The problem with monitoring a specific area (3 feet by 3 feet) from a specific distance away (1 foot) is field-of-view (FOV) of the sensor. The FOV must be large enough to capture the entire area, yet not too large lest the sensor also "sees" the surrounding area or the background behind the monitored area. Perhaps for the sake of minimizing unwanted contributions, the FOV could somehow be restricted to slightly less than the desired area.
Which brings up the problem of how to match a small finite-area sensor to a much larger area. Typically this requires a lens, and if a thermal radiation sensor is required, a lens that passes infrared radiation and (depending on the sensor) blocks shorter wavelengths. There are a number of stand-off radiation sensors that can be used to remotely "measure" surface temperature. The Melexis thermopile that
@Harald Kapp linked to is quite accurate, but perhaps a bit difficult to use. The Texas Instruments TMP006 that is part of the link
@OBW0549 referenced is a similar device to the Melexis Technologies MLX90614 series of sensors, but with perhaps the advantage of a stronger user community. Either one should do the job if the FOV of the sensor can be matched to the target area. TI has a nice little
application note that describes the problem.
Years ago, when they first became available, I acquired a
patented pyroelectric sensor from Harshaw Chemical here in Ohio to "play" with. The company I worked for in the 1970s was just getting involved with high-energy laser research and we spent a good deal of time investigating different types of infrared sensors, both cryogenically cooled and un-cooled sensors. The pyroelectric looked very promising because of it's simplicity and sensitivity. It later went on to become an integral part of passive infra-red (PIR) intrusion detectors, which use two pyroelectric sensors staring at slightly different FOVs by means of plastic Fresnel lenses placed in front of the two sensors. Motion within the two slightly different FOVs causes a differential output signal from the two sensors.
I like Harald's idea of re-purposing a PIR as a stand-off temperature sensor, but there is a big problem with that. The pyroelectric sensor only "sees"
changes in radiation. A static input radiation field produces zero output. To use a single pyroelectric sensor as a temperature sensor requires that it's FOV be alternated between a reference FOV and the measured FOV. This is typically done with a mechanical shutter, such as a rotating disk with alternating cut-outs and (sometimes) mirrors. Mirrors are often used to direct the sensor FOV to a temperature-controlled reference surface, but for this project a simple two-bladed shutter, painted black on the side facing the sensor, would be sufficient. That's what I used, back in the day, to detect people from across the room. No lens.
Because the FOV was huge, the signal to noise ratio was terrible. The optical chopper ensured that the sensor saw the entire background infrared radiation, but the additional radiation from a human body had to be detected in the presence of the radiation from the room. As it turned out, the only way to do this was to use a lock-in amplifier and suppress the steady-state background. So, if the background was at 300 K (approximately room temperature), then a human being was only a few degrees warmer but their "signal" was only a small fraction of the FOV of the sensor. IIRC, the lock-in required almost of minute of integration time to "recognize" the appearance of someone standing in a doorway I had roughly aimed the sensor at. Compare this with how fast a modern PIR, using differential FOVs to detect motion, works. You have to walk realllly slooow to fool a PIR motion sensor. Or wear a suit that matches your radiation profile to the background, which is not an easy task.
Still, I think it would be worthwhile to open up a PIR and block one of the sensors. Use a small motor with a two-bladed "chopper" wheel to provide an output signal from the un-blocked sensor. Depending on how "warm" the target area is compared to ambient temperature, you should easily be able to "see" a 3' x 3' surface from one foot away and "measure" it's temperature. Although I had to use a lock-in amplifier to "see" people, this should not be necessary to measure the temperature of a static area. The pyroelectric sensor produces a substantial current output (a few microamperes) when the radiation impinging on it, modulated by the optical chopper, is a few degrees higher (or lower) than the chopper blade temperature. You can probably use the existing high-impedance amplifier already connected to the two pyroelectric sensor elements. In any event, a simple current-to-voltage circuit using an op-amp with a FET input should suffice. Construction, however, require minimizing the leakage paths associated with the sensor connections and preventing microphonics that can be introduced by the megohm or so feedback resistor. "Dead bug" construction techniques and Teflon insulation (where necessary) recommended.
People do make thermistor-based radiometers, usually in a temperature-controlled bridge configuration, IIRC. But you can purchase pyroelectric sensor based PIRs for just a few bux and they are much more sensitive. If you decide you need more sensitivity, you can add an optical interrupter to the chopper and use it's signal to synchronize a lock-in amplifier.