Asked Questions
Having exactly two sides of equal length. The angles opposite those equal sides are also equal.
If you know the base b and leg a, the height h = √(a² − (b/2)²). The altitude drops from the apex perpendicular to the base.
Yes — when the apex angle is 90°. The two base angles are each 45°, giving a 45-45-90 triangle.
An equilateral triangle has all three sides equal (and all angles 60°). An isosceles has exactly two equal sides.
R = (a² × b) / (4 × Area), where a is the leg and b is the base.
The inradius is the radius of the inscribed circle: r = Area / semi-perimeter.
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