The area of a triangle · Code with Martin

The area of a triangle = a side times the height to it, divided by two. Why it holds and how not to slip.

A triangle is half a rectangle

You know the area of a rectangle: sides a times b. Today we derive the area of a triangle from it — and I mean the word “derive” seriously. A formula you understand cannot be forgotten.

Take any triangle. Add a second, the same, just turned — and a parallelogram is born. That has area side × height to it. A triangle is its half. And that is why:

S = (a × vₐ) : 2

A side and the height to it — that is all. The height vₐ is that perpendicular from lesson 20 that lands on side a.

A triangle has three sides, and so three side–height pairs. You can use any of them and the result always comes out the same. Pick the one you know, or the one that is easy to measure. The only ban: you must not take a side and a height that do not belong together.

Adam and Jonáš paint a triangular set and need to know how much paint to buy. Base 2 m, height of the set 1.5 m. Jonáš: “If it were a rectangle 2 by 1.5, that is 3 square metres.” Adam: “And a triangle is half — 1.5 m².” In the shop they read: a tin covers 2 m². “One is enough even for a second coat of the corners,” Jonáš counts. On the way back Adam thinks: “We did not really need a formula. It was enough to see the rectangle around it.”

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Before you plug into the formula, point at the side and point at its height. If your two numbers are not perpendicular to each other, they do not belong together and the result will be wrong.

A formula you can see

A derivation you believe: draw a triangle, trace it and glue the turned copy to the original along one side. A parallelogram with the same base and height is born. The area of a parallelogram is base × height (next lesson), a triangle is half. So:

S = (side × height to this side) : 2

The calculating method:

Step 1: pick a pair side + height to it (perpendicular!). Step 2: multiply. Step 3: divide by two. Step 4: units — sides in cm give area in cm².

With a right-angled triangle it is easiest: the legs are perpendicular, so one is the “side” and the other the “height”. S = (leg × leg) : 2.

Reverse questions: from the formula you can make anything the subject. You know area 24 cm² and side 8 cm? Height = 2 × 24 : 8 = 6 cm. (First double the area — you return that half — then divide by the side.)

Watch units twice: side 2 m and height 50 cm must not be multiplied until you match them (2 m × 0.5 m = 1 m², or 200 cm × 50 cm = 10 000 cm² — the same).

Side 12 cm, height to it 7 cm

Step 1: the pair belongs together (the height is perpendicular to the side 12 cm).

Step 2: I multiply: 12 × 7 = 84.

Step 3: I halve: 84 : 2 = 42.

Result: S = 42 cm². Check by thinking: a rectangle 12 × 7 would have 84 cm², a triangle is its half. Do not forget the unit — area is always in square units, because you count how many 1 × 1 cm squares fit in the shape.

A right-angled triangle with legs 6 cm and 9 cm

In a right-angled triangle the legs are perpendicular — a ready pair side + height.

Calculation: S = (6 × 9) : 2 = 54 : 2 = 27 cm².

Result: 27 cm². Watch the bait: if the question also gave the hypotenuse (here 10.8 cm), it does not belong in the formula. The hypotenuse has its own height, which the question did not give. Use only pairs that are perpendicular.

Area 30 cm², height 5 cm. How long is the side?

A reverse question — I go through the formula backwards.

Step 1: I return the halving: 2 × 30 = 60. (That much the rectangle around it would have.)

Step 2: I divide by the height: 60 : 5 = 12.

Result: the side measures 12 cm. Check by calculating forwards: (12 × 5) : 2 = 30 cm². That checks out. Reverse questions are not new material — it is the same formula, just read from right to left.

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Do not forget to divide by two and do not take the wrong height. Two most common mistakes in one sentence: who does not halve calculated a parallelogram; who multiplies a side by a height to another side calculated nonsense. Finger on the side, finger on the perpendicular — then calculate.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Start with ten. When it goes well, add more.

Calculate the area of a triangle: side times height, divided by two. Write only the number in cm².

Practise the questions

On paper: areas around us

For each question write the formula, the plugging-in and the result with the unit.

Calculate the area of a triangle with side 14 cm and height to it 6 cm.
A triangular flag has a base of 40 cm and a height of 90 cm. How many cm² of fabric do you need? And how many for 10 flags?
Draw any triangle, measure one side and construct and measure the height to it (a mark!). Calculate the area of your triangle.
The area of a triangle is 36 cm² and one side measures 9 cm. Calculate the height to this side and check by calculating forwards.

Now you 💪

I know the formula S = (a × vₐ) : 2 and I know why we divide by two.
I recognise which height belongs to which side (perpendicular).
I can do a reverse question: from the area and a side I calculate the height.

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Done when: Four questions with plugging-in and units, your drawn triangle has a measured area and the reverse question gave a height of 8 cm.

What to take from this lesson

A triangle is half a parallelogram: S = (side × height to it) : 2.
A side and a height must belong together — they are perpendicular.
In a right-angled triangle the legs serve.
Area is in square units; match the sides to the same unit.
A reverse question = the same formula read backwards.