Math
Pentagon Calculator
The pentagon calculator solves a regular pentagon from one known measurement — side, apothem, circumradius, area, or perimeter.
It uses the exact formula A = (1/4)√(5(5 + 2√5))·s² ≈ 1.7205·s² and reports every angle and length a homework problem could ask for.
Apothem = center to the middle of a side. Circumradius = center to a vertex.
Area is in square units; the rest are lengths.
Pick the unit your measurements use. Area is reported in square units and volume in cubic units.
Try an example
Result
Area
—
- Side
- —
- Perimeter
- —
- Apothem (inradius)
- —
- Circumradius
- —
- Interior angle
- —
- Exterior angle
- —
More details (1 more)
- Number of diagonals
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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What the pentagon calculator solves
The pentagon calculator solves a regular pentagon from one known measurement — side, apothem, circumradius, area, or perimeter. It uses the exact formula A = (1/4)√(5(5 + 2√5))·s² ≈ 1.7205·s² and reports every angle and length a homework problem could ask for.
Formula
How to use it
- Choose which pentagon measurement you know: side, apothem, circumradius, area, or perimeter.
- Enter the value and choose its unit.
- Read the area, side, perimeter, apothem, circumradius, angles, and diagonal count; open the work for each formula.
How to read the answer
A regular pentagon has 5 equal sides and 5 equal angles of 108°. The apothem is the radius of the circle that fits inside it and the circumradius is the radius of the circle through its corners; the area is half the perimeter times the apothem.
Common mistakes and edge cases
- Mixing up apothem (center to the middle of a side) with circumradius (center to a corner). The circumradius is always longer.
- Using the interior angle formula with the wrong n. Interior angle = (n − 2) × 180° / n.
- Forgetting that area is in square units while every other result is a length.
- Zero or negative measurements cannot form a polygon and return an error.
Worked examples
Side 4
Area
27.5276
Apothem 5 cm
Area
90.8178
Circumradius 10 in
Area
237.7641
Area 100 square units
Area
100
Perimeter 35 ft (side 7)
Area
84.3034
Decimal side 7.5 m
Area
96.7769
Negative side (impossible)
Area
Error
Frequently asked questions
What is the area of a regular pentagon?+
A = (1/4)·sqrt(5(5 + 2√5))·s² ≈ 1.7205·s², or equivalently half the perimeter times the apothem. Side 4 gives about 27.53.
What are the angles of a pentagon?+
Each interior angle of a regular pentagon is 108° (they sum to 540°), and each exterior angle is 72°.
How many diagonals does a pentagon have?+
Five: n(n − 3)/2 = 5 × 2 / 2. Drawing all five makes a pentagram, and each diagonal is φ ≈ 1.618 times the side (the golden ratio).
What is the apothem of a pentagon?+
The distance from the center to the midpoint of a side: a = s / (2·tan 36°) ≈ 0.688·s. Area = ½ × perimeter × apothem.
How do I find the side of a pentagon from its area?+
s = sqrt(A / 1.7205) = sqrt(4A·tan 36° / 5). Area 100 gives side ≈ 7.624.
Does this work for an irregular pentagon?+
No. Irregular pentagons have no single formula; split them into triangles and add the areas. This page assumes all five sides and angles are equal.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026