Problem
Equilateral triangle inside a regular pentagon
A regular pentagon ABCDE is drawn. An equilateral triangle stands inside it on the side DE. The pentagon's sides and the triangle's sides are equal in length. Find the marked angle.
Reasoning · Grade 4-2 Polygons and Angles
Your answer
Watch out Guessing the pentagon's corner is easy to get wrong: it is exactly 108 degrees.
How to solve
Strategy Identify Subproblems — The angle I want lives in triangle CDF, so the whole job is to learn enough about that one triangle. Two smaller questions have to be answered first: how big is a corner of a regular pentagon (which I get by cutting the pentagon into triangles, a problem I already know how to do), and why is triangle CDF isosceles (which I get by marking equal lengths on the picture). With the apex angle of an isosceles triangle known, its two base angles follow immediately.
1STEP 1
Find one interior angle of a regular pentagon
A regular pentagon's interior angle is 108 degrees.
180° × (5-2) = 540°, 540° ÷ 5 = 108°
A pentagon is hard, but a triangle is easy — and three triangles make a pentagon, so the one fact "a triangle's angles add to 180°" is enough to unlock the bigger shape.
8.G.A.5Solve An Easier Related Problem2STEP 2
Mark the segments that have the same length
Equal sides create an isosceles triangle.
DC = DE = DF → △ CDF is isosceles with DC = DF
"Regular" and "equilateral" are both promises that lots of segments are equal; drawing tick marks on all of them shows which triangle is secretly isosceles.
2.G.A.1Draw A Diagram3STEP 3
Find the apex angle at D
Its apex is 108 − 60 = 48 degrees.
∠ CDF = ∠ CDE - ∠ FDE = 108° - 60° = 48°
Angles behave like lengths when they sit side by side: the leftover wedge between the triangle and the pentagon's side is found by one subtraction.
4.MD.C.7Identify Subproblems4STEP 4
Split what is left between the two equal base angles
Halving what is left gives 66 degrees.
∠ DCF = ∠ DFC = (180° - 48°) ÷ 2 = 132° ÷ 2 = 66°
In an isosceles triangle the two angles facing the equal sides must be the same size, so once the apex is known the rest of the 180° is shared out evenly — one subtraction and one halving.
8.G.A.5Identify SubproblemsWhat is left after the apex angle splits evenly between the two base angles, because the two sides are equal.
Why?
Equal sides face equal angles, so the two base angles have to be the same size whatever the apex turns out to be.
🧱Equal sides face equal anglesFold an isosceles triangle down its middle: the two equal sides land on each other and carry the two base angles together, so those angles are equal — and the reverse holds too.
Why?
The three angles add to one straight angle, so subtracting the apex leaves a fixed amount for the pair to share.
🧱A triangle's angles sum to 180 degreesThe three corners of any flat triangle always add up to one straight angle, 180 degrees.
Answer
66 degrees
180 − 48 = 132, 132 ÷ 2 = 66
The answer is an angle, so degrees are correct, and 66° is the right sort of size: it is smaller than the pentagon's own corner at C, which is 108°, leaving ∠ BCF = 108° - 66° = 42° on the other side of CF. Both leftovers are positive, which confirms that F really is inside the pentagon and that the segment CF really does pass through the interior, just as the figure shows. The triangle also closes: 48° + 66° + 66° = 180°. Finally, 66° is a bit more than a right angle's two thirds and the drawn angle does look distinctly larger than 45° but smaller than a right angle, which matches.
Takeaway
A regular pentagon's side and an equilateral triangle's side are the same length, so the leftover triangle is isosceles — subtract 60° from 108°, then share the rest of 180° evenly.
- Find one interior angle of a regular pentagon
- Mark the segments that have the same length
- Find the apex angle at D
- Split what is left between the two equal base angles